Mechanics
1
CHAPTER 1
MECHANICS
OBJECTIVES
After reading this chapter, you should be able to:
•
Understand the measurements of mass, volume, density
•
Explain the determining the coefficient of friction
•
Focus on vector forces
•
Define particles and rigid body
•
Identify work and energy
•
Understand the collision, projectile motion, ballistic pendulum and
centripetal force
INTRODUCTION
Mechanics, science concerned with the motion of bodies under the
action of forces, including the special case in which a body remains at
rest. Of first concern in the problem of motion are the forces that bodies
exert on one another. This leads to the study of such topics as gravity,
electricity, and magnetism, according to the nature of the forces involved.
Given the forces, one can seek the manner in which bodies move under
the action of forces; this is the subject matter of mechanics proper.
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Experimental Physics I
Historically, mechanics was among the first of the exact sciences
to be developed. Its internal beauty as a mathematical discipline and
its early remarkable success in accounting in quantitative detail for the
motions of the Moon, Earth, and other planetary bodies had enormous
influence on philosophical thought and provided impetus for the
systematic development of science.
Mechanics may be divided into three branches: statics, which deals
with forces acting on and in a body at rest; kinematics, which describes
the possible motions of a body or system of bodies; and kinetics, which
attempts to explain or predict the motion that will occur in a given
situation. Alternatively, mechanics may be divided according to the
kind of system studied. The simplest mechanical system is the particle,
defined as a body so small that its shape and internal structure are of no
consequence in the given problem. More complicated is the motion of
a system of two or more particles that exert forces on one another and
possibly undergo forces exerted by bodies outside of the system.
1.1 MEASUREMENTS OF MASS, VOLUME,
DENSITY
1.1.1 Mass
Mass can be best understood as the
amount of matter present in any object or
body. Everything we see around us has
mass. For example, a table, a chair, your
bed, a football, a glass, and even air has
mass. That being said, all objects are light or
heavy because of their mass. We will learn
what is mass, how to calculate it, and its
examples while discovering interesting facts
about it.
Mass is the most basic property of matter
and it is one of the fundamental quantities.
Mass is defined as the amount of matter
present in a body. The SI unit of mass is the
kilogram (kg). The formula of mass can be
written as:
Mass = Density × Volume
Note
The mass of a body
is constant; it doesn’t
change at any time.
Only in certain
extreme cases when
a huge amount of energy is given or taken
from a body, the mass
may be impacted. For
example, in a nuclear reaction, a tiny
amount of matter is
converted into a huge
amount of energy, this
reduces the mass of
the substance.
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1.1.2 Volumes
Volumes of liquids are measured with the aid of graduated
cylinders, pipets, and burets. For the time being you will only be using
the graduated cylinder for direct determination of liquid volumes. The
measurements of volumes of solid objects is not as straightforward as it
is with liquids. In some cases, when dealing with regular solids such as
cubes, cylinders, or prisms, it is quite possible to calculate the volumes
by measuring the pertinent dimensions of the object. Some formulas for
the calculation of volumes of regular solids are given below:
where:
Ρ = length in centimeters
w = width in centimeters
h = height in centimeters
r = radius in centimeters
π = 3.14 ...
Volumes of liquids are expressed in units of milliliters (mL) while
volumes calculated from the geometry of the object will be given in
cubic centimeters (cm3 or cc). For our purposes 1 mL is equal to 1 cm3.
In the case of irregularly shaped objects a method known as volume
by liquid displacement can be used. If we fill a 100 mL graduated
cylinder to the 50 mL mark and if we then introduce the solid object
into the cylinder, and assuming that the object does not float, the liquid
level will then rise to a new mark. The difference between the two liquid
levels represents the desired volume. In another version of the same
method, a container is filled completely with liquid, usually water, and
when the object is submerged a certain amount of liquid overflows and
can be captured. The volume of liquid can either be measured directly
or can be weighed. Knowing the mass of liquid and the density we can
get the volume. This method of determining volume is easy to carry out
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Experimental Physics I
and is more accurate due to the high accuracy possible in weighing. In
the case of objects that float, it is of course necessary to make sure that
all of the substance is completely submerged in the liquid by pushing
them down.
1.1.3 Density
Knowing the mass and volume of an object allows the calculation
of its density. Density is defined as the mass divided by the volume of
the object.
Mass should be expressed in units of grams and volume in units of
mL or cm3. Note, that the density of water is 1 g≅ mL-1.
The general physical law relating to floating objects is known as
Archimedes’ Principle. It states that a body wholly or partly immersed in
a fluid is buoyed up by a force equal to the weight of the fluid displaced.
buoyant force=(den.of fluid)(acc.due to gravity)(vol.of immersed
object)
In the special case of floating objects one can also express the
principle as: “A floating body displaces a volume of liquid whose weight
equals that of the floating body.” Archimedes’ Principle can be used to
determine specific gravities of solids.
1.2 DETERMINING THE COEFFICIENT OF
FRICTION
Friction is a resistive force that prevents two objects from sliding
freely against each other. The coefficient of friction (fr) is a number that
is the ratio of the resistive force of friction (Fr) divided by the normal or
perpendicular force (N) pushing the objects together. It is represented
by the equation:
fr = Fr/N.
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There are different types and values for the coefficient of friction,
depending on the type of resistive force. You can determine the coefficient
of friction through experiments, such as measuring the force required to
overcome friction or measuring the angle at which an object will start
to slide off an incline. There are also charts of common coefficients of
friction available.
1.2.1 Different types of Coefficient
The different types of friction are static, kinetic, deformation,
molecular and rolling. Each has its own coefficient of friction.
Static Coefficient
Static friction is the force that holds back a stationary object up to
the point that it just starts moving. Thus, the static coefficient of friction
concerns the force restricting the movement of an object that is stationary
on a relatively smooth, hard surface.
Kinetic Coefficient
Once you overcome static friction, kinetic friction is the force
holding back regular motion. This, kinetic fiction coefficient of friction
concerns the force restricting the movement of an object that is sliding
on a relatively smooth, hard surface.
Deformation Coefficient
The deformation coefficient of friction concerns the force restricting
the movement of an object that is sliding or rolling and one or both
surfaces are relatively soft and deformed by the forces.
Molecular Coefficient
Molecular coefficient of friction concerns the force restricting the
movement of an object that is sliding on an extremely smooth surface or
where a fluid is involved.
Rolling coefficient
The rolling coefficient of friction combines static, deformation and
molecular coefficients of friction. This coefficient of friction can be made
quite low.
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Experimental Physics I
1.2.2 Experiments to Determine Coefficient
There are a number of experiments you can do to determine the
coefficient of friction between two materials. You can directly measure
the forces involved or you can use some indirect methods, measuring
such things as incline of a ramp or time to stop.
Direct Measurements
An experiment to determine the coefficient of friction would be to
use some force to push to materials together and then measure that force.
You could apply this force by squeezing a pair of pliers, by applying the
brakes in your car, or by using the force of gravity to apply a weight on
an object.
Then you can try to move one object and measure the necessary
force. This could be trying to pull a strip of wood from the grip of the
pliers, trying to move a car wheel when the brakes are applied, or pulling
a weighted object along the floor. A scale or similar device can be used to
measure the forces.
Measuring the Squeezing Force
If you can measure the force you apply
to push the materials together, you can
determine their static coefficient of friction.
Thus, if you took a pair of pliers and squeezed
it so that it applied 18 pounds of force on a
piece of wood, and it took 9 pounds of force
to pull the wood from the squeezed pliers,
then the static coefficient of friction of the
wood and the pliers would be 9 pounds / 18
pounds = 0.5. Thus, no matter how hard you
squeezed the pliers, it would always take 0.5
times that force to pull the wood out.
Important
A pair pliers
is a simple machine.
It has a mechanical
advantage of about
3 times. Thus you
would only have to
squeeze the handles with 6 pounds
of force to create a
force of 18 pounds
at the pliers head.
Using the Force of Gravity
Since it is difficult to measure the force that you squeeze, a more
common way to measure the force between objects is to use the weight
of one object. An object’s weight is the force is exerts on another object,
caused by gravity. If the weight is W in pounds or newtons, the friction
equation for an object sliding across a material on the ground can be
rewritten as:
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Once you know the weight of the object you are sliding, you
can use a scale to measure the force it takes to move the object. For
example, measure the weight of a book. Then use the scale to measure
the force required to start the book sliding along a table. From these
two measurements, you can determine the static coefficient of friction
between the book and the table.
You can verify that the friction equation is true by adding a second
book and repeating the measurement. The force required to pull two
books should be twice as much as for one book. To measure the static
coefficient of friction, you take the value of the force just as the object
starts to move. Doing the same experiment with sliding or kinetic
friction, you want to take your reading when the object is sliding at an
even velocity. Otherwise, you will be adding in acceleration force effects.
Indirect Measurements
There are several indirect methods to determine the coefficient of
friction. A method to determine the static coefficient of friction is to
measure the angle at which an object starts to slide on an incline or ramp.
A method to determine the kinetic coefficient of friction is to measure
the time is takes to stop an object.
Using an Incline
You can use an object on an incline to determine the static coefficient
of friction by finding the angle at which the force of gravity overcomes
the static friction.
Perpendicular Force Reduced:
When an object is placed on an incline, the force perpendicular
between the surfaces is reduced, according to the angle of the incline.
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Experimental Physics I
The force required to overcome friction (Fr) equals the coefficient of
friction (u) times the cosine of the incline angle (cos(a)) times the weight
of the object (W). There are mathematical tables that give the values of
cosines for various angles.
Gravity contributes to sliding:
Note that when an object is on an incline, the force of gravity
contributes to causing the object to slide down the ramp or incline. Let’s
call that force (FG), and it is equal to the weight of the object (W) times
the sine of the angle (sin(a))
Tangent of angle determines coefficient:
If you put the ramp at a steep enough angle, Fg will become greater
than Fr and the object will slide down the incline. The angle at which it
just starts to slide is determined from the equation:
Dividing both sides of the equation by W and cos(a), we get the
equation for the static coefficient of friction fr
where tan(a) is the tangent of angle (a) and equals sin(a)/cos(a).
There are mathematical tables for determining the tangent, sine and
cosine of various angles.
Using Time
You can also use a stopwatch to determine the kinetic or rolling
coefficient of friction. But it is not easy to do. If you have an object
moving at some velocity v and you let it roll or slide along a surface
until it stopped. You could then measure the time t it takes to stop to
determine its coefficient of friction.
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From the Force Equation, F = m*a, where a is the acceleration. Since
the object is starting at some velocity v and decelerating until v = 0, then
the force of friction can be written as:
Fr = m*v/t
If the object weighs W pounds, and W = m*g, where g is the gravity
constant 32 ft/sec/sec (9.8 m/s2, then the Friction Equation is:
Combining the two equations for Fr, we get:
Thus, if a car is moving at 64 feet per second and takes 4 seconds to
come to a stop, its coefficient of friction is:
fr = 64 / (32 x 4) = 0.5
1.3 VECTOR FORCES
A force is defined as any cause that tends to alter the state of rest
of a body or its state of uniform motion in a straight line. A force can be
defined quantitatively as the product of the mass of the body that the
force is acting on and the acceleration of the force.
P = ma
where
P = applied force
m = mass of the body (kg)
a = acceleration caused by the force (m/s2)
The SI units for force are therefore kg m/s2, which is designated a
Newton (N). The following multiples are often used: 1 kN = 1 000 N, 1
MN = 1 000 000 N
All objects on earth tend to accelerate toward the center of the earth
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Experimental Physics I
due to gravitational attraction; hence the force of gravitation acting on a
body with the mass (m) is the product of the mass and the acceleration
due to gravity (g), which has a magnitude of 9.81 m/s2:
F = mg = v rg
where:
F = force (N)
m = mass (kg)
g = acceleration due to gravity (9.81m/s2)
v = volume (m³)
r = density (kg/m³)
All forces (interactions) between objects can be placed into two
broad categories:
•
•
contact forces, and
forces resulting from action-at-a-distance
Contact forces are those types of forces that result when the two
interacting objects are perceived to be physically contacting each other.
Examples of contact forces include frictional forces, tensional forces,
normal forces, air resistance forces, and applied forces.
Action-at-a-distance forces are those types of forces that result even
when the two interacting objects are not in physical contact with each
other, yet are able to exert a push or pull despite their physical separation.
Examples of action-at-a-distance forces include gravitational forces. For
example, the sun and planets exert a gravitational pull on each other
despite their large spatial separation.
Even when your feet leave the earth and you are no longer in
physical contact with the earth, there is a gravitational pull between
you and the Earth. Electric forces are action-at-a-distance forces. For
example, the protons in the nucleus of an atom and the electrons outside
the nucleus experience an electrical pull towards each other despite their
small spatial separation.
And magnetic forces are action-at-a-distance forces. For example,
two magnets can exert a magnetic pull on each other even when
separated by a distance of a few centimeters.
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Vector
Most forces have magnitude and direction and can be shown as
a vector. The point of application must also be specified. A vector is
illustrated by a line, the length of which is proportional to the magnitude
on a given scale, and an arrow that shows the direction of the force.
Vector Addition
The sum of two or more vectors is called the resultant. The resultant
of two concurrent vectors is obtained by constructing a vector diagram
of the two vectors. The vectors to be added are arranged in tip-totail fashion. Where three or more vectors are to be added, they can be
arranged in the same manner, and this is called a polygon. A line drawn
to close the triangle or polygon (from start to finishing point) forms the
resultant vector. The subtraction of a vector is defined as the addition of
the corresponding negative vector.
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Experimental Physics I
1.3.1 Types of Force
In simple terms, a force is a push, a pull, or a drag on an object. There
are three main types of force:
•
•
•
An applied force is an interaction of one object on another that
causes the second object to change its velocity.
A resistive force passively resists motion and works in a
direction opposite to that motion.
An inertial force resists a change in velocity. It is equal to and in
an opposite direction of the other two forces.
There is no such thing as a unidirectional force or a force that acts
on only one object. There must always be two objects involved, acting
on each other. One object acts on the other, while the second resists the
action of the first.
Applied Force
An applied force is an interaction that causes the second object to
change its velocity.
Force Equation
The force required to overcome the inertia of an object is according
to the equation:
F = ma
where:
•
•
•
F is the force
m is the mass of the object
a is the acceleration caused by the force
Types of Applied Force
There are several types of applied force:
The most common form of force is a push through physical contact.
For example, you can push on a door to open it. An object can also collide
with another object, exerting a force and causing the second object to
accelerate. This is another type of push and can be called an impulse
force, since the time interval is very short. You can pull on an object
to change its velocity. Gravitation, magnetism, and static electricity
are some of the pulling forces that act at a distance with no physical
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contact required to move objects. Finally, if two objects or materials are
in contact, one can drag the other along by friction or other means.
Resistive Force
A resistive force passively inhibits or resists the motion of an object.
It is a form of push-back. It is considered passive, since it only responds
to actions on the object. Friction and fluid resistance are the major
resistive forces.
Friction
When an object is being pushed along the surface of another object
or material, the resistive force of friction pushes back on the first object
to resist its motion.
Fluid Resistance
Fluid resistance pushes back on the moving object, which is basically
trying to plow through the fluid. It also included friction on the surface
of the object.
Air resistance and water resistance are common forms of fluid
resistance.
Inertial Force
An inertial force works against a change in velocity, caused by an
applied force, as well as a resistive force.
Against Applied Force
According to Newton’s Third Law of
Motion or the Action-Reaction Law:
Whenever one body exerts force upon
a second body, the second body exerts an
equal and opposite force upon the first body.
This is often stated as: “For every action
there is an equal and opposite reaction.”
When you push on an object, an equal
inertial force pushes back. This is the
resistance to acceleration.
Hints
The more
mass something has
the more inertia it
has. You can think
of inertia as a property that makes it
hard to push something around.
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Experimental Physics I
Likewise, when swinging an object on a rope around you in a circle,
you pull on the rope to change the direction of motion. In turn, you can
feel a pull on the rope.
Against Resistive Force
When a resistive force like friction, slows down the motion of an
object, the inertial force will push in the opposite direction and tend to
keep the object moving.
The main type of force is an applied force, which is an interaction of
one object on another that causes the second object to change its velocity.
Other types of forces include a resistive force that passively resists
motion and an inertial force that resists a change in velocity.
There must always be two objects involved in a force, acting on each
other.
1.3.2 Principle for Number of Forces Acting On a Body
or Point
If a number of forces acting simultaneously on a body then the effect
produced by all the forces will be same as produced by a single resultant
of all the forces.
FR = F1 + F2
FR = 30 + (-20) F2 is –ve, is opposite or in –ve direction
FR = 10 KN
Two forces F1 and F2 are acting simultaneously opposite to each
other on a body (a). The resultant of both, FR will produce the same
effect on the body as produced by F1 and F2 (b).
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1.3.3 Causes of Force
Forces may arise from a number of different effects, including
i.
ii.
iii.
iv.
v.
vi.
Gravity;
Electromagnetism or electrostatics;
Pressure exerted by fluid or gas on part of a structure
Wind or fluid induced drag or lift forces;
Contact forces, which act wherever a structure or component
touches anything;
Friction forces, which also act at contacts.
Some of these forces can be described by universal laws. For example,
gravity forces can be calculated using Newton’s law of gravitation;
electrostatic forces acting between charged particles are governed by
Coulomb’s law; electromagnetic forces acting between current carrying
wires are governed by Ampere’s law; buoyancy forces are governed by
laws describing hydrostatic forces in fluids.
Some forces have to be measured. For example, to determine friction
forces acting in a machine, you may need to measure the coefficient of
friction for the contacting surfaces. Similarly, to determine aerodynamic
lift or drag forces acting on a structure, you would probably need to
measure its lift and drag coefficient experimentally.
Contact forces are pressures that act on the small area of contact
between two objects. Contact forces can either be measured, or they
can be calculated by analyzing forces and deformation in the system of
interest.
1.3.4 Forces in Translational Equilibrium
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Experimental Physics I
There are many objects we do not want to see in motion. In
the Figure above, the mountain climbers want their ropes to keep them
from moving downward. We construct buildings and bridges to be
as motionless as possible. We want the acceleration (and velocity) of
these objects to be zero. For an object to be in static equilibrium (that is,
motionless) the right-hand side of Newton’s Second Law, ∑F=ma, must
be zero. Thus, ∑F=0. This equation is simple enough when an object
is held with a single support. In an earlier example, we depicted Joe
Loose hanging by a single rope (Figure below). Joe’s goal was to remain
hanging in equilibrium (just like the climbers in the photograph). The
force of gravity pulling Joe down was exactly balanced by the tension in
the rope that supported him.
But Joe won’t be hanging for very long, will he? You can see that
the rope is slowly fraying against the mountainside (recall the original
problem). Soon it will snap. But Joe’s in luck, because a rescue team has
come to his aid. They arrive just in time to secure two more ropes to the
mountain side and toss Joe the slack to tie around his waist before his
rope snaps! Joe is saved. But does Joe thank the rescue team like any
sane person would? No. Instead, still in midair, he pulls out a pad and
pencil from his back pocket in order to analyze the forces acting on him
(Figure below).
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In order for Joe to remain in equilibrium, he must not move in
the x− or y−directions. This means that the sum of all forces in the x−
direction must add to zero. And the sum of all forces in the y−direction
must add to zero.
The procedure for solving problems with forces in equilibrium is as
follows:
1.
2.
3.
Place Figure above in a coordinate plane with the object at the
origin.
Resolve the tension vectors T1 and T2 into their x− and y−
components.
Use Newton’s Second Law: ∑Fx=0 and ∑Fy=0.
In order to solve this problem, we’ll need more information, including
the angles that the ropes make with the vertical. The information is
provided below, along with Figure below.
θ1=45°
θ2=30°
Mg=800 N
Find T1 and T2
The solution requires solving a set of simultaneous equations.
First, we find the components of vectors T1 and T2.
T1x=T1 sin 45∘ and T1y=T1 cos 45∘
T2= T2 sin 30∘ and T2y=T2 cos 30∘
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Experimental Physics I
Next we apply Newton’s Second Law.
∑Fx:T2 sin 30° −T1 sin45° = 0
∑Fy:T2 cos 30°+T1 cos45° = 800
The first equation can be quickly simplified to give T2=2–√T1. T2 is
then substituted in the second equation and T1 is found. Once T1 is
found, T2 can easily be computed using T2=2–√T1
T1 = 414.11 = 414 = 410 N
T2 = 585.56 = 586 = 590 N
1.4 PARTICLES AND RIGID BODY
Particles are objects that have mass, position, and velocity, and
respond to forces, but that have no spatial extent. Because they are
simple, particles are by far the easiest objects to simulate. Despite their
simplicity, particles can be made to exhibit a wide range of interesting
behavior. For example, a wide variety of non-rigid structures can be
built by connecting particles with simple damped springs.
A rigid body is an idealized model of an object that has a definite
and unchanging shape and size. In reality, real-world bodies are
constantly interacting with the environment, undergoing forces that
can twist, stretch, or squeeze them in ways that would make precise
calculations involving them quite impractical. The concept of a rigid
body is particularly crucial when considering rotational motion. Some
key points related to a rigid body and rotational motion are:
•
Every part of a rigid body has the same angular velocity.
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•
19
In a rotating rigid body, the velocity and acceleration of
any point can be calculated using the distance from the
axis of rotation and the body’s angular velocity and angular
acceleration.
You can calculate the kinetic energy of a rotating rigid body
using the angular velocity of the body and its moment of
inertia.
1.4.1 Systems of Particles
Systems of particles means such things
as a swarm of bees, a star cluster, a cloud
of gas, an atom, a brick. A brick is indeed
composed of a system of particles – atoms
− which are constrained so that there is very
little motion (apart from small amplitude
vibrations) of the particles relative to each
other. In a system of particles there may
be very little or no interaction between the
particles (as in a loose association of stars
separated from each other by large distances)
or there may be (as in the brick) strong forces
between the particles.
Momentum
Hints
Rigid body
analysis is more
complex and also
has to take into
account moments
and rotational
motions. In actuality, no bodies are
truly particles, but
some bodies can be
approximated as
particles to simplify
analysis.
The momentum of a particle is defined
as the product of its mass times its velocity.
It is a vector quantity. The momentum of a
system is the vector sum of the momenta
of the objects which make up the system. If the system is an isolated
system, then the momentum of the system is a constant of the motion
and subject to the principle of conservation of momentum.
The basic definition of momentum applies even at relativistic
velocities but then the mass is taken to be the relativistic mass.
The most common symbol for momentum is p. The SI unit for
momentum is kg m/s.
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Experimental Physics I
The mathematical definition of momentum is consistent with our
intuitive, everyday notion of “momentum.” If two cars have equal masses
but one has twice the velocity of the other, it has twice the momentum.
And if a truck has three times the mass of a car and the same velocity, it
has three times the momentum.
Newton’s First Law states that, in the absence of external forces, the
velocity of a particle remains constant. Expressed in terms of momentum,
the First Law therefore states that the momentum remains constant:
P = [constant]
(no external forces)
Thus, we can say that the momentum of the particle is conserved.
Of course, we could equally well say that the velocity of this particle is
conserved; but the deeper significance of momentum will emerge when
we study the motion of a system of several particles exerting forces on
one another. We will find that the total momentum of such a system is
conserved—any momentum lost by one particle is compensated by a
momentum gain of some other particle or particles.
To express the Second Law in terms of momentum, we note that
since the mass is constant, the time derivative of above equation is
or
But, according to Newton’s Second Law, ma equals the force; hence,
the rate of change of the momentum with respect to time equals the force:
dp
=F
dt
This equation gives the Second Law a concise and elegant form.
We can also express Newton’s Third Law in terms of momentum.
Since the action force is exactly opposite to the reaction force, the rate
of change of momentum generated by the action force on one body is
exactly opposite to the rate of change of momentum generated by the
reaction force on the other body. Hence, we can state the Third Law as
follows:
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Whenever two bodies exert forces on each other, the resulting
changes of momentum are of equal magnitudes and of opposite
directions.
This balance in the changes of momentum leads us to a general law
of conservation of the total momentum for a system of particles. The
total momentum of a system of n particles is simply the (vector) sum
of all the individual momenta of all the particles. Thus, if p1 = m1v1, p2
= m2v2, . . . , and pn = mnvn are the individual momenta of the particles,
then the total momentum is
P = p1 + p2 + ………. + pn
The simplest of all many-particle systems consists of just two
particles exerting some mutual forces on one another (Figure 1). Let us
assume that the two particles are isolated from the rest of the Universe
so that, apart from their mutual forces, they experience no extra forces of
any kind. According to the above formulation of the Third Law, the rates
of change of p1 and p2 are then exactly opposite:
Figure 1. Two particles exerting mutual forces on each other. The net change
of momentum of the isolated particle pair is zero.
The rate of change of the sum p1+ p2 is therefore zero, since the rate
of change of the first term in this sum is canceled by the rate of change
of the second term:
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Experimental Physics I
This means that the sum p1+ p2 is a constant of the motion:
p1+ p2 = [constant]
This is the Law of Conservation of Momentum. Note that Newton’s
Third Law is an essential ingredient for establishing the conservation
of momentum: the total momentum is constant because the equality of
action and reaction keeps the momentum changes of the two particles
exactly equal in magnitude but opposite in direction—the particles
merely exchange some momentum by means of their mutual forces.
Thus, for our particles, the total momentum P at some instant equals the
total momentum P’ at some other instant, so
P = P’
Conservation of momentum is a powerful tool which permits us to
calculate some general features of the motion even when we are ignorant
of the detailed properties of the interparticle forces.
Center of Mass
The motion of a rotating ax thrown between two jugglers looks
rather complicated, and very different from the standard projectile
motion. Experiments have shown that one point of the ax follows a
trajectory described by the standard equations of motion of a projectile.
This special point is called the center of mass of the ax.
The position of the center of mass of a system of two particles with
mass m1 and m2, located at position x1 and x2, respectively, is defined as
=
xcm
m1 x1 + m2 x2
1
=
∑m x
m1 + m2
M i i i
Since we are free to define our coordinate system in whatever way
is convenient, we can define the origin of our coordinate system to
coincide with the left most object (Figure 2). The position of the center
of mass is now
Mechanics
xcm =
23
m2 d
m1 + m2
This equation shows that the center of mass lies between the two
masses, closest to the heavier mass.
Figure 2. Position of the center of mass in one dimension.
In general, for a system with more than two particles, the position
of the center of mass will satisfy the following relation
xmin ≤ xcm ≤ xmax
The definition of the center of mass in one dimension can be easily
generalized to three dimensions
xcm =
1
∑m x
M i i i
ycm =
1
∑m y
M i i i
zcm =
1
∑m z
M i i i
or in vector notation
rcm =
1
∑ m r
M i ii
24
Experimental Physics I
For a rigid body, the summation will be replaced by an integral
1
M
rcm =

∫ rdm
v
Suppose we are dealing with a number of objects. Figure 3 shows
a system consisting of 4 masses, m1, m2, m3 and m4, located at x1, x2, x3
and x4, respectively. The position of the center of mass of m1 and m2 is
given by
xcm1,2 =
m1 x1 + m2 x2
m1 + m2
The position of the center of mass of m3 and m4 is given by
xcm 3,4 =
m3 x3 + m4 x4
m3 + m4
The position of the center of mass of the whole system is given by
xcm =
m1 x1 + m2 x2 + m3 x3 + m4 x4
m1 + m2 + m3 + m4
This can be rewritten as
=
xcm
1
m1 + m2 + m3 + m4

m x + m4 x4
m1 x1 + m2 x2
+ ( m3 + m4 ) 3 3
 ( m1 + m2 )
m
+
m
m3 + m4
1
2




Using the center of mass of m1 and m2 and of m3 and m4 we can
express the center of mass of the whole system as follows
xcm =
(m
1
+ m2 ) xcm1,2 + ( m3 + m4 ) xcm 3,4
(m
1
+ m2 ) + ( m3 + m4 )
Figure 3. Location of 4 masses.
Mechanics
25
This shows that the center of mass of a system can be calculated
from the position of the center of mass of all objects that make up the
system.
The Motion of the Center of Mass
The definition of the center of mass of a system of particles can be
rewritten as
Mrcm = ∑ mi ri
i
where M is the total mass of the system. Differentiating this equation
with respect to time shows
Mvcm = ∑ mi vi
i
where vcm is the velocity of the center of mass and vi is the velocity
of mass mi. The acceleration of the center of mass can be obtained by
once again differentiating this expression with respect to time
Macm = ∑ mi ai
i
where acm is the acceleration of the center of mass and ai is the
acceleration of mass mi. Using Newton’s second law we can identify mi
ai with the force acting on mass mi. This shows that
Ma
=
cm
F ∑F
∑=
i
i
ext
This equation shows that the motion of the center of mass is only
determined by the external forces. Forces exerted by one part of the
system on other parts of the system are called internal forces. According
to Newton’s third law, the sum of all internal forces cancel out (for each
interaction there are two forces acting on two parts: they are equal in
magnitude but pointing in an opposite direction and cancel if we take
the vector sum of all internal forces).
26
Experimental Physics I
Figure 4. Internal and External Forces acting on a System of Particles.
The previous equations show that the center of mass of a system of
particles acts like a particle of mass M, and reacts like a particle when
the system is exposed to external forces. They also show that when the
net external force acting on the system is zero, the velocity of the center
of mass will be constant.
Energy of a System of Particles
The total kinetic energy of a system of particles is simply the sum of
the individual kinetic energies of all the particles,
1
1
1
K = m1 v12 + m2 v22 + mn vn2
2
2
2
Since for the momentum of a system of particles resembles the
expression for the momentum of a single particle, we might be tempted
to guess that the equation for the kinetic energy for a system of particles
also can be expressed in the form of the translational kinetic energy of
1
2
MvCM
2
the center of mass
, resembling the kinetic energy of a single
particle. The total kinetic energy of a system of particles is usually larger
1
2
MvCM
2
than
. We can see this in the following simple example: Consider
two automobiles of equal masses moving toward each other at equal
speeds .The velocity of the center of mass is then zero, and consequently
1
2
MvCM
2
= 0. However, since each automobile has a positive kinetic energy,
the total kinetic energy is not zero.
Mechanics
27
If the internal and external forces acting on a system of particles
are conservative, then the system will have a potential energy. We saw
above that for the specific example of the gravitational potential energy
near the Earth’s surface, the potential energy of the system took the
same form as for a single particle, U = MgyCM. But this form is a result of
the particular force (uniform and proportional to mass); in general, the
potential energy for a system does not have the same form as for a single
particle. Unless we specify all of the forces, we cannot write down an
explicit formula for the potential energy; but in any case, this potential
energy will be some function of the positions of all the particles. The
total mechanical energy is the sum of the total kinetic energy and the
total potential energy. This total energy will be conserved during the
motion of the system of particles.
Note that in reckoning the total potential energy of the system, we
must include the potential energy of both the external forces and the
internal forces. We know that the internal forces do not contribute to
the changes of total momentum of the system, but these internal forces,
and their potential energies, contribute to the total energy. For instance,
if two particles are falling toward each other under the influence of their
mutual gravitational attraction, the momentum gained by one particle is
balanced by momentum lost by the other, but the kinetic energy gained
by one particle is not balanced by kinetic energy lost by the other—both
particles gain kinetic energy. In this example the gravitational attraction
plays the role of an internal force in the system, and the gain of kinetic
energy is due to a loss of mutual gravitational potential energy
1.4.2 Rotation of a Rigid Body
A body is rigid if the particles in the body do not move relative to
one another. Thus, the body has a fixed shape, and all its parts have a
fixed position relative to one another. A hammer is a rigid body, and so
is a baseball bat. A baseball is not rigid—when struck a blow by the bat,
the ball suffers a substantial deformation; that is, different parts of the
ball move relative to one another. However, the baseball can be regarded
as a rigid body while it flies through the air—the air resistance is not
sufficiently large to produce an appreciable deformation of the ball. This
example indicates that whether a body can be regarded as rigid depends
on the circumstances. Nobody is absolutely rigid; when subjected to
a sufficiently large force, anybody will suffer some deformation or
perhaps even break into several pieces.
28
Experimental Physics I
Motion of a Rigid Body
A rigid body can simultaneously have
two kinds of motion: it can change its position
in space, and it can change its orientation in
space. Change of position is translational
motion, this motion can be conveniently
described as motion of the center of mass.
Change in orientation is rotational motion;
that is, it is rotation about some axis.
Hints
When a rigid
body moves, both its
position and orientation vary with time. In
the kinematic sense,
these changes are
referred to as translation and rotation,
respectively. Indeed,
the position of a rigid
body can be viewed as
a hypothetic translation and rotation
(roto-translation) of
the body starting from
a hypothetic reference
position
As an example, consider the motion of
a hammer thrown upward (Figure 5). The
orientation of the hammer changes relative
to fixed coordinates attached to the ground.
Instantaneously, the hammer rotates about
a horizontal axis, say, a horizontal axis that
passes through the center of mass. In figure
5, this horizontal axis sticks out of the plane
of the page and moves upward with the
center of mass. The complete motion can
then be described as a rotation of the hammer about this axis and a
simultaneous translation of the axis along a parabolic path.
Figure 5. A hammer in free fall under the influence of gravity. The center of
mass of the hammer moves with constant vertical acceleration g, just like a
particle in free fall.
Mechanics
29
In this example of the thrown hammer, the axis of rotation always
remains horizontal, out of the plane of the page. In the general case of
motion of a rigid body, the axis of rotation can have any direction and
can also change its direction. To describe such complicated motion, it is
convenient to separate the rotation into three components along three
perpendicular axes. The three components of rotation are illustrated by
the motion of an aircraft (Figure 6): the aircraft can turn left or right
(yaw), it can tilt to the left or the right (roll), and it can tilt its nose up or
down (pitch).
Figure 6. Pitch, roll, and yaw motions of an aircraft.
Rotation about a Fixed Axis
Figure 7 shows a rigid body rotating about a fixed axis, which
coincides with the z axis. During this rotational motion, each point of
the body remains at a given distance from this axis and moves along a
circle centered on the axis. To describe the orientation of the body at any
instant, we select one particle in the body and use it as a reference point;
any particle can serve as reference point, provided that it is not on the
axis of rotation. The circular motion of this reference particle (labeled P
in figure 7) is then representative of the rotational motion of the entire
body, and the angular position of this particle is representative of the
angular orientation of the entire body.
30
Experimental Physics I
Figure 7. The four blades of this fan are a rigid body rotating about a fixed
axis, which coincides with the z axis. The reference particle P in this rigid
body moves along a circle around this axis.
Figure 8. Motion of a reference particle P in the rigid body rotating about a
fixed axis. The axis is indicated by the circled dot O. The radius of the circle
traced out by the motion of the reference particle is R.
Figure 8 shows the rotating rigid body as seen from along the axis
of rotation. The coordinates in figure 8 have been chosen so the z axis
coincides with the axis of rotation, whereas the x and y axes are in the
plane of the circle traced out by the motion of the reference particle.
The angular position of the reference particle—and hence the angular
orientation of the entire rigid body—can be described by the position
Mechanics
31
angle f between the radial line OP and the x axis. Conventionally, the
angle is taken as positive when reckoned in a counterclockwise direction
(as in figure 8). We will usually measure this position angle in radians,
rather than degrees. By definition, the angle in radians is the length s of
the circular arc divided by the radius R, or
φ=
s
R
In Figure 8, the length s is the distance traveled by the reference
particle from the x axis to the point P. Note that if the length s is the
circumference of a full circle, then s = 2pR, and f = s/R = 2pR/R =2p. Thus,
there are 2p radians in a full circle; that is, there are 2p radians in 360°
2p radians = 360°
Accordingly, 1 radian equals 360/2p, or
1 radian = 57.3°
When a rigid body rotates, the position angle f changes in time.
The body then has an angular velocity w.The definition of the angular
velocity for rotational motion is mathematically analogous to the
definition of velocity for translational motion. The average angular
velocity ω is defined as
ω=
∆φ
∆t
where Df is the change in the angular position and Dt the
corresponding change in time. The instantaneous angular velocity is
defined as
ω=
dφ
dt
According to these definitions, the angular velocity is the rate of
change of the angle with time. The unit of angular velocity is the radian
per second (1 radian/s).The radian is the ratio of two lengths, and hence
it is a pure number; thus, 1 radian/s is the same thing as 1/s. However, to
prevent confusion, it is often useful to retain the vacuous label radian as
a reminder that angular motion is involved.
32
Experimental Physics I
If the body rotates with constant angular velocity, then we can also
measure the rate of rotation in terms of the ordinary frequency f, or
the number of revolutions per second. Since each complete revolution
involves a change f of by 2p radians, the frequency of revolution is
smaller than the angular velocity by a factor of 2p:
f =
ω
2π
This expresses the frequency in terms of the angular velocity. The
unit of rotational frequency is the revolution per second (1 rev/s). Like
the radian, the revolution is a pure number, and hence 1 rev/s is the
same thing as 1/s. But we will keep the label rev to prevent confusion
between rev/s and radian/s.
As in the case of planetary motion, the time per revolution is called
the period of the motion. If the number of revolutions per second is f,
then the time per revolution is 1/f, that is,
T=
1
f
If the angular velocity of a rigid body is changing, the body has
an angular acceleration a. The rotational motion of a ceiling fan that is
gradually building up speed immediately after being turned on is an
example of accelerated rotational motion. The mathematical definition
of the average angular acceleration is, again, analogous to the definition
of acceleration for translational motion. If the angular velocity changes
by Dw in a time Dt, then the average angular acceleration is
α=
∆ω
∆t
and the instantaneous angular acceleration is
α=
dω
dt
Thus, the angular acceleration is the rate of change of the angular
velocity. The unit of angular acceleration is the radian per second per
second, or radian per second squared (1 radian/s2).
Mechanics
33
Since the angular velocity w is the rate of change of the angular
position f, the angular acceleration given by above equation can also be
written
d 2φ
α= 2
dt
The angular velocity and acceleration of the rigid body; that is, they
give the angular velocity and acceleration of every particle in the body. It
is interesting to focus on one of the particles and evaluate its translational
speed and acceleration as it moves along its circular path around the
axis of rotation of the rigid body. If the particle is at a distance R from
the axis of rotation (figure 9), then the length along the circular path of
the particle is, according to the definition of angle,
s = fR
Figure 9. The instantaneous translational velocity of a particle in a rotating
rigid body is tangent to the circular path.
Since R is a constant, the rate of change of s is entirely due to the rate
of change of f, so
ds dφ
= R
dt dt
Here ds/dt is the translational speed v with which the particle moves
along its circular path, and df/dt is the angular velocity w; hence above
equation is equivalent to
34
Experimental Physics I
v = wR
This shows that the translational speed of the particle along its
circular path around the axis is directly proportional to the radius: the
farther a particle in the rigid body is from the axis, the faster it moves.
We can understand this by comparing the motions of two particles,
one on a circle of large radius R1, and the other on a circle of smaller
radius R2 (figure 10). For each revolution of the rigid body, both of these
particles complete one trip around their circles. But the particle on the
larger circle has to travel a larger distance, and hence must move with a
larger speed.
Figure 10. Several particles in a rigid body rotating about a fixed axis and
their velocities.
For a particle at a given R, the translational speed is constant if
the angular velocity is constant. This speed is the distance around the
circular path (the circumference) divided by the time for one revolution
(the period), or
Since 2p/T = 2pf = w.
If v is changing, that the rate of change of v is proportional to the
rate of change of w:
Mechanics
35
A rate of change of the speed along the circle implies that the particle
has an acceleration along the circle, called a tangential acceleration.
According to the last equation, this tangential acceleration is
atangential = aR
Note that, besides this tangential acceleration directed along the
circle, the particle also has a centripetal acceleration directed toward
the center of the circle. We know that the centripetal acceleration for
uniform circular motion is
With v = wR, this becomes
acentripital = w2R
The net translational acceleration of the particle is the vector sum of
the tangential and the centripetal accelerations, which are perpendicular
(figure 11); thus, the magnitude of the net acceleration is
Figure 11. A particle in a rotating rigid body with an angular acceleration
has both a centripetal acceleration acentripetal and a tangential acceleration
atangential. The net instantaneous translational acceleration anet is then the
vector sum of acentripetal and atangential.
36
Experimental Physics I
Although we have here introduced the concept of tangential
acceleration in the context of the rotational motion of a rigid body, this
concept is also applicable to the translational motion of a particle along
a circular path or any curved path. For instance, consider an automobile
(regarded as a particle) traveling around a curve. If the driver steps on
the accelerator (or on the brake), the automobile will suffer a change of
speed as it travels around the curve. It will then have both a tangential
and a centripetal acceleration.
Review of Vector Operations
Addition
(a1 , b1 ) + (a2 , b2 ) =
(a1 + a2 , b1 + b2 )
Subtraction
(a1 , b1) − (a2 , b2 ) =
(a1 − a2 , b1 − b2 )
Scalarmultiplication k ⋅ (a, b) =(k ⋅ a, k ⋅ b)
In vector addition, we add the corresponding components. In vector
subtraction, we subtract the corresponding components.
1.4 ROTATIONAL EQUILIBRIUM/TORQUE
Equilibrium is the state of being balanced physically, and it is an
important factor for being in motion. Explore the relationship between
motion and equilibrium, focusing on the two types: translational and
rotational.
1.4.1 Torque
Up to this point in physics, we haven’t allowed forces to act in such
a way as to cause rotation. The forces acting on a grocery cart traveling
along an aisle could only be exerted in such a way that no rotation
of the cart was produced. As you know from experience, this can be
accomplished by pushing straight ahead on the center of the handle
of the cart or by pushing equally on the two ends of the handle. (This
assumes no wobbly, squeaky wheels—generally a bad assumption.) But
as we know, bad things can happen at the end of the aisle when paths
cross and carts can end up spinning off into the produce section. When
this occurs, the cart moves along some new path and often spins. Forces
that tend to cause rotation are said to apply a torque to the objects they
act on. Consider the three forces, F, in Figures 12a–c. All three act in the
plane of the page. The rod is hinged so that it can rotate around its axis
of rotation, which is normal (perpendicular) to the page.
Mechanics
37
Figure 12a. Torque on a rod.
Figure 12b. Force perpendicular to r. Maximum torque.
Figure 12c. Force collinear to r. Zero torque.
The axis of rotation (×) is the point (or line) around which we want
to calculate the torque. We can choose any point we like, but there will
usually be a best choice, which will make our calculations simplest. Or
there may be a point specified in an assignment (e.g., «Find the torque
around the door hinge.»).The line of action of the force is an extended
line collinear with the force. This is usually shown as a dotted line in
figures. The lever arm, r⊥, is the shortest distance between the axis of
rotation and the line of action of the force. It is a line drawn from the axis
of rotation so as to hit the line of action at a right angle. (Sometimes the
character ℓ is used to represent the lever arm.)We define torque, τ (tau),
as the product of the lever arm, r⊥, and the force, F.
38
Experimental Physics I
τ = r ⊥F
Another quantity, which sadly has no name, is r. r is the distance
between the axis and the point of application of the force. Notice in
Figures 12a–c that r and r⊥ are sometimes, but not always, equal.
In Figure 12b, where the force is perpendicular to r, r and r⊥ are equal.
Thus, the torque is simply rF. This is the maximum torque possible for
a given r and F. The other extreme occurs when the line of action passes
through the axis of rotation. In this case, r and r⊥ = 0. Thus, the torque is
also zero. Often, 0 < r⊥ < r ; thus, 0 < r⊥F < rF, as in Figure 12a.
1.4.2 Rotational Equilibrium
We know that a body with no net force acting on it will be
in translational equilibrium and have zero translational acceleration. As
shown in Figure 13a, two equal but opposite forces are able to cancel,
resulting in no translational acceleration.
Figure 13a.
But if you give a spin to a ball by giving it a twist with your fingers,
you can give it a rotational acceleration even though the net force
acting is zero. In Figure 13b, the two equal forces, F1 and F2, acting on
the ball will not cause a translational acceleration, but they will cause
a rotational acceleration. Each force produces a counterclockwise torque,
thus producing a nonzero torque.
Figure 13b.
Mechanics
39
For a body to be in rotational equilibrium, the net torque on it must
equal zero. This is the case in both Figures 13c and 13d.
Figure 13c.
In Figure 13c, the two equal forces F1 and F2 act with equal lever
arms r⊥.
Force F2 applies a negative, clockwise (cw), torque of –r⊥F2.
Force F1 applies a positive, counterclockwise (ccw), torque of r⊥F1.
∑ τ=
( −r⊥ F2 ) + r⊥ F1=
0
Figure 13d.
In Figure 13d, the two nonequal forces F1 and F2 act with nonequal
lever arms r⊥ and 2r⊥.
F2 = 2F1.
Force F2 applies a negative, clockwise, torque of –r⊥F2.
Force F1 applies a positive, counterclockwise, torque of 2r⊥F1.
∑ τ = ( −r F )
⊥ 2
∑τ=0
+ 2r⊥ F1 =
( −r
⊥
2F1 ) + 2r⊥ F1
40
Experimental Physics I
We will be working with systems that are in rotational equilibrium;
that is, where the vector sum of the torques about any axis equals zero.
∑τ
any axis
=
0
This is a vector equation where positive torques are defined as
torques that would tend to produce a counterclockwise rotation and
negative torques would tend to produce a clockwise rotation. The First
Condition Equilibrium.
∑F
any direction
= 0
The two equations can also be used together in situations where
forces act to put a system in both types of equilibrium. For example,
when you lean against a wall, there are forces exerted on you by the
wall, the ground, and Earth’s gravity.
1.5 WORK AND ENERGY
The concepts of work and energy are closely tied to the concept of
force because an applied force can do work on an object and cause a
change in energy. Energy is defined as the ability to do work.
1.5.1 Work
The concept of work in physics is much more narrowly defined than
the common use of the word. Work is done on an object when an applied
force moves it through a distance. In our everyday language, work is
related to expenditure of muscular effort, but this is not the case in the
language of physics. A person that holds a heavy object does no physical
work because the force is not moving the object through a distance. Work,
according to the physics definition, is being accomplished while the
heavy object is being lifted but not while the object is stationary. Another
example of the absence of work is a mass on the end of a string rotating
in a horizontal circle on a frictionless surface. The centripetal force is
directed toward the center of the circle and, therefore, is not moving
the object through a distance; that is, the force is not in the direction
of motion of the object. (However, work was done to set the mass in
motion.) Mathematically, work is W = F · x, where F is the applied force
and x is the distance moved, that is, displacement. Work is a scalar. The
SI unit for work is the joule (J), which is newton‐meter or kg m/s 2.
Mechanics
41
If work is done by a varying force, the above equation cannot be
used. Figure shows the force‐versus‐displacement graph for an object
that has three different successive forces acting on it. The force is
increasing in segment I, is constant in segment II, and is decreasing in
segment III. The work performed on the object by each force is the area
between the curve and the x axis. The total work done is the total area
between the curve and the x axis. For example, in this case, the work
done by the three successive forces is shown in below Figure.
In this example, the total work accomplished is (1/2)(15)(3) + (15)
(2) + (1/2)(15)(2) = 22.5 + 30 + 15; work = 67.5 J. For a gradually changing
force, the work is expressed in integral form, W = ∫ F · dx.
1.5.2 Dot or Scalar Product
The dot product, also called the scalar product, of two vector s is a
number ( scalar quantity) obtained by performing a specific operation
on the vector components. The dot product has meaning only for pairs
of vectors having the same number of dimensions. The symbol for dot
product is a heavy dot (  ).
In the two-dimensional Cartesian plane, vectors are expressed
in terms of the x -coordinates and y -coordinates of their end points,
assuming they begin at the origin ( x , y ) = (0,0).
The dot product of two vectors is determined by multiplying
their x -coordinates, then multiplying their y -coordinates, and finally
adding the two products. Thus, in the above example:
42
Experimental Physics I
A  B = (2 x -4) + (5 x -3) = -8 - 15 = -23
B  C = (-4 x 5) + (-3 x -5) = -20 + 15 = -5
C  A = (5 x 2) + (-5 x 5) = 10 - 25 = -15
In polar coordinates, vectors are expressed in terms of length
(magnitude) and direction. When expressed in this format, the dot
product of two vectors is equal to the product of their lengths, multiplied
by the cosine of the angle between them.
For any two vectors A and B , A B = B A . That is, the dot product
operation is commutative; it does not matter in which order the operation
is performed.
Work done by a force
Work is done whenever a force moves something over a distance.
You can calculate the energy transferred, or work done, by multiplying
the force by the distance moved in the direction of the force.
Energy transferred = work done = force x distance moved in the direction
of the force
When energy is transferred from chemical energy stored in muscles
to ‘uphill energy’ in a raised load, or to ‘elastic energy’ in stretched
springs, the energy transferred is a measure of how much work has been
done.
Energy transferred = mgh
This second equation is illustrated by raising kilograms onto
different height shelves. You can show that the equation is a good
summary of what happens. It takes account of the mass, the height
Mechanics
raised and whether the kilogram is raised on
the Earth or the Moon.
The useful thing which you get from
fuels by burning them is the transfer of
energy released, to some other energy store
such as a raised load, or a moving body.
However, not all the energy available
does a useful job. If you lift a lot of bricks,
you can get too hot. As well as transferring
energy to the raised bricks, some of the
energy generated in your muscles warms
you up. The transfer of energy is not 100%
efficient and not all the energy transferred is
represented by mgh. Nor do you know how
much total energy is stored by things being
‘uphill’. You can only calculate energy that is
transferred.
43
Important
The term
work was introduced in 1826 by
the French mathematician GaspardGustave Coriolis
as “weight lifted
through a height”,
which is based on
the use of early
steam engines to
lift buckets of water
out of flooded ore
mines. The SI unit of
work is the joule (J).
1.5.3 Potential Energy
An object can store energy as the result of its position. For example,
the heavy ball of a demolition machine is storing energy when it is held
at an elevated position. This stored energy of position is referred to as
potential energy.
Similarly, a drawn bow is able to store energy as the result of its
position. When assuming its usual position (i.e., when not drawn), there
is no energy stored in the bow. Yet when its position is altered from its
usual equilibrium position, the bow is able to store energy by virtue
of its position. This stored energy of position is referred to as potential
energy. Potential energy is the stored energy of position possessed by
an object.
44
Experimental Physics I
Gravitational Potential Energy
Gravitational potential energy is the energy stored in an object
as the result of its vertical position or height. The energy is stored as
the result of the gravitational attraction of the Earth for the object.
The gravitational potential energy of the massive ball of a demolition
machine is dependent on two variables - the mass of the ball and the
height to which it is raised. There is a direct relation between gravitational
potential energy and the mass of an object. More massive objects have
greater gravitational potential energy. There is also a direct relation
between gravitational potential energy and the height of an object. The
higher that an object is elevated, the greater the gravitational potential
energy. These relationships are expressed by the following equation:
PEgrav = mass • g • height
PEgrav = m *• g • h
In the above equation, m represents the mass of the object, h represents
the height of the object and g represents the gravitational field strength
(9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity.
Since many of our labs are done on tabletops, it is often customary
to assign the tabletop to be the zero height position. Again this is merely
arbitrary. If the tabletop is the zero position, then the potential energy of
an object is based upon its height relative to the tabletop. For example, a
pendulum bob swinging to and from above the tabletop has a potential
energy that can be measured based on its height above the tabletop.
By measuring the mass of the bob and the height of the bob above the
tabletop, the potential energy of the bob can be determined.
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45
Since the gravitational potential energy of an object is directly
proportional to its height above the zero position, a doubling of the height
will result in a doubling of the gravitational potential energy. A tripling of
the height will result in a tripling of the gravitational potential energy.
Elastic Potential Energy
The second form of potential energy that we will discuss is elastic
potential energy. Elastic potential energy is the energy stored in elastic
materials as the result of their stretching or compressing. Elastic potential
energy can be stored in rubber bands, bungee chords, trampolines,
springs, an arrow drawn into a bow, etc. The amount of elastic potential
energy stored in such a device is related to the amount of stretch of the
device - the more stretch, the more stored energy.
Springs are a special instance of a device that can store elastic
potential energy due to either compression or stretching. A force is
required to compress a spring; the more compression there is, the
more force that is required to compress it further. For certain springs,
the amount of force is directly proportional to the amount of stretch or
compression (x); the constant of proportionality is known as the spring
constant (k).
Fspring = k • x
Such springs are said to follow Hooke’s Law. If a spring is not
stretched or compressed, then there is no elastic potential energy stored
in it. The spring is said to be at its equilibrium position. The equilibrium
position is the position that the spring naturally assumes when there
is no force applied to it. In terms of potential energy, the equilibrium
position could be called the zero-potential energy position. There is a
special equation for springs that relates the amount of elastic potential
46
Experimental Physics I
energy to the amount of stretch (or compression) and the spring constant.
The equation is:
PEspring = 0.5 • k • x2
Where
k = spring constant
x = amount of compression (relative to equilibrium position)
To summarize, potential energy is the energy that is stored in an object
due to its position relative to some zero position. An object possesses
gravitational potential energy if it is positioned at a height above (or
below) the zero height. An object possesses elastic potential energy if it
is at a position on an elastic medium other than the equilibrium position.
1.5.4 Kinetic Energy
Kinetic energy, form of energy that an object or a particle has by
reason of its motion. If work, which transfers energy, is done on an object
by applying a net force, the object speeds up and thereby gains kinetic
energy. Kinetic energy is a property of a moving object or particle and
depends not only on its motion but also on its mass. The kind of motion
may be translation (or motion along a path from one place to another),
rotation about an axis, vibration, or any combination of motions.
Translational kinetic energy of a body is equal to one-half the
product of its mass, m, and the square of its velocity, v, or 1/2mv2.
This formula is valid only for low to relatively high speeds; for
extremely high-speed particles it yields values that are too small.
When the speed of an object approaches that of light (3 × 108 metres
per second, or 186,000 miles per second), its mass increases, and the
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47
laws of relativity must be used. Relativistic kinetic energy is equal to the
increase in the mass of a particle over that which it has at rest multiplied
by the square of the speed of light.
The unit of energy in the metre-kilogram-second system is the joule.
A two-kilogram mass (something weighing 4.4 pounds on Earth) moving
at a speed of one metre per second (slightly more than two miles per
hour) has a kinetic energy of one joule. In the centimetre-gram-second
system the unit of energy is the erg, 10−7 joule, equivalent to the kinetic
energy of a mosquito in flight. Other units of energy also are used,
in specific contexts, such as the still smaller unit, the electron volt, on
the atomic and subatomic scale.
For a rotating body, the moment of inertia, I, corresponds to
mass, and the angular velocity (omega), ω, corresponds to linear, or
translational, velocity. Accordingly, rotational kinetic energy is equal
to one-half the product of the moment of inertia and the square of the
angular velocity, or 1/2Iω2.
1.5.5 Mechanical Energy
Mechanical energy is often confused with Kinetic and Potential
Energy. We will try to make it very easy to understand and know the
difference. Before that, we need to understand the word ‘Work’.
Work’ is done when a force acts on an object to cause it to move,
change shape, displace, or do something physical.
For, example, if I push a door open for my pet dog to walk in, work is
done on the door (by causing it to open). But what kind of force caused
the door to open? Here is where Mechanical Energy comes in.
Mechanical energy is the sum of kinetic and potential energy in an object that is used to do work. In other words, it is energy in an object due
to its motion or position, or both. In the ‘open door’ example above, I
possess potential chemical energy (energy stored in me), and by lifting
my hands to push the door, my action also had kinetic energy (energy in
the motion of my hands). By pushing the door, my potential and kinetic
energy was transferred into mechanical energy, which caused work to
be done (door opened). Here, the door gained mechanical energy, which
caused the door to be displaced temporarily. Note that for work to be
done, an object has to supply a force for another object to be displaced.
Here is another example of a boy with an iron hammer and nail. In the
illustration below…
48
Experimental Physics I
(1)
(2)
(3)
The iron hammer on its own has no kinetic energy, but it has
some potential energy (because of its weight).
To drive a nail into the piece of wood (which is work), he has
to lift the iron hammer up, (this increases its potential energy
because if it is high position).
And force it to move at great speed downwards (now has
kinetic energy) to hit the nail.
The sum of the potential and kinetic energy that the hammer
acquired to drive in the nail is called the Mechanical energy, which
resulted in the work done.
1.5.6 Power
The quantity work has to do with a force causing a displacement.
Work has nothing to do with the amount of time that this force acts to
cause the displacement. Sometimes, the work is done very quickly and
other times the work is done rather slowly.
Power is the rate at which work is done. It is the work/time ratio.
Mathematically, it is computed using the following equation.
Power = Work / time
or
P=W/t
The standard metric unit of power is the Watt. As is implied by
the equation for power, a unit of power is equivalent to a unit of work
divided by a unit of time. Thus, a Watt is equivalent to a Joule/second.
For historical reasons, the horsepower is occasionally used to describe
the power delivered by a machine. One horsepower is equivalent to
approximately 750 Watts.
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49
Most machines are designed and built to do work on objects. All
machines are typically described by a power rating. The power rating
indicates the rate at which that machine can do work upon other objects.
Thus, the power of a machine is the work/time ratio for that particular
machine. A car engine is an example of a machine that is given a power
rating. The power rating relates to how rapidly the car can accelerate the
car. Suppose that a 40-horsepower engine could accelerate the car from 0
mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four
times the horsepower could do the same amount of work in one-fourth
the time. That is, a 160-horsepower engine could accelerate the same
car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same
amount of work, power and time are inversely proportional. The power
equation suggests that a more powerful engine can do the same amount
of work in less time.
A person is also a machine that has a power rating. Some people
are more power-full than others. That is, some people are capable of
doing the same amount of work in less time or more work in the same
amount of time. A common physics lab involves quickly climbing a
flight of stairs and using mass, height and time information to determine
a student’s personal power. Despite the diagonal motion along the
staircase, it is often assumed that the horizontal motion is constant and
all the force from the steps is used to elevate the student upward at a
constant speed. Thus, the weight of the student is equal to the force
that does the work on the student and the height of the staircase is the
upward displacement. Suppose that Ben Pumpiniron elevates his 80kg body up the 2.0-meter stairwell in 1.8 seconds. If this were the case,
then we could calculate Ben’ spower rating. It can be assumed that Ben
must apply an 800-Newton downward force upon the stairs to elevate
his body. By so doing, the stairs would push upward on Ben’s body with
just enough force to lift his body up the stairs. It can also be assumed that
the angle between the force of the stairs on Ben and Ben’s displacement
is 0 degrees. With these two approximations, Ben’s power rating could
be determined as shown below.
Work 784 N2.0 m
=
Time 1.8 seconds
Power = 871 Watts
Power
=
Ben’s power rating is 871 Watts. He is quite a horse.
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Experimental Physics I
Another Formula for Power
The expression for power is work/time. And since the expression
for work is force*displacement, the expression for power can be
rewritten as (force*displacement)/time. Since the expression for velocity
is displacement/time, the expression for power can be rewritten once
more as force*velocity. This is shown below.
Work Force  Displcement
=
Time
Time
Displcement
Power = Force 
Time
Power = Force  Velocity
Power
=
This new equation for power reveals that a powerful machine is
both strong (big force) and fast (big velocity). A powerful car engine
is strong and fast. A powerful piece of farm equipment is strong and
fast. A powerful weightlifter is strong and fast. A powerful lineman on a
football team is strong and fast. A machine that is strong enough to apply
a big force to cause a displacement in a small amount of time (i.e., a big
velocity) is a powerful machine.
1.5.7 Conservative and Nonconservative Forces
It is important to know the difference between conservative and
nonconservative forces. The work a conservative force does on an object
is path-independent; the actual path taken by the object makes no
difference. Fifty meters up in the air has the same gravitational potential
energy whether you get there by taking the steps or by hopping on a
Ferris wheel. That is different from the force of friction, which dissipates
kinetic energy as heat. When friction is involved, the path you take
matters — a longer path will dissipate more kinetic energy than a short
one. For that reason, friction is a nonconservative force.
For example, suppose you and some buddies arrive at Mt. Newton,
a majestic peak that rises h meters into the air. You can take two ways up
the quick way or the scenic route. Your friends drive up the quick route,
and you drive up the scenic way, taking time out to have a picnic and
to solve a few physics problems. They greet you at the top by saying,
“Guess what — our potential energy compared to before is mgh greater.”
“Mine, too,” you say, looking out over the view. You pull out this
equation:
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51
∆PE
= mg(h f − h i )
This equation basically states that the actual path you take when
going vertically from hi to hf does not matter. All that matters is your
beginning height compared to your ending height. Because the
path taken by the object against gravity does not matter, gravity is a
conservative force.
Here is another way of looking at conservative and nonconservative
forces. Say you are vacationing in the Alps and your hotel is at the top of
Mt. Newton. You spend the whole day driving around — down to a lake
one minute, to the top of a higher peak the next. At the end of the day,
you end up back at the same location: your hotel on top of Mt. Newton.
What is the change in your gravitational potential energy? In other
words, how much network did gravity perform on you during the
day? Gravity is a conservative force, so the change in your gravitational
potential energy is 0. Because you have experienced no net change in
your gravitational potential energy, gravity did no network on you
during the day.
The road exerted a normal force on your car as you drove around,
but that force was always perpendicular to the road (meaning no force
parallel to your motion), so it did not do any work, either.
Conservative forces are easier to work with in physics because they
do not “leak” energy as you move around a path — if you end up in the
same place, you have the same amount of energy. If you have to deal
with nonconservative forces such as friction, including air friction, the
situation is different. If you are dragging something over a field carpeted
with sandpaper, for example, the force of friction does different amounts
of work on you depending on your path. A path that is twice as long will
involve twice as much work to overcome friction.
What is really not being conserved around a track with friction is
the total potential and kinetic energy, which taken together is mechanical
energy. When friction is involved, the loss in mechanical energy goes into
heat energy. You can say that the total amount of energy does not change
if you include that heat energy. However, the heat energy dissipates into
the environment quickly, so it is not recoverable or convertible. For that
and other reasons, physicists often work in terms of mechanical energy.
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Experimental Physics I
Gravity
Gravity is the most common conservative force, and to demonstrate
that it is conservative is relatively simple. Consider first a ball thrown up
into the air. On the ball’s trip upward, gravity works against the motion
of the ball, producing a total work of - mgh. This negative work causes
the ball to slow down until it stops, reverses direction and begins to
fall. During its fall, the force of gravity is in the same direction as the
motion of the ball, and the gravitational force does positive work of
magnitude mgh , accelerating ball until it reaches the ground with the
same speed with which it left. What is the net work done by gravity on
the ball over this closed loop? Zero, as we expect by our first principle
of conservative forces.
What about our second principle? Let us construct two alternative
paths for a ball being thrown up into the air:
Friction
Friction is the most common nonconservative force, and we will
demonstrate why it is not conservative. Consider a crate on a rough
floor, of weight W. The crate is pushed from one end of the floor to the
other, a distance of h meters, and then back to its original spot. What
is the net work done on the crate? At all times the friction opposes the
motion of the crate, exerting a force of μk W at all times. Thus the total
work done over the trip is simply (- 2) (μk W) (h) = - 2hwμk , clearly not
equal to zero. The network by friction over a closed path is not zero, and
it is nonconservative.
Is friction path independent? We expect not, because we know it is
nonconservative. To prove the suspicion, simply consider two possible
ways to move a crate between two points on a rough floor. One is a
straight line, one is a somewhat longer route. No matter the path, the
force is the same at all times that the crate is moving. The difference,
however, is that friction acts over a longer distance in the case of the
second path, causing a greater network to be done. Thus friction is not
path independent, and we confirm that it is nonconservative.
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53
1.6 COLLISION
Collision means two objects coming into contact with each other
for a very short period. In other words, collision is a reciprocative
interaction between two masses for a very short interval wherein the
momentum and energy of the colliding masses changes. While playing
carroms, you might have noticed the effect of a striker on coins when
they both collide.
Collision involves two masses m1 and m2. The v1i is the speed
of particle m1, where the subscript ‘i’ implies initial. The particle with
mass m2 is at rest. In this case, the object with mass m1 collides with the
stationary object of mass m2.
As a result of this collision the masses m1 and m2 move in different
directions.
1.6.1 Elastic and Inelastic Collisions
In an elastic collision, the objects separate after impact and don’t lose
any of their kinetic energy. Kinetic energy is the energy of motion and
is covered in detail elsewhere. The law of conservation of momentum is
very useful here, and it can be used whenever the net external force on a
system is zero. Figure 14 shows an elastic collision where momentum is
conserved.
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Experimental Physics I
Figure 14. The diagram shows a one-dimensional elastic collision between
two objects.
Perfectly elastic collisions can happen only with subatomic particles.
Everyday observable examples of perfectly elastic collisions don’t
exist—some kinetic energy is always lost, as it is converted into heat
transfer due to friction. However, collisions between everyday objects
are almost perfectly elastic when they occur with objects and surfaces
that are nearly frictionless, such as with two steel blocks on ice.
Now, to solve problems involving one-dimensional elastic collisions
between two objects, we can use the equation for conservation of
momentum. First, the equation for conservation of momentum for two
objects in a one-dimensional collision is
p1 + p2 = p′1 + p′ 2 (Fnet = 0).
Substituting the definition of momentum p = mv for each initial and
final momentum, we get
m1 v1 + m2 v 2 = m1 v′1 + m2 v′ 2 ,
where the primes (‘) indicate values after the collision; In some texts,
you may see i for initial (before collision) and f for final (after collision).
The equation assumes that the mass of each object does not change
during the collision.
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55
Now, let us turn to the second type of collision. An inelastic
collision is one in which objects stick together after impact, and kinetic
energy is not conserved. This lack of conservation means that the forces
between colliding objects may convert kinetic energy to other forms
of energy, such as potential energy or thermal energy. The concepts of
energy are discussed more thoroughly elsewhere. For inelastic collisions,
kinetic energy may be lost in the form of heat. Figure 15 shows an
example of an inelastic collision. Two objects that have equal masses
head toward each other at equal speeds and then stick together. The two
objects come to rest after sticking together, conserving momentum but
not kinetic energy after they collide. Some of the energy of motion gets
converted to thermal energy, or heat.
Figure 15. A one-dimensional inelastic collision between two objects. Momentum is conserved, but kinetic energy is not conserved. (a) Two objects
of equal mass initially head directly toward each other at the same speed.
(b) The objects stick together, creating a perfectly inelastic collision. In the
case shown in this figure, the combined objects stop; This is not true for all
inelastic collisions.
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Experimental Physics I
Since the two objects stick together after colliding, they move
together at the same speed. This lets us simplify the conservation of
momentum equation from
m1 v1 + m2 v 2 = m1 v′1 + m2 v′ 2
to
m1 v1 + m2 v 2 =( m1 + m2 ) v′
for inelastic collisions, where v′ is the final velocity for both objects
as they are stuck together, either in motion or at rest.
Solving Collision Problems
In one-dimensional collisions, the incoming and outgoing velocities
are all along the same line. But what about collisions, such as those
between billiard balls, in which objects scatter to the side? These are twodimensional collisions, and just as we did with two-dimensional forces,
we will solve these problems by first choosing a coordinate system and
separating the motion into its x and y components.
One complication with two-dimensional collisions is that the objects
might rotate before or after their collision. For example, if two ice skaters
hook arms as they pass each other, they will spin in circles. We will not
consider such rotation until later, and so for now, we arrange things
so that no rotation is possible. To avoid rotation, we consider only the
scattering of point masses—that is, structureless particles that cannot
rotate or spin.
We start by assuming that Fnet = 0, so that momentum p is conserved.
The simplest collision is one in which one of the particles is initially at
rest. The best choice for a coordinate system is one with an axis parallel
to the velocity of the incoming particle, as shown in Figure 16. Because
momentum is conserved, the components of momentum along the xand y-axes, displayed as px and py, will also be conserved. With the
chosen coordinate system, py is initially zero and px is the momentum of
the incoming particle.
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57
Figure 16. A two-dimensional collision with the coordinate system chosen
so that m2 is initially at rest and v1 is parallel to the x-axis.
Now, we will take the conservation of momentum
equation, p1 + p2 = p′1 + p′2 and break it into its x and y components.
Along the x-axis, the equation for conservation of momentum is
p1x + p2x = p′1x + p′ 2x .
In terms of masses and velocities, this equation is
m1 v 1 x + m2 v 2 x = m1 v ′1 x + m2 v ′ 2 x .
But because particle 2 is initially at rest, this equation becomes
m
=
v
m1 v ′1 x + m2 v ′ 2 x .
1 1x
The components of the velocities along the x-axis have the
form v cos θ. Because particle 1 initially moves along the x-axis, we
find v1x = v1. Conservation of momentum along the x-axis gives the
equation
where θ1 and θ2 are as shown in Figure 16.
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Experimental Physics I
Along the y-axis, the equation for conservation of momentum is
p1 y + p2 y = p′1 y + p′ 2 y ,
or
m1 v 1 y + m2 v 2 y = m1 v ′1 y + m2 v ′ 2 y .
But v1y is zero, because particle 1 initially moves along the x-axis.
Because particle 2 is initially at rest, v2y is also zero. The equation for
conservation of momentum along the y-axis becomes
=
0 m1 v ′1 y + m2 v ′ 2 y.
The components of the velocities along the y-axis have the
form v sin θ. Therefore, conservation of momentum along the y-axis
gives the following equation:
1.6.2 Conservation of Momentum
It is important we realize that momentum is conserved during
collisions, explosions, and other events involving objects in motion. To
say that a quantity is conserved means that it is constant throughout the
event. In the case of conservation of momentum, the total momentum in
the system remains the same before and after the collision.
where forces acting on the objects produced large changes in
momentum. Why is this? The systems of interest considered in those
problems were not inclusive enough. If the systems were expanded
to include more objects, then momentum would in fact be conserved
in those sample problems. It is always possible to find a larger system
where momentum is conserved, even though momentum changes for
individual objects within the system.
For example, if a football player runs into the goalpost in the end
zone, a force will cause him to bounce backward. His momentum is
obviously greatly changed, and considering only the football player, we
would find that momentum is not conserved. However, the system can
be expanded to contain the entire Earth. Surprisingly, Earth also recoils—
conserving momentum—because of the force applied to it through the
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59
goalpost. The effect on Earth is not noticeable because it is so much more
massive than the player, but the effect is real.
Next, consider what happens if the masses of two colliding objects
are more similar than the masses of a football player and Earth—in the
example shown in Figure 17 of one car bumping into another. Both
cars are coasting in the same direction when the lead car, labeled m2, is
bumped by the trailing car, labeled m1. The only unbalanced force on each
car is the force of the collision, assuming that the effects due to friction
are negligible. Car m1 slows down as a result of the collision, losing
some momentum, while car m2 speeds up and gains some momentum.
If we choose the system to include both cars and assume that friction is
negligible, then the momentum of the two-car system should remain
constant. Now we will prove that the total momentum of the two-car
system does in fact remain constant, and is therefore conserved.
Figure 17. Car of mass m1 moving with a velocity of v1 bumps into another
car of mass m2 and velocity v2. As a result, the first car slows down to a velocity of v′1 and the second speeds up to a velocity of v′2. The momentum of
each car is changed, but the total momentum ptot of the two cars is the same
before and after the collision if you assume friction is negligible.
Using the impulse-momentum theorem, the change in momentum
of car 1 is given by
Äp1 = F1Ät ,
where F1 is the force on car 1 due to car 2, and Δt is the time the force
acts, or the duration of the collision.
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Experimental Physics I
Similarly, the change in momentum of car 2 is Δp2=F2Δt where F2 is
the force on car 2 due to car 1, and we assume the duration of the
collision Δt is the same for both cars. We know from Newton’s third law
of motion that F2 = –F1, and so Δp2=−F1Δt=−Δp1.
Therefore, the changes in momentum are equal and opposite,
and Δp1+Δp2=0.
Because the changes in momentum add to zero, the total momentum
of the two-car system is constant. That is,
p1+p2=constant
p1+p2=p′1+p′2,
where p′1 and p′2 are the momenta of cars 1 and 2 after the collision.
This result that momentum is conserved is true not only for this
example involving the two cars, but for any system where the net
external force is zero, which is known as an isolated system. The law of
conservation of momentum states that for an isolated system with any
number of objects in it, the total momentum is conserved. In equation
form, the law of conservation of momentum for an isolated system is
written as
ptot=constant
or
ptot = p′tot ,
where ptot is the total momentum, or the sum of the momenta of
the individual objects in the system at a given time, and p′tot is the total
momentum some time later.
The conservation of momentum principle can be applied to systems
as diverse as a comet striking the Earth or a gas containing huge numbers
of atoms and molecules. Conservation of momentum appears to be
violated only when the net external force is not zero. But another larger
system can always be considered in which momentum is conserved by
simply including the source of the external force. For example, in the
collision of two cars considered above, the two-car system conserves
momentum while each one-car system does not.
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61
1.7 PROJECTILE MOTION
Projectile motion is the motion of an object thrown or projected into
the air, subject to only the acceleration of gravity. The object is called
a projectile, and its path is called its trajectory. The motion of falling
objects, as covered in Problem-Solving Basics for One-Dimensional
Kinematics, is a simple one-dimensional type of projectile motion in
which there is no horizontal movement. We consider two-dimensional
projectile motion, such as that of a football or other object for which air
resistance is negligible.
The most important fact to remember here is that motions along
perpendicular axes are independent and thus can be analyzed separately.
Where vertical and horizontal motions were seen to be independent.
The key to analyzing two-dimensional projectile motion is to break it
into two motions, one along the horizontal
axis and the other along the vertical. (This
choice of axes is the most sensible, because
Hints
acceleration due to gravity is vertical—
thus, there will be no acceleration along
We used
the horizontal axis when air resistance is
the notation A to
negligible.) As is customary, we call the
represent a vechorizontal axis the x-axis and the vertical
tor with compoaxis the y-axis. Figure 18 illustrates the
nents Ax and Ay.
notation for displacement, where s is defined
If we continued
this format, we
to be the total displacement and x and y are
would call displaceits components along the horizontal and
ment s with comvertical axes, respectively. The magnitudes
ponents sx and sy.
of these vectors are s, x, and y.
However, to simpli-
fy the notation, we
To describe motion we must deal
will simply reprewith velocity and acceleration, as well as
sent the component
with displacement. We must find their
vectors as x and y.
components along the x– and y-axes, too. We
will assume all forces except gravity (such as
air resistance and friction, for example) are
negligible. The components of acceleration are then very simple: ay = –g =
–9.80 m/s2. (Note that this definition assumes that the upwards direction
is defined as the positive direction. If you arrange the coordinate system
instead such that the downwards direction is positive, then acceleration
due to gravity takes a positive value.) Because gravity is vertical, ax=0.
Both accelerations are constant, so the kinematic equations can be used.
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Experimental Physics I
Figure 18. The total displacement s of a soccer ball at a point along its path.
The vector s has components x and y along the horizontal and vertical axes.
Its magnitude is s, and it makes an angle θ with the horizontal.
Given these assumptions, the following steps are then used to
analyze projectile motion:
Step 1. Resolve or break the motion into horizontal and vertical
components along the x- and y-axes. These axes are perpendicular, so Ax =
A cos θ and Ay = A sin θ are used. The magnitude of the components
of displacement s along these axes are x and y. The magnitudes of the
components of the velocity v are Vx = V cos θ and Vy = v sin θ where v is
the magnitude of the velocity and θ is its direction, as shown in 2. Initial
values are denoted with a subscript 0, as usual.
Step 2.Treat the motion as two independent one-dimensional
motions, one horizontal and the other vertical. The kinematic equations
for horizontal and vertical motion take the following forms:
Step 3. Solve for the unknowns in the two separate motions—
one horizontal and one vertical. Note that the only common variable
between the motions is time t. The problem solving procedures here are
the same as for one-dimensional kinematics and are illustrated in the
solved examples below.
Step 4. Recombine the two motions to find the total displacement s and
velocity v. Because the x – and y -motions are perpendicular, we
determine these vectors by using the techniques outlined in the Vector
Addition and Subtraction: Analytical Methods and employing
=
A
Ax 2 + Ay 2
Mechanics
63
and θ = tan−1 (Ay/Ax) in the following form, where θ is the direction
of the displacement s and θv is the direction of the velocity v:
Figure 19. (a) We analyze two-dimensional projectile motion by breaking it
into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because ax=0 and vx is thus
constant. (c) The velocity in the vertical direction begins to decrease as the
object rises; at its highest point, the vertical velocity is zero. As the object
falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d)
The x – and y -motions are recombined to give the total velocity at any given
point on the trajectory.
In solving part (a) of the preceding example, the expression we
found for y is valid for any projectile motion where air resistance is
negligible. Call the maximum height y=h; then,
h=
v0 y 2
2g
64
Experimental Physics I
This equation defines the maximum height of a projectile and depends
only on the vertical component of the initial velocity.
One of the most important things illustrated by projectile motion is
that vertical and horizontal motions are independent of each other. Galileo
was the first person to fully comprehend this characteristic. He used it
to predict the range of a projectile. On level ground, we define range to
be the horizontal distance R traveled by a projectile. Galileo and many
others were interested in the range of projectiles primarily for military
purposes—such as aiming cannons. However, investigating the range
of projectiles can shed light on other interesting phenomena, such as
the orbits of satellites around the Earth. Let us consider projectile range
further.
Figure 20. Trajectories of projectiles on level ground. (a) The greater the initial speed v0, the greater the range for a given initial angle. (b) The effect of
initial angle θ0 on the range of a projectile with a given initial speed. Note
that the range is the same for 15º and 75º, although the maximum heights of
those paths are different.
How does the initial velocity of a projectile affect its range?
Obviously, the greater the initial speed v0, the greater the range, as
shown in Figure 20(a). The initial angle θ0 also has a dramatic effect on
the range, as illustrated in Figure 20(b). For a fixed initial speed, such
as might be produced by a cannon, the maximum range is obtained
with θ0 = 45º. This is true only for conditions neglecting air resistance.
Mechanics
65
If air resistance is considered, the maximum angle is approximately
38º. Interestingly, for every initial angle except 45º, there are two angles
that give the same range—the sum of those angles is 90º. The range
also depends on the value of the acceleration of gravity g. The lunar
astronaut Alan Shepherd was able to drive a golf ball a great distance
on the Moon because gravity is weaker there. The range R of a projectile
on level ground for which air resistance is negligible is given by
R=
v 0 2 sin2θ0
g
where v0 is the initial speed and θ0 is the initial angle relative to the
horizontal. When we speak of the range of a projectile on level ground,
we assume that R is very small compared with the circumference of the
Earth. If, however, the range is large, the Earth curves away below the
projectile and acceleration of gravity changes direction along the path.
The range is larger than predicted by the range equation given above
because the projectile has farther to fall than it would on level ground.
(See Figure 21.) If the initial speed is great enough, the projectile goes
into orbit. This is called escape velocity. This possibility was recognized
centuries before it could be accomplished. When an object is in orbit, the
Earth curves away from underneath the object at the same rate as it falls.
The object thus falls continuously but never hits the surface. In Addition
of Velocities, we will examine the addition of velocities, which is another
important aspect of two-dimensional kinematics and will also yield
insights beyond the immediate topic.
Figure 21. Projectile to satellite. In each case shown here, a projectile is
launched from a very high tower to avoid air resistance. With increasing
initial speed, the range increases and becomes longer than it would be on
level ground because the Earth curves away underneath its path. With a
large enough initial speed, orbit is achieved.
66
Experimental Physics I
1.8 BALLISTIC PENDULUM
A ballistic pendulum is a device for measuring a bullet’s momentum,
from which it is possible to calculate the velocity and kinetic energy.
Ballistic pendulums have been largely rendered obsolete by modern
chronographs, which allow direct measurement of the projectile velocity.
It can be used to measure any transfer of momentum. For example, a
ballistic pendulum was used by physicist C. V. Boys to measure the
elasticity of golf balls and by physicist Peter Guthrie Tait to measure the
effect that spin had on the distance a golf ball traveled.
In a perfectly inelastic collision, a bullet is fired into the stationary
pendulum, which captures the bullet and absorbs its energy. The
stationary pendulum now moves with a new velocity just after the
collision. While not all of the energy from the bullet is transformed into
kinetic energy for the pendulum (some is used as heat and deformation
energy) , the momentum of the system is conserved. By measuring the
height of the pendulum’s swing, the potential energy of the pendulum
when it stops can be measured. In the case of a pendulum total mechanical
energy is conserved. So kinetic energy of the pendulum (after firing) is
fully converted to potential energy. Thus the pendulum’s initial velocity
can be calculated. Using the law of conservation of momentum, the
velocity of the bullet can be computed.
From the law of conservation of mechanical energy of the pendulum;
1
(m + M)V 2 =
(m + M)gh
2
where,
m- Mass of bullet
M- Mass of Pendulum
h- Maximum height reached by the pendulum
v- Velocity of the bullet
V- Velocity of pendulum
g-Gravity of earth
According to the law of conservation of momentum;
Mechanics
67
mv
= (M + m)V
mv
= (M + m) (2gh)
Velocity of the bullet is given by;
1
(M + m) (2gh)
m
M
=
v ( + m) (2gh)
m
=
v
1.9 CENTRIPETAL FORCE
A centripetal force is a force that makes a body follow a curved path.
Its direction is always orthogonal to the motion of the body and towards
the fixed point of the instantaneous center of curvature of the path. Isaac
Newton described it as «a force by which bodies are drawn or impelled, or
in any way tend, towards a point as to a centre”. In Newtonian mechanics,
gravity provides the centripetal force causing astronomical orbits.
Formula
The magnitude of the centripetal force on an object of mass m moving
at tangential speed v along a path with radius of curvature r is:
=
Fc ma
=
c
a c = lim
∆t → 0
mv 2
r
| ∆v |
∆t
68
Experimental Physics I
Where ac is the centripetal acceleration and ∆v
is
the difference between the velocity vectors. Since the velocity vectors
in the above diagram have constant magnitude and since each one is
perpendicular to its respective position vector, simple vector subtraction
implies two similar isosceles triangles with congruent angles – one
comprising a base of ∆v and a leg length of v, and the other a base of ∆r
(position vector difference) and a leg length of r.
| ∆v | | ∆r |
=
v
r
∆v |=
v
| ∆r |
r
v
| ∆r |.
r
| ∆v | v
| ∆r |
| ∆r |
v2
a c = lim
= lim
= ω lim
= vω =
∆t → 0 ∆t
∆t → 0 ∆t
r ∆t →0 ∆t
r
Therefore, || ∆v | can be substituted with
The direction of the force is toward the center of the circle in which
the object is moving, or the osculating circle (the circle that best fits
the local path of the object, if the path is not circular). The speed in the
formula is squared, so twice the speed needs four times the force. The
inverse relationship with the radius of curvature shows that half the
radial distance requires twice the force. This force is also sometimes
written in terms of the angular velocity ω of the object about the center
of the circle, related to the tangential velocity by the formula
v = ωr
so that
F=
mrω2 .
c
Expressed using the orbital period T for one revolution of the circle,
ω=
2π
T
the equation becomes
Mechanics
69
2
 2π 
Fc = mr   .
 T 
In particle accelerators, velocity can be very high (close to the speed
of light in vacuum) so the same rest mass now exerts greater inertia
(relativistic mass) thereby requiring greater force for the same centripetal
acceleration, so the equation becomes:
Fc =
γmv 2
r
where
γ=
1
1−
v2
c2
is the Lorentz factor.
Thus the centripetal force is given by:
Fc =
γmvω
which is the rate of change of relativistic momentum γmv.
EXERCISE
Answer the following questions:
1.
Certain force acting on a 20 kg mass changes its velocity from
5 m s–1 to 2 m s–1. Calculate the work done by the force.
2.
The potential energy of a freely falling object decreases
progressively. Does this violate the law of conservation of
energy? Why?
3.
What is the volume of 35.7 g of water?
4.
What is a Collision? Explain the types of collision.
5.
What is projectile motion?
6.
Discuss about the centripetal force
7.
An object of mass, m is moving with a constant velocity, v.
How much work should be done on the object in order to
bring the object to rest?
70
Experimental Physics I
8.
9.
10.
Define centripetal force. Can any type of force (for example,
tension, gravitational force, friction, and so on) be a centripetal
force? Can any combination of forces be a centripetal force?
Calculate the centripetal force on the end of a 100 m (radius)
wind turbine blade that is rotating at 0.5 rev/s. Assume the
mass is 4 kg.
Give an estimate of the accuracy with which you can measure
volumes with your 50 mL graduated cylinder.
MULTIPLE CHOICE QUESTIONS
Tick the correct answer.
1. What is termed as the quantity of matter contained in a body?
a.
Density
b.
Volume
c.
Mass
d.
Specific gravity
2. Which is called mass per unit volume of a substances?
a.
Mass
b.
Weight
c.
Volume
d.
Density
3. An iron sphere of mass 10 kg has the same diameter as an
aluminium sphere of mass is 3.5 kg. Both spheres are dropped
simultaneously from a tower. When they are lo m above the
ground, they have the same.
a.
acceleration
b.
momenta
c.
potential energy
d.
kinetic energy
4. The work done on an object does not depend upon the
a.
displacement
b.
force applied
c.
angle between force and displacement
d.
initial velocity of the object
Mechanics
5.
71
If speed of a car becomes 2 times, its kinetic energy becomes
a.
4 times
b.
8 times
c.
16 times
d.
12 times
6. Work done by friction
a.
increases kinetic energy of body
b.
decreases kinetic energy of body
c.
increases potential energy of body
d.
decreases potential energy of body.
7. Which of the following is a type of motion?
a.
Circular
b.
Rectilinear
c.
Periodic
d.
All the above
8. A body of mass m, projected at an angle of θ from the ground with
an initial velocity of v, acceleration due to gravity is g, what is the
maximum horizontal range covered?
a.
R = v2 (sin 2θ)/g
b.
R = v2 (sin θ)/2g
c.
R = v2 (sin 2θ)/2g
d.
R = v2 (sin θ)/g
9. On calculating which of the following quantities, the mass of the
body has an effect in simple projectile motion?
a.
Velocity
b.
Force
c.
Time of flight
d.
Range
10. When do we get maximum range in a simple projectile motion?
a.
When θ = 45°
b.
When θ = 60°
c.
When θ = 90°
d.
When θ = 0°
72
Experimental Physics I
ANSWER
1. (c)
2. (d)
3. (a)
4. (d)
5. (a)
6. (b)
7. (a)
8. (a)
9. (b)
10. (a)
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Carroll, Sean M., “From Eternity to Here”, Penguin Group, 2010
Eastlake, Charles N., “An Aerodynamicist’s View of Lift, Bernoulli,
and Newton”, The Physics Teacher 40, 166 (March 2002).
Gonzalez, Guillermo and Richards, Jay W., The Privileged Planet,
Regnery Publishing, 2004.
Klarreich, Erica, “Navigating Celestial Currents”, Science News
167, 250, April 16, 2005.
Sears, Zemansky,Young and Freedman, University Physics, 10th
Ed., Addison-Wesley, 2000
von Arx, William S., An Introduction to Physical Oceanography,
Addison-Wesley, 1962.
Watts, Robert G. and Ferrer, Ricardo, The lateral force on a spinning
sphere: Aerodynamics of a curveball, American Journal of Physics
55, 40, Jan 1987.
Weinberg, S. (May 1, 2005). The Quantum Theory of Fields, Volume
1: Foundations (1st ed.). Cambridge University Press. p. xxi.
Young, H.D. and Freedman, R. A., University Physics, 11th Ed.,
Pearson, 2004.
Heat
39
CHAPTER 2
HEAT
OBJECTIVES
After reading this chapter, you should be able to:
•
Define thermal linear expansion
•
Explain calorimetry and the specific heat of a metal
•
Discuss about heat of fusion of ice
•
Elaborate heat of vaporization of water
Heat, energy that is transferred from one body to another as the result
of a difference in temperature. If two bodies at different temperatures
are brought together, energy is transferred—i.e., heat flows—from the
hotter body to the colder. The effect of this transfer of energy usually, but
not always, is an increase in the temperature of the colder body and a
decrease in the temperature of the hotter body. A substance may absorb
heat without an increase in temperature by changing from one physical
state (or phase) to another, as from a solid to a liquid (melting), from a
solid to a vapour (sublimation), from a liquid to a vapour (boiling), or
40
Experimental Physics I
from one solid form to another (usually called a crystalline transition).
The important distinction between heat and temperature (heat being a
form of energy and temperature a measure of the amount of that energy
present in a body) was clarified during the 18th and 19th centuries.
Heat is one of the essential energy forms on Earth for the survival
of different lives. Heat transfer takes place from one body to the other
because of the difference in temperature according to thermodynamics.
We use heat energy in our day to day activities such as cooking,
transportation, ironing, recreation and much more. Heat energy also
plays a crucial role when it comes to nature. The occurrence of rain,
wind, change in the seasons, etc. is all dependent on the gradient that is
created because of the uneven heating of various regions. In this article,
we will discuss what is meant by heat, what is Latent heat and what is
used to measure heat.
2.1 THERMAL LINEAR EXPANSION
Thermal linear expansion is the process by which solid objects
expand in length as a result of a transfer of energy into that object due
to heat. When heat flows into a solid object, the individual molecules
and atoms composing that solid begin to vibrate with greater energies
thereby increasing the distance between neighboring atoms. This
expansion between atoms and molecules collectively increases the length
of any long solid object, such as a metal rod. The length that the solid
object elongates upon heating or compresses upon cooling is dependent
upon three factors. It depends on the original length, which means that
a solid object with a longer length will elongate a greater distance. It
depends on the change in temperature, which basically means that
objects will expand more for greater increases in temperatures. Finally,
the expansion also depends on the type of object we are using, which is
given by the coefficient of linear expansion.
2.1.1 Linear Expansion
Thermal expansion is the tendency of matter to change in volume
in response to a change in temperature. (An example of this is the
buckling of railroad track, as seen in. ) Atoms and molecules in a solid,
for instance, constantly oscillate around its equilibrium point. This kind
of excitation is called thermal motion. When a substance is heated, its
constituent particles begin moving more, thus maintaining a greater
average separation with their neighboring particles. The degree of
Heat
41
expansion divided by the change in temperature is called the material’s
coefficient of thermal expansion; it generally varies with temperature.
Figure 1: Thermal expansion of long continuous sections of rail tracks is the
driving force for rail buckling.
Expansion, Not Contraction
Why does matter usually expand when heated? The answer
can be found in the shape of the typical particle-particle potential in
matter. Particles in solids and liquids constantly feel the presence
of other neighboring particles. This interaction can be represented
mathematically as a potential curve. Fig 2 illustrates how this interparticle potential usually takes an asymmetric form rather than a
symmetric form, as a function of particle-particle distance. Note that the
potential curve is steeper for shorter distance. In the diagram, (b) shows
that as the substance is heated, the equilibrium (or average) particleparticle distance increases. Materials which contract or maintain their
shape with increasing temperature are rare. This effect is limited in size,
and only occurs within limited temperature ranges.
Linear Expansion
To a first approximation, the change in length measurements of
an object (linear dimension as opposed to, for example, volumetric
dimension) due to thermal expansion is related to temperature change
by a linear expansion coefficient. It is the fractional change in length per
degree of temperature change. Assuming negligible effect of pressure,
we may write:
42
Experimental Physics I
Figure 2: Typical inter-particle potential in condensed matter (such as solid
or liquid).
where L is a particular length measurement and dL/dT is the rate
of change of that linear dimension per unit change in temperature.
From the definition of the expansion coefficient, the change in the linear
dimension ΔL over a temperature range ΔT can be estimated to be:
This equation works well as long as the linear-expansion coefficient
does not change much over the change in temperature. If it does, the
equation must be integrated.
2.1.2 Area Expansion
Objects expand in all dimensions. That is, their areas and volumes,
as well as their lengths, increase with temperature.
We learned about the linear expansion (in one dimension) in the
previous Atom. Objects expand in all dimensions, and we can extend
the thermal expansion for 1D to two (or three) dimensions. That is, their
areas and volumes, as well as their lengths, increase with temperature.
Heat
43
Area thermal expansion coefficient
The area thermal expansion coefficient relates the change in a
material’s area dimensions to a change in temperature. It is the fractional
change in area per degree of temperature change. Ignoring pressure, we
may write:
, where is some area of interest on the object, and
dA/dT is the rate of change of that area per unit change in temperature.
The change in the linear dimension can be estimated as:
. This equation works well as long as the linear expansion coefficient
does not change much over the change in temperature ΔT. If it does, the
equation must be integrated.
Figure 3: In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown
with solid lines, and the expanded boundaries with dashed lines. (a) Area
increases because both length and width increase. The area of a circular
plug also increases. (b) If the plug is removed, the hole it leaves becomes
larger with increasing temperature, just as if the expanding plug were still
in place.
Relationship to linear thermal expansion coefficient
For isotropic materials, and for small expansions, the linear thermal
expansion coefficient is one half of the area coefficient. To derive the
relationship, let’s take a square of steel that has sides of length L. The
original area will be A = L2,and the new area, after a temperature increase,
44
Experimental Physics I
will be
The approximation holds for a sufficiently small ΔL campared to L.
Since
from the equation above (and from the definitions of
the thermal coefficients), we get
.
2.1.3 Volume Expansion
Substances expand or contract when their temperature changes,
with expansion or contraction occurring in all directions.
The volumetric thermal expansion coefficient is the most basic
thermal expansion coefficient. illustrates that, in general, substances
expand or contract when their temperature changes, with expansion or
contraction occurring in all directions. Such substances that expand in
all directions are called isotropic. For isotropic materials, the area and
linear coefficients may be calculated from the volumetric coefficient
(discussed below).
Figure 5: Volumetric Expansion: In general, objects expand in all directions
as temperature increases. In these drawings, the original boundaries of
the objects are shown with solid lines, and the expanded boundaries with
dashed lines. (a) Area increases because both length and width increase.
The area of a circular plug also increases. (b) If the plug is removed, the
hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. (c) Volume also increases, because all three
dimensions increase.
Heat
45
Mathematical definitions of these coefficients are defined below for
solids, liquids, and gasses:
The subscript p indicates that the pressure is held constant during
the expansion. In the case of a gas, the fact that the pressure is held
constant is important, as the volume of a gas will vary appreciably with
pressure as well as with temperature.
For a solid, we can ignore the effects of pressure on the material,
thus the volumetric thermal expansion coefficient can be written:
where V is the volume of the material, and is dV/dT the rate of
change of that volume with temperature. This means that the volume
of a material changes by some fixed fractional amount. For example, a
steel block with a volume of 1 cubic meter might expand to 1.002 cubic
meters when the temperature is raised by 50 °C. This is an expansion of
0.2%. The volumetric expansion coefficient would be 0.2% for 50 °C, or
0.004% per degree C.
Relationship to Linear Thermal Expansion Coefficient
For isotropic material, and for small expansions, the linear thermal
expansion coefficient is one third the volumetric coefficient. To derive
the relationship, let’s take a cube of steel that has sides of length L. The
original volume will be V = L3,and the new volume, after a temperature
increase, will be:
The approximation holds for a sufficiently small ΔLΔL compared
to L. Since:
46
Experimental Physics I
Important
(and from the definitions of the thermal
coefficients), we arrive at:
Special Properties of Water
Objects will expand with increasing
temperature, but water is the most important
exception to the general rule.
In the case
of a gas, expansion
depends on how the
pressure changed in
the process because
the volume of a gas
will vary appreciably with pressure as
well as temperature.
Special Properties of Water
In general, objects will expand with increasing temperature.
However, a number of materials contract on heating within certain
temperature ranges; this is usually called negative thermal expansion,
rather than “thermal contraction. ” Water is the most important exception
to the general rule. Water has this unique characteristic because of the
particular nature of the hydrogen bond in H2O.
Density of Water as Temperature Changes
At temperatures greater than 4ºC (40ºF) water expands with
increasing temperature (its density decreases). However, it expands
with decreasing temperature when it is between +4ºC and 0ºC (40ºF to
32ºF). Water is densest at +4ºC.
Figure 6: Water Density vs. Temperature: The density of water as a function
of temperature. Note that the thermal expansion is actually very small. The
maximum density at +4ºC is only 0.0075% greater than the density at 2ºC,
and 0.012% greater than that at 0ºC.
Heat
47
Perhaps the most striking effect of this phenomenon is the freezing
of water in a pond. When water near the surface cools down to 4ºC it is
denser than the remaining water and thus will sink to the bottom. This
“turnover” results in a layer of warmer water near the surface, which is
then cooled. Eventually the pond has a uniform temperature of 4ºC. If
the temperature in the surface layer drops below 4ºC, the water is less
dense than the water below, and thus stays near the top.
As a result, the pond surface can completely freeze over, while
the bottom may remain at 4ºC. The ice on top of liquid water provides
an insulating layer from winter’s harsh exterior air temperatures. Fish
and other aquatic life can survive in 4ºC water beneath ice, due to this
unusual characteristic of water. It also produces circulation of water in
the pond that is necessary for a healthy ecosystem of the body of water.
Figure 7: Temperature in a Lake: Temperature distribution in a lake on
warm and cold days in winter.
Ice Versus Water
The solid form of most substances is denser than the liquid phase;
thus, a block of most solids will sink in the liquid. However, a block of
ice floats in liquid water because ice is less dense. Upon freezing, the
density of water decreases by about 9%.
2.1.4 Thermal Expansion of Solids and Liquids
The expansion of alcohol in a thermometer is one of many commonly
encountered examples of thermal expansion, the change in size or volume
of a given mass with temperature. Hot air rises because its volume
increases, which causes the hot air’s density to be smaller than the
density of surrounding air, causing a buoyant (upward) force on the
48
Experimental Physics I
hot air. The same happens in all liquids and gases, driving natural heat
transfer upwards in homes, oceans, and weather systems. Solids also
undergo thermal expansion. Railroad tracks and bridges, for example,
have expansion joints to allow them to freely expand and contract with
temperature changes.
What are the basic properties of thermal expansion? First, thermal
expansion is clearly related to temperature change. The greater the
temperature change, the more a bimetallic strip will bend. Second, it
depends on the material.
In a thermometer, for example, the expansion of alcohol is much
greater than the expansion of the glass containing it.
An increase in temperature implies an increase in the kinetic
energy of the individual atoms. In a solid, unlike in a gas, the atoms or
molecules are closely packed together, but their kinetic energy (in the
form of small, rapid vibrations) pushes neighboring atoms or molecules
apart from each other.
This neighbor-to-neighbor pushing results in a slightly greater
distance, on average, between neighbors, and adds up to a larger size for
the whole body. For most substances under ordinary conditions, there is
no preferred direction, and an increase in temperature will increase the
solid’s size by a certain fraction in each dimension.
2.1.5 Linear Thermal Expansion—Thermal Expansion in
One Dimension
The change in length ΔL is proportional to length L. The dependence
of thermal expansion on temperature, substance, and length is
summarized in the equation ΔL = αLΔT,where ΔL is the change in
length L, ΔT is the change in temperature, and α is the coefficient of linear
expansion, which varies slightly with temperature.
Table 1 lists representative values of the coefficient of linear
expansion, which may have units of 1/ºC or 1/K. Because the size of a
kelvin and a degree Celsius are the same, both α and ΔT can be expressed
in units of kelvins or degrees Celsius. The equation ΔL = αLΔT is accurate
for small changes in temperature and can be used for large changes in
temperature if an average value of α is used.
Heat
49
Table 1: Thermal Expansion Coefficients at 20ºC
Material
Coefficient of linear Coefficient of volume expansion β(1/
expansion α(1/ºC)
ºC)
Solids
Aluminum
25 × 10– 6
75 × 10– 6
Brass
19 × 10– 6
56 × 10– 6
Copper
17 × 10– 6
51 × 10– 6
Gold
14 × 10–
6
42 × 10– 6
Iron or Steel
12 × 10– 6
35 × 10– 6
Invar (Nickel-iron alloy) 0.9 × 10– 6
2.7 × 10– 6
Lead
29 × 10– 6
87 × 10– 6
Silver
18 × 10– 6
54 × 10– 6
Glass (ordinary)
9 × 10– 6
27 × 10– 6
Glass (Pyrex®)
3 × 10–
9 × 10– 6
Quartz
0.4 × 10– 6
1 × 10– 6
Concrete, Brick
~12 × 10– 6
~36 × 10– 6
Marble (average)
2.5 × 10– 6
7.5 × 10– 6
6
Liquids
Ether
1650 × 10– 6
Ethyl alcohol
1100 × 10– 6
Petrol
950 × 10– 6
Glycerin
500 × 10– 6
Mercury
180 × 10– 6
Water
210 × 10– 6
Gases
Air and most other gases
at atmospheric pressure
3400 × 10– 6
Example. Calculating linear thermal expansion: the golden gate
bridge
The main span of San Francisco’s Golden Gate Bridge is 1275 m long
at its coldest. The bridge is exposed to temperatures ranging from –15ºC
to 40ºC. What is its change in length between these temperatures?
Assume that the bridge is made entirely of steel.
50
Experimental Physics I
Strategy
Use the equation for linear thermal expansion ΔL = αLΔT to calculate
the change in length , ΔL. Use the coefficient of linear expansion, α, for
steel from Table 1, and note that the change in temperature, ΔT, is 55ºC.
Solution
Plug all of the known values into the equation to solve for ΔL.
Although not large compared with the length of the bridge,
this change in length is observable. It is generally spread over many
expansion joints so that the expansion at each joint is small.
2.1.6 Thermal Expansion in Two and Three Dimensions
Objects expand in all dimensions, That is, their areas and volumes,
as well as their lengths, increase with temperature. Holes also get larger
with temperature. If you cut a hole in a metal plate, the remaining
material will expand exactly as it would if the plug was still in place.
The plug would get bigger, and so the hole must get bigger too. (Think
of the ring of neighboring atoms or molecules on the wall of the hole as
pushing each other farther apart as temperature increases. Obviously,
the ring of neighbors must get slightly larger, so the hole gets slightly
larger).
In general, objects will expand with increasing temperature.
Water is the most important exception to this rule. Water expands with
increasing temperature (its density decreases) when it is at temperatures
greater than 4ºC (40ºF). However, it expands with decreasing temperature
when it is between +4ºC and 0ºC (40ºF to 32ºF). Water is densest at +4ºC.
Perhaps the most striking effect of this phenomenon is the freezing of
water in a pond. When water near the surface cools down to 4ºC it is
denser than the remaining water and thus will sink to the bottom. This
“turnover” results in a layer of warmer water near the surface, which
is then cooled. Eventually the pond has a uniform temperature of 4ºC.
If the temperature in the surface layer drops below 4ºC, the water
is less dense than the water below, and thus stays near the top. As a
result, the pond surface can completely freeze over. The ice on top of
liquid water provides an insulating layer from winter’s harsh exterior
air temperatures. Fish and other aquatic life can survive in 4ºC water
Heat
51
beneath ice, due to this unusual characteristic of water. It also produces
circulation of water in the pond that is necessary for a healthy ecosystem
of the body of water.
2.1.7 Thermal Stress
Thermal stress is created by thermal expansion or contraction.
Thermal stress can be destructive, such as when expanding gasoline
ruptures a tank. It can also be useful, for example, when two parts are
joined together by heating one in manufacturing, then slipping it over the
other and allowing the combination to cool. Thermal stress can explain
many phenomena, such as the weathering of rocks and pavement by the
expansion of ice when it freezes.
Forces and pressures created by thermal stress are typically as great
as that in the example above. Railroad tracks and roadways can buckle
on hot days if they lack sufficient expansion joints. Power lines sag
more in the summer than in the winter, and will snap in cold weather
if there is insufficient slack. Cracks open and close in plaster walls
as a house warms and cools. Glass cooking pans will crack if cooled
rapidly or unevenly, because of differential contraction and the stresses
it creates. (Pyrex® is less susceptible because of its small coefficient of
thermal expansion.) Nuclear reactor pressure vessels are threatened by
overly rapid cooling, and although none have failed, several have been
cooled faster than considered desirable. Biological cells are ruptured
when foods are frozen, detracting from their taste. Repeated thawing
and freezing accentuate the damage. Even the oceans can be affected.
A significant portion of the rise in sea level that is resulting from global
warming is due to the thermal expansion of sea water.
Metal is regularly used in the human body for hip and knee
implants. Most implants need to be replaced over time because, among
other things, metal does not bond with bone. Researchers are trying to
find better metal coatings that would allow metal-to-bone bonding. One
challenge is to find a coating that has an expansion coefficient similar to
that of metal. If the expansion coefficients are too different, the thermal
stresses during the manufacturing process lead to cracks at the coatingmetal interface.
Another example of thermal stress is found in the mouth. Dental
fillings can expand differently from tooth enamel. It can give pain when
eating ice cream or having a hot drink. Cracks might occur in the filling.
52
Experimental Physics I
Metal fillings (gold, silver, etc.) are being replaced by composite fillings
(porcelain), which have smaller coefficients of expansion, and are closer
to those of teeth.
2.2 CALORIMETRY AND THE SPECIFIC HEAT
OF A METAL
A calorimeter is a device that is in use for measuring the warmth
of chemical reactions or physical changes also as heat capacity.
The most common types of calorimeters are differential scanning
calorimeters, titration calorimeters, isothermal micro calorimeters, and
accelerated rate calorimeters. A normal calorimeter usually consists of a
thermometer. This thermometer is again attached to a metal container
filled with water suspended above a combustion chamber. It is one
of the measurement devices useful in the study of thermodynamics,
chemistry, and biochemistry.
2.2.1 Procedure of Calorimeter
Now let us find the enthalpy change per mole of a substance A in
a reaction between two substances A and B. Here, both the substances
A and B, are separately added to a calorimeter and the initial and final
temperatures (before the reaction has started and after it’s finished) are
noted. Multiplying the natural process by the mass and heat capacities of
the substances gives worth for the energy given off or absorbed during
the reaction. Dividing the energy change by what percentage moles of A
were present gives its enthalpy change of the reaction.
q=Cv(Tf−Ti)q=Cv(Tf−Ti)
Heat
53
Where q is that the amount of warmth consistent with the change
in temperature measured in joules and Cv is that the heat capacity of
the calorimeter which is measured in units of energy per temperature
(Joules/Kelvin).
2.2.2 History
In 1761 Black introduced the thought of heat of transformation which
causes the creation of the primary ice-calorimeters. In the year 1780, a
French nobleman and chemist Lavoisier performed an experiment in
which he used the warmth from the guinea pig’s respiration to melt
snow surrounding his apparatus, showing that respiratory gas exchange
is the combustion, almost like a candle burning.
2.2.3 Types of Calorimeter
Adiabatic Calorimeters
An adiabatic calorimeter is a calorimeter which helps to examine
a runaway reaction. As a result of an adiabatic environment, any heat
generated by the fabric sample under test causes the sample to extend
in temperature, thus fueling the reaction. The adiabatic calorimeter is
actually a wrong term because it’s not fully adiabatic. Some amount of
heat is usually lost by the sample to the sample holder. A mathematical
correction factor, referred to as the phi-factor, are often wont to adjust
the calorimetric result to account for these heat losses.
Reaction Calorimeters
A reaction calorimeter may be a calorimeter during which a reaction
is initiated within a closed insulated container. Reaction heats are
measured and therefore the heat content is obtained by integrating heat
flow versus time. This is the quality utilized in industry to live heats since
industrial processes are engineered to run at constant temperatures.
There are four, primary types of methods for measuring the warmth
in reaction calorimeter:
1) Heat Flow Calorimeter – The cooling/heating jacket plays an
important role in controlling either the temperature of the
method or the temperature of the jacket. Heat is measured by
monitoring the temperature difference between heat transfer
fluid and therefore the process fluid.
54
Experimental Physics I
2)
3)
4)
Heat Balance Calorimeter – The cooling/heating jacket
controls the temperature of the method. Heat is measured by
monitoring the warmth gained or lost by the warmth transfer
fluid.
Power Compensation – Power compensation uses a
heater placed within the vessel to take care of a continuing
temperature. The energy supplied to the present heater is
often varied as reactions require and therefore the calorimetry
signal is only derived from this electric power.
Constant Flux Calorimeter – Constant flux calorimetry (or
COFLUX) springs from heat balance calorimetry and uses
specialized control mechanisms to take care of a continuing
heat flow (or flux) across the vessel wall.
Bomb Calorimeters
A bomb may be a sort of constant-volume calorimeter utilized in
measuring the warmth of combustion of a specific reaction. Electrical
energy is employed to ignite the fuel; because the fuel is burning, it’ll
heat up the encompassing air, which expands and escapes through a
tube that leads the air out of the calorimeter. When the air is escaping
through the copper tube it’ll also heat up the water outside the tube. The
temperature difference of the water allows for calculating the calorie
content of the fuel.
In brief, a bomb consists of a little cup to contain the sample,
oxygen, a chrome steel bomb, water, a stirrer, a thermometer, the dewar
or insulating container (to prevent heat be due to the calorimeter to the
surroundings) and ignition circuit connected to the bomb. By using
chrome steel for the bomb, the reaction will occur with no volume
change observed.
Calvet-type Calorimeters
The detection is predicated on a three-dimensional fluxmeter sensor.
The fluxmeter element consists of a hoop of several thermocouples
serial. The alternative thermopile having high thermal conductivity
surrounds the experimental space within the calorimetric block.
The thermopiles arranged radially guarantees an almost complete
integration of the warmth. The calibration of the calorimetric detectors
may be a key parameter and has got to be performed very carefully.
Heat
55
For this Calvet-type calorimeters, a selected calibration, named as Joule
effect or electrical calibration, has been developed and used to beat all
sorts of issues.
The main advantages of this sort of calibration are as follows:
1.
2.
3.
4.
It is an absolute calibration.
The use of standard materials for calibration is not necessary.
The calibration is often performed at a continuing temperature,
within the heating mode and within the cooling mode.
It can be applied to any experimental vessel volume.
It is a very accurate calibration.
Adiabatic Calorimeters
Adiabatic calorimeters measure the change in enthalpy of a reaction
occurring in solution. During the reaction, the no heat exchange with
the surroundings is allowed and the atmospheric pressure remains
constant.
Differential Scanning Calorimeter
In this differential scanning calorimeter or DSC, heat flows into a
sample which is usually contained within a small aluminium capsule or
‘pan’. This heat flow is measured differentially, i.e., by comparing it to
the flow into an empty reference pan.
Isothermal Titration Calorimeter
In an isothermal titration calorimeter, the warmth of reaction is
employed to follow a titration experiment. This technique is gaining high
importance mainly within the field of biochemistry because it facilitates
the determination of substrate binding to enzymes. The technique is
usually utilized in the pharmaceutical industry to characterize potential
drug candidates.
2.2.4 Specific Heat and Heat Capacity
Heat capacity is a measure of the amount of heat energy required to
change the temperature of a pure substance by a given amount.
Heat Capacity
Heat capacity is an intrinsic physical property of a substance
that measures the amount of heat required to change that substance’s
56
Experimental Physics I
temperature by a given amount. In the International System of Units
(SI), heat capacity is expressed in units of joules per kelvin (J⋅K−1). Heat
capacity is an extensive property, meaning that it is dependent upon
the size/mass of the sample. For instance, a sample containing twice the
amount of substance as another sample would require twice the amount
of heat energy (Q) to achieve the same change in temperature (ΔT) as
that required to change the temperature of the first sample.
Molar and Specific Heat Capacities
There are two derived quantities that specify heat capacity as
an intensive property (i.e., independent of the size of a sample) of a
substance. They are:
•
•
the molar heat capacity, which is
the heat capacity per mole of a pure
substance. Molar heat capacity is
often designated CP, to denote
heat capacity under constant
pressure conditions, as well as CV,
to denote heat capacity under
constant volume conditions. Units
Important
Specific heat
capacity is a measure of the amount
of heat necessary to
raise the temperature of one gram of
a pure substance by
one degree K.
of molar heat capacity are
.
the specific heat capacity, often
simply called specific heat, which
is the heat capacity per unit mass
of a pure substance. This is designated cP and cV and its units
are given in
.
Heat, Enthalpy, and Temperature
Given the molar heat capacity or the specific heat for a pure
substance, it is possible to calculate the amount of heat required to raise/
lower that substance’s temperature by a given amount. The following
two formulas apply:
q=mcpΔT
q=nCPΔT
In these equations, m is the substance’s mass in grams (used when
calculating with specific heat), and n is the number of moles of substance
(used when calculating with molar heat capacity).
Heat
57
Constant-Volume Calorimetry
Constant-volume calorimeters, such as bomb calorimeters, are used
to measure the heat of combustion of a reaction.
The Bomb Calorimeter
Bomb calorimetry is used to measure the heat that a reaction absorbs
or releases, and is practically used to measure the calorie content of
food. A bomb calorimeter is a type of constant-volume calorimeter used
to measure a particular reaction’s heat of combustion. For instance, if
we were interested in determining the heat content of a sushi roll, for
example, we would be looking to find out the number of calories it
contains. In order to do this, we would place the sushi roll in a container
referred to as the “bomb”, seal it, and then immerse it in the water inside
the calorimeter. Then, we would evacuate all the air out of the bomb
before pumping in pure oxygen gas (O2). After the oxygen is added, a
fuse would ignite the sample causing it to combust, thereby yielding
carbon dioxide, gaseous water, and heat. As such, bomb calorimeters
are built to withstand the large pressures produced from the gaseous
products in these combustion reactions.
A schematic representation of a bomb calorimeter used for the
measurement of heats of combustion. The weighed sample is placed in
a crucible, which in turn is placed in the bomb. The sample is burned
completely in oxygen under pressure. The sample is ignited by an iron
wire ignition coil that glows when heated. The calorimeter is filled with
fluid, usually water, and insulated by means of a jacket. The temperature
of the water is measured with the thermometer. From the change in
temperature, the heat of reaction can be calculated.
Once the sample is completely combusted, the heat released in the
reaction transfers to the water and the calorimeter. The temperature
change of the water is measured with a thermometer. The total heat
given off in the reaction will be equal to the heat gained by the water
and the calorimeter:
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Experimental Physics I
qrxn=−qcal
Keep in mind that the heat gained by the calorimeter is the sum of
the heat gained by the water, as well as the calorimeter itself. This can be
expressed as follows:
qcal=mwaterCwaterΔT+CcalΔT
where Cwater denotes the specific heat capacity of the water
, and Ccal is the heat capacity of the calorimeter (typically in
).
Therefore, when running bomb calorimetry experiments, it is necessary
to calibrate the calorimeter in order to determine Ccal.
Since the volume is constant for a bomb
calorimeter, there is no pressure-volume
work. As a result:
ΔU=qV
Important
Change in
enthalpy can be
calculated based
on the change in
temperature of the
solution, its specific
heat capacity, and
mass.
where ΔU is the change in internal
energy, and qV denotes the heat absorbed
or released by the reaction measured
under conditions of constant volume. (This
expression was previously derived in the
“Internal Energy and Enthalpy ” section.)
Thus, the total heat given off by the reaction
is related to the change in internal energy
(ΔU), not the change in enthalpy (ΔH) which is measured under
conditions of constant pressure.
The value produced by such experiments does not completely
reflect how our body burns food. For example, we cannot digest fiber,
so obtained values have to be corrected to account for such differences
between experimental (total) and actual (what the human body can
absorb) values.
Constant-Pressure Calorimetry
A constant-pressure calorimeter measures the change in enthalpy of
a reaction at constant pressure.
Heat
59
A constant-pressure calorimeter measures the change in enthalpy
of a reaction occurring in a liquid solution. In that case, the gaseous
pressure above the solution remains constant, and we say that the
reaction is occurring under conditions of constant pressure. The heat
transferred to/from the solution in order for the reaction to occur is equal
to the change in enthalpy (ΔH=qP), and a constant-pressure calorimeter
thus measures this heat of reaction. In contrast, a bomb calorimeter ‘s
volume is constant, so there is no pressure-volume work and the heat
measured relates to the change in internal energy (ΔU=qV).
A simple example of a constant-pressure calorimeter is a coffee-cup calorimeter, which is constructed from two nested Styrofoam cups and a
lid with two holes, which allows for the insertion of a thermometer and
a stirring rod. The inner cup holds a known amount of a liquid, usually
water, that absorbs the heat from the reaction. The outer cup is assumed
to be perfectly adiabatic, meaning that it does not absorb any heat whatsoever. As such, the outer cup is assumed to be a perfect insulator.
A styrofoam cup with an inserted thermometer can be used as a
calorimeter, in order to measure the change in enthalpy/heat of reaction
at constant pressure.
Calculating Specific Heat
Data collected during a constant-pressure calorimetry experiment
can be used to calculate the heat capacity of an unknown substance. We
already know our equation relating heat (q), specific heat capacity (C),
and the change in observed temperature (ΔTΔT):
q=mCΔTq=mCΔT
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Experimental Physics I
2.3 HEAT OF FUSION OF ICE
Latent heat, energy absorbed or released by a substance during
a change in its physical state (phase) that occurs without changing its
temperature. The latent heat associated with melting a solid or freezing
a liquid is called the heat of fusion; that associated with vaporizing a
liquid or a solid or condensing a vapour is called the heat of vaporization.
The latent heat is normally expressed as the amount of heat (in units of
joules or calories) per mole or unit mass of the substance undergoing a
change of state.
For example, when a pot of water is kept boiling, the temperature
remains at 100 °C (212 °F) until the last drop evaporates, because all the
heat being added to the liquid is absorbed as latent heat of vaporization
and carried away by the escaping vapour molecules. Similarly, while ice
melts, it remains at 0 °C (32 °F), and the liquid water that is formed with
the latent heat of fusion is also at 0 °C. The heat of fusion for water at
0 °C is approximately 334 joules (79.7 calories) per gram, and the heat
of vaporization at 100 °C is about 2,230 joules (533 calories) per gram.
Because the heat of vaporization is so large, steam carries a great deal
of thermal energy that is released when it condenses, making water an
excellent working fluid for heat engines.
Latent heat arises from the work required to overcome the forces
that hold together atoms or molecules in a material. The regular
structure of a crystalline solid is maintained by forces of attraction
among its individual atoms, which oscillate slightly about their average
positions in the crystal lattice. As the temperature increases, these
motions become increasingly violent until, at the melting point, the
attractive forces are no longer sufficient to maintain the stability of the
crystal lattice. However, additional heat (the latent heat of fusion) must
be added (at constant temperature) in order to accomplish the transition
to the even more-disordered liquid state, in which the individual
particles are no longer held in fixed lattice positions but are free to move
about through the liquid. A liquid differs from a gas in that the forces
of attraction between the particles are still sufficient to maintain a longrange order that endows the liquid with a degree of cohesion. As the
temperature further increases, a second transition point (the boiling
point) is reached where the long-range order becomes unstable relative
to the largely independent motions of the particles in the much larger
volume occupied by a vapour or gas. Once again, additional heat (the
latent heat of vaporization) must be added to break the long-range order
Heat
61
of the liquid and accomplish the transition to the largely disordered
gaseous state.
Latent heat is associated with processes other than changes among
the solid, liquid, and vapour phases of a single substance. Many solids
exist in different crystalline modifications, and the transitions between
these generally involve absorption or evolution of latent heat. The
process of dissolving one substance in another often involves heat; if
the solution process is a strictly physical change, the heat is a latent heat.
Sometimes, however, the process is accompanied by a chemical change,
and part of the heat is that associated with the chemical reaction.
2.3.1 Heat of Fusion Formula
The heat of fusion of any substance is the important calculation
of the heat. It is the change in the value of the enthalpy by providing
energy i.e. heat, for a specific quantity of the substance. It will change its
state from a solid to a liquid keeping the pressure constant. The heat of
fusion of any sample will measure the amount of heat that needs to be
introduced to convert its crystalline fraction into the disordered state.
‘Heat of fusion’ measures the amount of energy required to melt a
given amount of a solid at its melting point temperature. In other words,
it also represents the amount of energy given up when a given mass of
liquid solidifies. For example, water has a heat of fusion of 80 calories
per gram. It means that it takes 80 calories of energy to melt 1 gram of
ice at the temperature of zero degrees C into the water at zero degrees C.
Heat of fusion values will differ for the different materials.
For example, we may see that heat gained by ice is equal to the heat
lost by the water. We denote the Heat of fusion by the symbol ΔHf.
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Experimental Physics I
When a solid material turns into the liquid, then it is what we know
as melting. This melting process will need an increase in energy to allow
the solid-state particles to break free from each other. This energy input
is the heat of fusion. The heat of fusion is not the same for all substances,
but it is a constant value for each individual kind of substance.
The Formula for the Heat of Fusion:
We compute it as:
ΔHf
heat of fusion
q
Heat
m
mass
Examples for Heat of Fusion Formula
1: Calculate the heat in Joules which is required to melt 26 grams of
the ice. It is given here that heat of fusion of water is 334 J/g i.e. equals
to 80 cal per gram.
Solution: Given parameters are,
Mass, m = 26 g
We know that,
Rearranging the formula,
q=m×ΔHf,
=26×334
= 8684 Joules.
Thus heat required will be 8684 Joules.
2: What will be the heat of fusion for the water, if it takes 668 Joules
of the heat energy to melt 2 grams?
Solution: Known values are,
Q = 668 joules
Heat
63
M = 2 grams
Formula is:
= 334 J per gram.
Thus heat of fusion will be 334 J per gram.
3: What mass of water will be melted at zero degrees C, if 1500 J of
heat energy is applied? Use Heat of Fusion Formula.
Solution: The heat of fusion for water is applicable here and the
equation has to be rearranged to solve it for the mass.
Here, Hf=1500J
Q = 334 C per gram
Then,
= 4.49 gram
Mass of water will be 4.49 gram.
2.3.2 How to Measure Heat of Fusion of Ice
“Heat” represents the thermal energy of molecules in a substance.
Water freezes at 0 degrees Celsius. But the temperature of an ice cube
64
Experimental Physics I
can fall well below that. When an ice cube is removed from a freezer, the
cube’s temperature increases as it absorbs heat from its surroundings. But
once the ice cube reaches 0 C, it begins to melt and its temperature stays
at 0 throughout the melting process, even though the ice cube continues
to absorb heat. This occurs because the thermal energy absorbed by the
ice cube is consumed by water molecules separating from each other
during melting.
The amount of heat absorbed by a solid during its melting phase is
known as the latent heat of fusion and is measured via calorimetry.
Data Collection
Place an empty Styrofoam cup on a balance and record the mass of
the empty cup in grams. Then fill the cup with about 100 milliliters, or
about 3.5 ounces, of distilled water. Return the filled cup to the balance
and record the weight of the cup and water together.
Place a thermometer into the water in the cup, wait about 5 minutes
for the thermometer to come to thermal equilibrium with the water, then
record the temperature of the water as the initial temperature.
Place two or three ice cubes on a paper towel to remove any liquid
water on the surfaces of the cubes, then quickly transfer the cubes to
the Styrofoam cup. Use the thermometer to gently stir the mixture.
Observe the temperature reading on the thermometer. It should begin
to drop almost immediately. Continue stirring and record the lowest
temperature indicated on the thermometer before the temperature
begins to rise. Record this value as the “final temperature.”
Heat
65
Remove the thermometer and return the Styrofoam cup once again
to the balance and record the mass of the cup, water and melted ice
together.
Calculations
Determine the mass of water in the cup by subtracting the mass
of the empty cup from the weight of the cup and water together, as
collected in step 1. For example, if the empty cup weighed 3.1 grams
and the cup and water together weighed 106.5 grams, then the mass of
the water was 106.5 - 3.1 = 103.4 g.
Calculate the temperature change of the water by subtracting the
initial water temperature from the final water temperature. Thus, if the
initial temperature was 24.5 C and the final temperature was 19.2 C,
then deltaT = 19.2 - 24.5 = -5.3 C.
Calculate the heat, q, removed from the water according to the
equation q = mc(deltaT), where m and deltaT represent the mass and
temperature change of the water, respectively, and c represents water’s
specific heat capacity, or 4.184 joules per gram per degree Celsius, or
4.187 J/g-C. Continuing the example from steps 1 and 2, q = ms(deltaT) =
103.4 g * 4.184 J/g-C * -5.3 C = -2293 J. This represents the heat removed
from the water, hence its negative sign. By the laws of thermodynamics,
this means that the ice cubes in the water absorbed +2293 J of heat.
Determine the mass of the ice cubes by subtracting the mass of the
cup and water from the mass of the cup, water and ice cubes together. If
the cup, water and ice together weighed 110.4 g, then the mass of the ice
cubes was 110.4 g - 103.4 g = 7.0 g.
Find the latent heat of fusion, Lf, according to Lf = q ÷ m by dividing
the heat, q, absorbed by the ice, as determined in step 3, by the mass of
ice, m, determined in step 4. In this case, Lf = q / m = 2293 J ÷ 7.0 g = 328
J/g. Compare your experimental result to the accepted value of 333.5 J/g.
2.3.4 Heats of Fusion and Solidification
Suppose you hold an ice cube in your hand. It feels cold because heat
energy leaves your hand and enters the ice cube. What happens to the
ice cube? It melts. However, the temperature during a phase change remains constant. So the heat that is being lost by your hand does not
raise the temperature of the ice above its melting temperature of 0°C.
Rather, all the heat goes into the change of state. Energy is absorbed dur-
66
Experimental Physics I
ing the process of changing ice into water. The water that is produced
also remains at 0°C until all of the ice is melted.
2.3.5 Heats of Fusion and Solidification
All solids absorb heat as they melt to become liquids. The gain of
heat in this endothermic process goes into changing the state rather
than changing the temperature. The molar heat of fusion of
a substance is the heat absorbed by one mole of that substance as it is
converted from a solid to a liquid. Since the melting of any substance
absorbs heat, it follows that the freezing of any substance releases heat.
The molar heat of solidification of a substance is the heat released
by one mole of that substance as it is converted from a liquid to a solid.
Since fusion and solidification of a given substance are the exact opposite
processes, the numerical value of the molar heat of fusion is the same as
the numerical value of the molar heat of solidification, but opposite in
sign. In other words,
. The Figure below shows all
of the possible changes of state along with the direction of heat flow
during each process.
Heat
67
From left to right, heat is absorbed from the surroundings during
melting, evaporation, and sublimation. Form right to left, heat is released
to the surroundings during freezing, condensation, and deposition.
Every substance has a unique value for its molar heat of
fusion, depending on the amount of energy required to disrupt the
intermolecular forces present in the solid. When 1 mol of ice at 0°C is
converted to 1 mol of liquid water at 0°C, 6.01 kJ of heat are absorbed
from the surroundings. When 1 mol of water at 0°C freezes to ice at 0°C,
6.01 kJ of heat are released into the surroundings.
The molar heats of fusion and solidification of a given substance
can be used to calculate the heat absorbed or released when various
amounts are melted or frozen.
Sample Problem Heat of Fusion
Calculate the heat absorbed when 31.6 g of ice at 0°C is completely
melted.
Step 1: List the known quantities and plan the problem .
Known
•
•
•
mass = 31.6 g ice
molar mass H 2 O( s ) = 18.02 g/mol
molar heat of fusion = 6.01 kJ/mol
Unknown
•
The mass of ice is first converted to moles. This is then multiplied by
the conversion factor of in order to find the kJ of heat absorbed.
Step 2: Solve .
Step 3: Think about your result .
The given quantity is a bit less than 2 moles of ice, and so just less
than 12 kJ of heat is absorbed by the melting process.
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Experimental Physics I
2.3.6 Determining the Heat of Fusion of Ice
When a sample of ice of mass m completely undergoes a phase
change from solid into liquid, i.e. melts to water, the total energy Q
which the ice absorbs from its environment is proportional to the heat
of fusion Lf , the heat transfer for a unit mass, and the mass m:
In the other direction, when the phase change is from liquid to solid,
the sample must release the same amount of
energy. For water at its normal freezing or
melting temperature
Important
This lab employs a double-wall
calorimeter as shown in Figure to measure
the heat of fusion for ice. The calorimeter
consists of an aluminum container, a
reservoir, a plastic lid and an insulator ring.
The reservoir holds a maximum of 150 ml
water. The clear plastic lid has 3 access holes.
It includes a cork with a hole for holding a
thermometer, and a hole for a stirrer.
Water is poured into the reservoir, and
the initial equilibrium temperature T1 is
reached after heat transfer among all the
devices is completed. Ice of mass m is then
put into the water, and then absorbs heat from
water. In the meanwhile, the water releases
heat, and as a result, the temperature of the
whole system decreases until the system
It is important to prevent any
unnecessary heat
transfer between
the environment
and the system.
When you measure
the mass for ice or
put the ice into the
water, do it as soon
as possible because
you do not want the
ice to get heat from
the air. Also, when
you stir the water
with the ice don’t do
it too fast because
you don’t want to
give your energy to
the system
Heat
69
starts to absorb heat from air surrounding the system. At this turning
point, the temperature is T2. The temperature then increases again.
The total heat transfer can be broken into two parts: heat given off
and heat absorbed (if we assume the system is closed). The heat absorbed
goes into
•
•
the ice, to turn it into ice water
the resulting ice water, to raise it to T2.
The heat given off comes from
•
•
•
the reservoir
the water in the reservoir
the thermometer
all of which start out at T1. Assuming the system is closed, all of the
heat given off must be absorbed. The heat equation for the system can
then be expressed as:
where
•
•
•
•
•
mw is the mass of water
mA is the mass of the reservoir and stirrer
c is the specific heat of water
cA is the specific heat of the (aluminum) reservoir
q is the heat released by the thermometer which is equal to:
where V is the volume of the thermometer which can be determined
by Archimedes’ Principle.
Method
1.
2.
3.
Weigh the masses of all components: water, stirrer, and
reservoir. Make sure the water fills at least half of the reservoir
of the calorimeter.
Heat the water up to about 50◦C.
Measure the volume of the thermometer by use of the
graduated cylinder.
70
Experimental Physics I
4.
5.
6.
7.
8.
9.
Put ice into water in a beaker for a while, and bring the
temperature of the ice down to 0◦C.
Take the ice out of the water in the beaker, dry it, and then
measure its mass quickly by the scale. Then put the ice into
the reservoir, and cover the lid promptly.
Measure the temperature of the system every 30 seconds.
Plot a graph of temperature vs. time. A typical graph of
temperature vs. time for the system is shown in Figure.
Extrapolate from the graph as shown to get T1, T2, and their
uncertainties.
Use these values in Equation 3 to determine Lf , and compare
it to the expected value.
2.4 HEAT OF VAPORIZATION OF WATER
Water in its liquid form has an unusually high boiling point
temperature, a value close to 100°C. As a result of the network of
hydrogen bonding present between water molecules, a high input of
energy is required to transform one gram of liquid water into water
vapor, an energy requirement called the heat of vaporization. Water
has a heat of vaporization value of 40.65 kJ/mol. A considerable amount
of heat energy (586 calories) is required to accomplish this change in
water. This process occurs on the surface of water. As liquid water
heats up, hydrogen bonding makes it difficult to separate the water
molecules from each other, which is required for it to enter its gaseous
phase (steam). As a result, water acts as a heat sink, or heat reservoir,
and requires much more heat to boil than does a liquid such as ethanol
(grain alcohol), whose hydrogen bonding with other ethanol molecules
Heat
71
is weaker than water’s hydrogen bonding. Eventually, as water reaches
its boiling point of 100° Celsius (212° Fahrenheit), the heat is able to
break the hydrogen bonds between the water molecules, and the kinetic
energy (motion) between the water molecules allows them to escape
from the liquid as a gas. Even when below its boiling point, water’s
individual molecules acquire enough energy from each other such that
some surface water molecules can escape and vaporize; this process is
known as evaporation.
The fact that hydrogen bonds need to be broken for water to
evaporate means that a substantial amount of energy is used in the
process. As the water evaporates, energy is taken up by the process,
cooling the environment where the evaporation is taking place. In many
living organisms, including humans, the evaporation of sweat, which
is 90 percent water, allows the organism to cool so that homeostasis of
body temperature can be maintained.
The (latent) heat of vaporization (∆Hvap) also known as the enthalpy
of vaporization or evaporation, is the amount of energy (enthalpy) that
must be added to a liquid substance, to transform a given quantity of the
substance into a gas.
The enthalpy of vaporization is a function of the pressure at which
that transformation takes place. The heat of vaporization diminishes
with increasing temperature and it vanishes completely at a certain point
called the critical temperature (Critical temperature for water: 373.946
°C or 705.103 °F, Critical pressure: 220.6 bar = 22.06 MPa = 3200 psi ).
72
Experimental Physics I
Heat of vaporization for liquid water at saturation pressure at
temperatures from 0 to 374 °C:
Temperature
Vapor pressure
Heat of vaporization, ∆Hvap
[°C]
[kPa]
[100*bar]
[J/mol]
[kJ/kg]
[Wh/kg]
[Btu(IT)/lbm]
0.01
0.61165
45054
2500.9
694.69
1075.2
2
0.70599
44970
2496.2
693.39
1073.2
4
0.81355
44883
2491.4
692.06
1071.1
10
1.2282
44627
2477.2
688.11
1065.0
14
1.5990
44456
2467.7
685.47
1060.9
18
2.0647
44287
2458.3
682.86
1056.9
20
2.3393
44200
2453.5
681.53
1054.8
Heat
73
25
3.1699
43988
2441.7
678.25
1049.7
30
4.2470
43774
2429.8
674.94
1044.6
34
5.3251
43602
2420.3
672.31
1040.5
40
7.3849
43345
2406.0
668.33
1034.4
44
9.1124
43172
2396.4
665.67
1030.3
50
12.352
42911
2381.9
661.64
1024.0
54
15.022
42738
2372.3
658.97
1019.9
60
19.946
42475
2357.7
654.92
1013.6
70
31.201
42030
2333.0
648.06
1003.0
80
47.414
41579
2308.0
641.11
992.26
90
70.182
41120
2282.5
634.03
981.30
96
87.771
40839
2266.9
629.69
974.59
100
101.42
40650
2256.4
626.78
970.08
110
143.38
40167
2229.6
619.33
958.56
120
198.67
39671
2202.1
611.69
946.73
140
361.54
38630
2144.3
595.64
921.88
160
618.23
37508
2082.0
578.33
895.10
180
1002.8
36286
2014.2
559.50
865.95
200
1554.9
34944
1939.7
538.81
833.92
220
2319.6
33462
1857.4
515.94
798.54
240
3346.9
31804
1765.4
490.39
758.99
260
4692.3
29934
1661.6
461.56
714.36
280
6416.6
27798
1543.0
428.61
663.37
300
8587.9
25304
1404.6
390.17
603.87
320
11284
22310
1238.4
344.00
532.42
340
14601
18507
1027.3
285.36
441.66
360
18666
12967
719.8
199.9
309.5
373.946
22064
0
0.0
0.0
0.0
Heat of vaporization for liquid water at saturation pressure at
temperatures from 0 to 705 °F:
Temperature
Vapor
pressure
Heat of vaporization, ∆Hvap
[°F]
[psi]
[Btu(IT)/mol]
[Btu(IT)/lbm]
[cal/g]
[kJ/kg]
32.2
0.0891
42.70
1075.2
597.33
2500.9
40
0.1219
42.52
1070.7
594.82
2490.4
50
0.1783
42.30
1065.0
591.67
2477.2
74
Experimental Physics I
60
0.2564
42.07
1059.4
588.54
2464.1
70
0.3632
41.85
1053.7
585.39
2450.9
80
0.5073
41.62
1048.0
582.25
2437.7
90
0.6990
41.40
1042.4
579.09
2424.5
100
0.9506
41.17
1036.7
575.92
2411.3
110
1.277
40.95
1030.9
572.74
2398.0
120
1.695
40.72
1025.2
569.55
2384.6
130
2.226
40.49
1019.4
566.34
2371.2
140
2.893
40.26
1013.6
563.13
2357.7
150
3.723
40.02
1007.7
559.86
2344.0
160
4.747
39.79
1001.8
556.58
2330.3
170
6.000
39.55
995.87
553.26
2316.4
180
7.519
39.31
989.85
549.92
2302.4
190
9.350
39.07
983.76
546.54
2288.2
200
11.54
38.83
977.60
543.11
2273.9
210
14.14
38.58
971.35
539.64
2259.4
212
14.71
38.53
970.08
538.93
2256.4
220
17.20
38.33
965.02
536.12
2244.6
240
24.99
37.81
952.05
528.92
2214.5
260
35.45
37.28
938.64
521.46
2183.3
280
49.22
36.73
924.71
513.73
2150.9
300
66.6
36.15
910.21
505.67
2117.1
350
135
34.59
870.97
483.87
2025.9
400
247
32.82
826.41
459.12
1922.2
450
422
30.78
774.93
430.51
1802.5
500
680
28.37
714.36
396.87
1661.6
550
1044
25.48
641.56
356.42
1492.3
600
1541
21.93
552.09
306.72
1284.2
625
1849
19.41
488.64
271.46
1136.6
650
2205
16.95
426.81
237.11
992.8
675
2615
13.47
339.22
188.46
789.0
705.103
3196
0.00
0.00
0.00
0.0
The heat of vaporization (here denoted as Lv) is defined as the
heat added (or given off) when unit mass undergoes isobaric phase
Heat
75
transformation in any closed two-phase, one-component liquid/vapor
system. In engineering and meteorology, Lv is used in a restricted sense
to mean the heat of vaporization of the two-phase liquid water/water
vapor system. Although much of the subsequent discussion focuses
on Lv for this specific system as an example, the concepts covered are
universally applicable to all fluid/vapor systems.
As illustrated below, for the liquid water/water vapor system,
Lv represents the heat gained when unit mass of water in the system
evaporates in the isobaric phase transformation H2O (liquid) → H2O
(vapor). For the reverse phase change H2O (vapor) → H2O (liquid) i.e.
condensation, Lv is lost from the system. This seemingly simple phase
transition H2O (liquid) ↔ H2O (vapor) is the fundamental driving process
of the earth’s hydrological cycle, the working principle of the steam
engine that ushered humanity into the industrial revolution along with
its (often negative) social and environmental pollution consequences,
and the physical mechanism that maintains the body temperature of
plants and warm-blooded animals.
In general, the state of any closed two-phase, one-component system
is defined by the state variables temperature (T in o K), saturation vapor
pressure (P in Pascal), and volume (V in m3 ). The behavior of any such
system (generally termed as PVT systems) is usually represented as a
family of experimental constant temperature curves (isotherms) on a
P-V coordinate plane called an Amagat-Andrews diagram. The general
shape of these experimental isotherms is illustrated below.
76
Experimental Physics I
For the liquid water/water vapor system the liquid and vapor phases
co-exist in equilibrium only at P-V coordinates between 2 and 3 along
an isotherm, provided that T1 is above the triple point temperature of
water (0.01 o C, i.e. the temperature at which ice, liquid water, and water
vapor can coexist in equilibrium), and below the critical temperature
(374 o C, i.e. the temperature above which it is impossible to produce
condensation by increasing the pressure). Between 2 and 1 the system
can exist as vapor only, and as liquid between 3 and 4. Thus the liquidvapor phase transition at a given temperature can only take place at
constant pressure or vice-versa. Consequently, as shown in Figure
8, isobaric liquid vaporization and condensation is necessarily an
isothermal process, implying that the triple point saturation vapor
pressure is fixed (it is 611 Pa), and so is the saturation vapor pressure at
the critical temperature (it is 2.21 x 107 Pa = 218.2 atm). Similar isotherms
and parameters exist for all liquid/vapor systems.
This observed behavior of closed two-phase, one-component
systems is of course predicted by Gibb’s Phase Rule namely, F + N = C
+ 2 where F = degrees of freedom i.e. the smallest number of intensive
variables (such as pressure, temperature, concentration of components
in each phase) that must be specified to completely describe the state of
the system, N = number of phases i.e. distinct subsystems of uniform
chemical composition and physical properties, and C = the number
of components i.e. the number of independent chemical constituents
meaning those constituents whose concentration can be varied
independently in the different phases. In a liquid / vapor system P =
2, and C =1, and therefore F =1 implying only one intensive variable is
needed to specify the state of the system. Therefore, temperature and
pressure cannot be fixed independently. For the liquid water/water
vapor system, this means physically that at a given temperature between
the triple point and critical temperature, water vapor will evaporate or
condense to achieve the equilibrium saturation vapor pressure as would
be evidenced in a complete Amagat-Andrews diagram for water.
Heat
77
The earth’s atmosphere and oceans can be considered as a vast
closed two phase liquid water/moist air system. Consequently for most
practical engineering and meteorological applications one is interested
in the heat of vaporization of the liquid water/moist air system rather
than a pure liquid water/water vapor system. Fortunately, the presence
of the other gases (collectively called dry air) in the liquid water/moist
air system has negligible effect on the saturation vapor pressure. The
reason is that the dry air component in the liquid water/moist air
system remains unchanged and is always in the gaseous state during
phase transition at temperatures and pressures of practical interest.
Therefore it can be considered as a closed sub-system as opposed to the
open liquid water and water vapor sub-systems. Consequently, results
obtained from an analysis of the thermodynamics of the pure system are
applicable to the natural liquid water/moist air system.
Energy conservation required under the first law of thermodynamics
implies that heat (Q) exchanged reversibly with the surroundings between
equilibrium states of any closed twophase PVT system is consumed
by any internal energy change (ΔU) of the liquid and vapor phases
associated with the mass change from one phase to the other, and any
mechanical work (± PΔV) realized as the volume of the system increases
(positive work) or decreases (negative work). Stated mathematically Q =
ΔU + PΔV or in differential form δQ = dU + pdV. Here P is the saturation
vapor pressure (Pvap). Since entropy (S) is defined as Q/T, then δQ = TdS.
The first law can therefore be restated in terms of exact differentials as
TdS = dU + PdV. Dividing by dV at constant T and rearranging gives dU/
dV = T (dS/dV) – P. Using the Maxwell relation (∂S/∂V)T = (∂P/∂T)V dS/
dV can be replaced (for fixed T) by dP/dT. The equation becomes dU/dV
= T (dP/dT) – P.
At a given pressure and temperature, the internal energy of the
system (U in Joules) can be partitioned as mwuw + mvap uvap where mw,
mvap and uw, uvap represent the masses and the specific internal energies
(internal energy per unit mass in J kg-1) of the water and water vapor in the
system. Similarly, the volume (V in m3 ) of the system can be partitioned
as mwvw + mvap vvap where vw, vvap represent the specific volume (volume
per unit mass in J kg-1) of the water and water vapor in the system. If, as
illustrated in Figure 8, the system internal energy changes by ΔU from
U to U + ΔU as a result of Lv Joules of heat absorption to convert unit
mass of water to water vapor, then U + ΔU = (mw – 1)uw + (mw + 1)uvap and
therefore ΔU = (uw – uvap). Similar reasoning shows that, if the volume
changes from V to V + ΔV in the process, then ΔV = (vvap – vw). The
78
Experimental Physics I
mechanical work due to volume change is PΔV where P (the saturation
vapor pressure) is a constant at a fixed temperature. Therefore, the heat
absorbed (or released) by the system for isobaric phase transition of unit
mass in the liquid water/water vapor system (Lv by definition) = ΔU + P
ΔV. Dividing by ΔV gives ΔU/ΔV = (Lv/ΔV) - P. Substituting ΔV = (vvap
– vw) gives ΔU/ΔV = [Lv / (vvap – vw)] – P or in differential form dU/dV =
[Lv / (vvap – vw)] – P. [It should be noted that since enthalpy (H) is defined
as H = U + PV, then ΔU + P ΔV = ΔH, and therefore Lv is the same as the
specific enthalpy change (Δh = ΔH per unit mass) for phase transition of
unit mass in the liquid water/water vapor system].
Combining the results for dU/dV from the two previous paragraphs
gives T (dP/dT) – P = [Lv / (vvap – vw)] – P and therefore T (dP/dT) = Lv
/ (vvap – vw) which can be rearranged to obtain the general forms of the
Clapeyron equation dP/dT = Lv/ [T(vvap – vw)] = Δh / [T(vvap – vw)] or Lv =
[T (vvap – vw)] dP/dT.
The Clapeyron equation can be used to obtain Lv at a given
temperature T for any liquid provided one can obtain values of (vvap –
vw) and an accurate representation of dP/dT. Values of (vvap – vw) can be
obtained from tabulated measurements. Alternatively, since vvap >> vw
at the low pressures (vvap – vw) can be taken as equal to vvap. Assuming
further that at low pressures water vapor behavior closely approximates
that of an ideal gas, then vvap = RT/P where R is the specific gas constant
for water = 8.314/0.018 = 461.9 J kg-1 o K-1. The Clapeyron equation
becomes Lv = (RT2 /P) dP/dT and this form is referred to as the ClausiusClapeyron equation.
2.4.1 Convective Heat Transfer
Heat energy transferred between a surface and a moving fluid with
different temperatures - is known as convection.
In reality this is a combination of diffusion and bulk motion of
molecules. Near the surface the fluid velocity is low, and diffusion
dominates. At distance from the surface, bulk motion increases the
influence and dominates.
Heat
79
Convective heat transfer can be
•
•
•
forced or assisted convection
natural or free convection
Conductive Heat Transfer
Forced or Assisted Convection
Forced convection occurs when a fluid flow is induced by an
external force, such as a pump, fan or a mixer.
Natural or Free Convection
Natural convection is caused by buoyancy forces due to density
differences caused by temperature variations in the fluid. At heating the
density change in the boundary layer will cause the fluid to rise and
be replaced by cooler fluid that also will heat and rise. This continues
phenomena is called free or natural convection.
Boiling or condensing processes are also referred to as a convective
heat transfer processes.
•
The heat transfer per unit surface through convection was
first described by Newton and the relation is known as
the Newton’s Law of Cooling.
80
Experimental Physics I
The equation for convection can be expressed as:
q = hc A dT (1)
where
q = heat transferred per unit time (W, Btu/hr)
A = heat transfer area of the surface (m2, ft2)
hc = convective heat transfer coefficient of the process (W/(m2oC, Btu/
(ft h oF))
2
dT = temperature difference between the surface and the bulk fluid
( C, F)
o
Heat Transfer Coefficients - Units
•
•
•
•
1 W/(m2K) = 0.85984 kcal/(h m2 oC) = 0.1761 Btu/(ft2 h oF)
1 Btu/(ft2 h oF) = 5.678 W/(m2 K) = 4.882 kcal/(h m2 oC)
1 kcal/(h m2 oC) = 1.163 W/(m2K) = 0.205 Btu/(ft2 h oF)
Overall Heat Transfer Coefficients
Convective Heat Transfer Coefficients
Convective heat transfer coefficients - hc - depends on type of media,
if its gas or liquid, and flow properties such as velocity, viscosity and
other flow and temperature dependent properties. Typical convective
heat transfer coefficients for some common fluid flow applications:
•
•
•
•
•
•
•
Free Convection - air, gases and dry vapors : 0.5 - 1000 (W/
(m2K))
Free Convection - water and liquids: 50 - 3000 (W/(m2K))
Forced Convection - air, gases and dry vapors: 10 - 1000 (W/
(m2K))
Forced Convection - water and liquids: 50 - 10000 (W/(m2K))
Forced Convection - liquid metals: 5000 - 40000 (W/(m2K))
Boiling Water : 3.000 - 100.000 (W/(m2K))
Condensing Water Vapor: 5.000 - 100.000 (W/(m2K))
Convective Heat Transfer Coefficient for Air
The convective heat transfer coefficient for air flow can be
approximated to
Heat
81
hc = 10.45 - v + 10 v1/2 (2)
where
hc = heat transfer coefficient (kCal/m2h°C)
v = relative speed between object surface and air (m/s)
Since
1 kcal/m2h°C = 1.16 W/m2°C
- (2) can be modified to
hcW = 12.12 - 1.16 v + 11.6 v1/2 (2b)
where
hcW = heat transfer coefficient (W/m2°C)
Example - Convective Heat Transfer
A fluid flows over a plane surface 1 m by 1 m. The surface temperature
is 50oC, the fluid temperature is 20oC and the convective heat transfer
coefficient is 2000 W/m2oC. The convective heat transfer between the
hotter surface and the colder air can be calculated as
q = (2000 W/(m2oC)) ((1 m) (1 m)) ((50 oC) - (20 oC))
= 60000 (W)
= 60 (kW)
82
Experimental Physics I
Convective Heat Transfer Chart
2.4.2 Mechanism of Forced Convection
Convection heat transfer is complicated since it involves fluid
motion as well as heat conduction. The fluid motion enhances heat
transfer (the higher the velocity the higher the heat transfer rate).
The rate of convection heat transfer is expressed by Newton’s law
of cooling:
The convective heat transfer coefficient h strongly depends on the
fluid properties and roughness of the solid surface, and the type of the
fluid flow (laminar or turbulent).
Heat
83
Figure 8: Forced convection.
It is assumed that the velocity of the fluid is zero at the wall, this
assumption is called no‐ slip condition. As a result, the heat transfer
from the solid surface to the fluid layer adjacent to the surface is by pure
conduction, since the fluid is motionless. Thus,
The convection heat transfer coefficient, in general, varies along the
flow direction. The mean or average convection heat transfer coefficient
for a surface is determined by (properly) averaging the local heat transfer
coefficient over the entire surface.
Velocity Boundary Layer
Consider the flow of a fluid over a flat plate, the velocity and the
temperature of the fluid approaching the plate is uniform at U∞ and T∞.
The fluid can be considered as adjacent layers on top of each others. Figure 9: Velocity boundary layer.
84
Experimental Physics I
Assuming no‐slip condition at the wall, the velocity of the fluid
layer at the wall is zero. The motionless layer slows down the particles
of the neighboring fluid layers as a result of friction between the two
adjacent layers. The presence of the plate is felt up to some distance from
the plate beyond which the fluid velocity U∞ remains unchanged. This
region is called velocity boundary layer. Boundary layer region is the region where the viscous effects and
the velocity changes are significant and the inviscid region is the region
in which the frictional effects are negligible and the velocity remains
essentially constant. The friction between two adjacent layers between two layers acts
similar to a drag force (friction force). The drag force per unit area is
called the shear stress:
where μ is the dynamic viscosity of the fluid kg/m.s or N.s/m2.
Viscosity is a measure of fluid resistance to flow, and is a strong
function of temperature.
The surface shear stress can also be determined from:
where Cf is the friction coefficient or the drag coefficient which is
determined experimentally in most cases.
The drag force is calculated from:
The flow in boundary layer starts as smooth and streamlined which
is called laminar flow. At some distance from the leading edge, the flow
turns chaotic, which is called turbulent and it is characterized by velocity
fluctuations and highly disordered motion. The transition from laminar to turbulent flow occurs over some
region which is called transition region.
Heat
85
The velocity profile in the laminar region is approximately parabolic,
and becomes flatter in turbulent flow.
The turbulent region can be considered of three regions: laminar
sublayer (where viscous effects are dominant), buffer layer (where both
laminar and turbulent effects exist), and turbulent layer.
The intense mixing of the fluid in turbulent flow enhances heat and
momentum transfer between fluid particles, which in turn increases the
friction force and the convection heat transfer coefficient.
Non‐dimensional Groups
In convection, it is a common practice to non‐dimensionalize the
governing equations and combine the variables which group together
into dimensionless numbers (groups).
Nusselt number: non‐dimensional heat transfer coefficient
where δ is the characteristic length, i.e. D for the tube and L for the
flat plate. Nusselt number represents the enhancement of heat transfer
through a fluid as a result of convection relative to conduction across the
same fluid layer.
Reynolds number: ratio of inertia forces to viscous forces in the fluid
At large Re numbers, the inertia forces, which are proportional to
the density and the velocity of the fluid, are large relative to the viscous
forces; thus the viscous forces cannot prevent the random and rapid
fluctuations of the fluid (turbulent regime). The Reynolds number at which the flow becomes turbulent is
called the critical Reynolds number. For flat plate the critical Re is
experimentally determined to be approximately Re critical = 5 x105 .
Prandtl number: is a measure of relative thickness of the velocity and
thermal boundary layer
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Experimental Physics I
where fluid properties are:
Thermal Boundary Layer
Similar to velocity boundary layer, a thermal boundary layer
develops when a fluid at specific temperature flows over a surface
which is at different temperature.
Figure 10: Thermal boundary layer. The thickness of the thermal boundary layer δt is defined as the
distance at which:
The relative thickness of the velocity and the thermal boundary
layers is described by the Prandtl number.
For low Prandtl number fluids, i.e. liquid metals, heat diffuses much
faster than momentum flow (remember Pr = ν/α<< 1) and the velocity
boundary layer is fully contained within the thermal boundary layer.
On the other hand, for high Prandtl number fluids, i.e. oils, heat diffuses
much slower than the momentum and the thermal boundary layer is
contained within the velocity boundary layer.
Heat
87
Flow Over Flat Plate
The friction and heat transfer coefficient for a flat plate can be
determined by solving the conservation of mass, momentum, and
energy equations (either approximately or numerically). They can also
be measured experimentally. It is found that the Nusselt number can be
expressed as:
where C, m, and n are constants and L is the length of the flat plate.
The properties of the fluid are usually evaluated at the film temperature
defined as:
Laminar Flow
The local friction coefficient and the Nusselt number at the location
x for laminar flow over a flat plate are where x is the distant from the leading edge of the plate and Rex
= ρV∞x / μ.
The averaged friction coefficient and the Nusselt number over the
entire isothermal plate for laminar regime are:
Taking the critical Reynolds number to be 5 x105 , the length of the
plate xcr over which the flow is laminar can be determined from
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Experimental Physics I
Turbulent Flow
The local friction coefficient and the Nusselt number at location x
for turbulent flow over a flat isothermal plate are:
The averaged friction coefficient and Nusselt number over the
isothermal plate in turbulent region are:
Combined Laminar and Turbulent Flow
If the plate is sufficiently long for the flow to become turbulent (and
not long enough to disregard the laminar flow region), we should use
the average values for friction coefficient and the Nusselt number.
where the critical Reynolds number is assumed to be 5x105 . After
performing the integrals and simplifications, one obtains:
Heat
89
The above relationships have been obtained for the case of
isothermal surfaces, but could also be used approximately for the case of
non‐isothermal surfaces. In such cases assume the surface temperature
be constant at some average value.
For isoflux (uniform heat flux) plates, the local Nusselt number for
laminar and turbulent flow can be found from:
Note the isoflux relationships give values that are 36% higher for
laminar and 4% for turbulent flows relative to isothermal plate case.
Example 1
Engine oil at 60°C flows over a 5 m long flat plate whose temperature
is 20°C with a velocity of 2 m/s. Determine the total drag force and the
rate of heat transfer per unit width of the entire plate.
We assume the critical Reynolds number is 5x105 . The properties of
the oil at the film temperature are:
The Re number for the plate is:
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Experimental Physics I
which is less than the critical Re. Thus we have laminar flow. The
friction coefficient and the drag force can be found from:
The Nusselt number is determined from:
Flow across Cylinders and Spheres
The characteristic length for a circular tube or sphere is the external
diameter, D, and the Reynolds number is defined:
The critical Re for the flow across spheres or tubes is 2x105 . The
approaching fluid to the cylinder (a sphere) will branch out and encircle
the body, forming a boundary layer.
Figure 11: Typical flow patterns over sphere and streamlined body and drag
forces.
Heat
91
At low Re (Re < 4) numbers the fluid completely wraps around the
body. At higher Re numbers, the fluid is too fast to remain attached to
the surface as it approaches the top of the cylinder. Thus, the boundary
layer detaches from the surface, forming a wake behind the body. This
point is called the separation point.
To reduce the drag coefficient, streamlined bodies are more suitable,
e.g. airplanes are built to resemble birds and submarine to resemble fish,
Fig. 11.
In flow past cylinder or spheres, flow separation occurs around 80°
for laminar flow and 140° for turbulent flow.
where frontal area of a cylinder is AN = L×D, and for a sphere is AN
= πD2 / 4.
The drag force acting on a body is caused by two effects: the friction
drag (due to the shear stress at the surface) and the pressure drag which
is due to pressure differential between the front and rear side of the
body. As a result of transition to turbulent flow, which moves the
separation point further to the rear of the body, a large reduction in the
drag coefficient occurs. As a result, the surface of golf balls is intentionally
roughened to induce turbulent at a lower Re number, see Fig. 12.
Figure 12: Roughened golf ball reduces CD. 92
Experimental Physics I
The average heat transfer coefficient for cross‐flow over a cylinder
can be found from the correlation presented by Churchill and Bernstein:
where fluid properties are evaluated at the film temperature Tf = (Ts
+ T∞) / 2.
For flow over a sphere, Whitaker recommended the following:
which is valid for 3.5 < Re < 80,000 and 0.7 < Pr < 380. The fluid
properties are evaluated at the free‐stream temperature T∞, except for μs
which is evaluated at surface temperature.
Example 2
The decorative plastic film on a copper sphere of 10‐mm diameter
is cured in an oven at 75°C. Upon removal from the oven, the sphere is
subjected to an air stream at 1 atm and 23°C having a velocity of 10 m/s,
estimate how long it will take to cool the sphere to 35°C.
Assumptions:
1. Negligible thermal resistance and capacitance for the plastic layer.
2. Spatially isothermal sphere.
3. Negligible Radiation.
Heat
93
The time required to complete the cooling process may be obtained
from the results for a lumped capacitance. Whitaker relationship can be used to find h for the flow over sphere:
where Re = ρVD / μ = 6510. Hence,
The required time for cooling is then
2.4.3 Forced Convection Heat Transfer
The fluid motion is, sustained by a difference of pressure created by
an external device such as a pump or fan, the term of “forced convection”
is used. On the other hand, if the fluid motion is predominantly sustained
by the presence of a thermally induced density gradient, then the term
of “natural convection” is used.
Heat transfer by convection occurs as the result of a moving fluid
encountering a fixed surface. The moving fluid carries the heat and
deposits it on the surface or draws it out of the surface. There are two
types of convection. In forced convection, the fluid is being driven or
forced along by some mechanism other than thermal gradients at the
surface. In free convection, the fluid is moved along by thermal gradients
or temperature differences at the surface. Convection obeys Newton’s
law of cooling given by
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Experimental Physics I
q in this case is the heat flux per unit area at the wall. The symbol
h is identified as the film heat transfer coefficient. It has units of W/
m2 K or Btu/h/ft2 /R. The thermal conductivity, is a function of only
the material and its temperature, h, the film heat transfer coefficient,
depends on the properties of the fluid, the temperature of the fluid, and
the flow characteristics. Multiple correlations have been determined for
calculating an appropriate h for most materials and flow situations. In
Eqs. 1a and 1b, the wall temperature is designated by Tw and T∞ is the
temperature of fluid far from the wall at free-stream condition.
To understand better the heat exchange between a solid and fluid,
consider a heated wall over which a fluid flows as sketched in Fig.12.
In this figure, U∞ is the velocity of the fluid under free-stream condition
and far away from the wall as well.
For a given stream velocity, the velocity of the fluid decreases as we
get closer to the wall. This is due to the viscous effects of the flowing fluid.
On the wall, because of the adherence (nonslip) condition the velocity
of the fluid is zero. The region in which the velocity of the fluid varies
from the free-stream value to zero is called “velocity boundary layer.”
Similarly, the region in which the fluid temperature varies from its freestream value to that on the wall is called the “thermal boundary layer,”
and both these boundary layers are defined in previous chapters. Since
the velocity of the fluid at the wall is zero, the heat must be transferred
by conduction at that point. Thus, we calculate the heat transfer by using
the Fourier’s heat conduction law, with thermal conductivity of the
fluid corresponding to the wall temperature and the fluid temperature
gradient at the wall.
Figure 13: Convection heat transfer to a flow over a heated wall.
Heat
95
The question at this point is that: since the heat flows by conduction
in this layer, why do we speak of convection heat transfer and need to
consider the velocity of the fluid. The short answer to this question is that
the temperature gradient of the fluid on the wall is highly dependent on
the flow velocity of the free-stream. As this velocity increases, the distance
from the wall we travel to reach fret stream temperature decreases. In
other words, the thickness of velocity and thermal boundary layers on
the wall decreases. The consequence of this decrease is to increase the
temperature gradient of the fluid at the wall, i.e., an increase in the rate
of heat transferred from the wall to the fluid. The effect of increasing fret
stream velocity on the fluid velocity and temperature profiles close to
the wall is illustrated in Fig. 8. Note also that the temperature gradient
of the fluid on the wall increases with increasing free-stream velocity.
Newton’s experiments end up finding the heat flux on the wall is based
on Eqs. 1a and 1b.
Table 1 gives the orders of magnitude of convective heat transfer
coefficients. We need to remind you that most flow that are occurring in
practical applications are turbulent. As we know so far, that turbulent
flow characterization is based on disorderly displacement of individual
volumes of fluid within the flow.
From the above discussion, we conclude that the basic laws of heat
conduction must be coupled with those of fluid motion to describe,
mathematically, the process of convection. The mathematical treatment
of the resulting system of differential equations is very complex.
Therefore, for engineering applications, the convection will be treated
by an ingenious combination of mathematical techniques, empirical
evidence, and experimentation.
Velocity, temperature, pressure, and other properties change
continuously in time at every point of turbulent flow and the governing
equations of mass, energy. Applies here and are valid for turbulent flow
as well as laminar flow even in transient mode. Therefore, we have to take
the note of the fact, that all the quantities such as velocity, pressure, and
temperature in these equations are instantaneous values. In this chapter
for time being, we concentrate on problems of heat transfer related to
laminar forced convection flow in pipes and ducts. For example, the rate
of heat transfer can become high at the location of reattachment of the
upstream flow on to the surface of the step, as is also the case at the
leading edge of a cylinder in cross-flow, but the detailed mechanisms
remain incompletely understood and research continues.
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Experimental Physics I
Table 2: Order of magnitude of convective heat transfer coefficients
EXERCISE
Answer the following questions:
1.
What is thermal linear expansion?
2.
Explain area thermal expansion coefficient.
3.
What are the types of calorimeter?
4.
What do you understand by the specific heat and heat
capacity?
5.
Discuss about the heat of fusion formula.
MULTIPLE CHOICE QUESTIONS
Tick the correct answer:
1. What is the value of coefficient of volume expansion for an ideal
gas?
a.
(ΔV/V)/Δ T
b.
2*coefficient of linear expansion
c.
1/3 * coefficient of linear expansion
d.
(V/ΔV)/ Δ T
2. What is the relation between coefficient of area expansion(A) and
coefficient of volume expansion(B)?
a.
B = 1.5A
b.
A = 1.5B
c.
A = 3B
d.
B = 3A
3. The latent heat of fusion of ice is:
a.
The sensible heat required for melting ice
b.
335 kJ per kg
Heat
4.
5.
97
c.
335 kJ per ton
d.
The latent heat required for evaporation of ice
Melting point is also known as
a.
fusion point
b.
constant point
c.
boiling point
d.
freezing point
Latent heat of fusion of ice is
a.
4.36 × 105 Jkg-1
b.
2.36 × 105 Jkg-1
c.
1.36 × 105 Jkg-1
d.
3.36 × 105 Jkg-1
ANSWERS
1. (a)
2. (a)
3. (b)
4. (a)
5. (d)
REFERENCES
1.
2.
3.
4.
Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and
Engineers - Volume 1 Mechanics/Oscillations and Waves/
Thermodynamics. New York, NY: Worth Publishers. pp. 666–670.
ISBN 978-1-4292-0132-2.
Papini, Jon J.; Dyre, Jeppe C.; Christensen, Tage (2012-11-29).
“Cooling by Heating---Demonstrating the Significance of the
Longitudinal Specific Heat”. Physical Review X. 2 (4): 041015.
arXiv:1206.6007. Bibcode:2012PhRvX...2d1015P.
Bönisch, Matthias; Panigrahi, Ajit; Stoica, Mihai; Calin, Mariana;
Ahrens, Eike; Zehetbauer, Michael; Skrotzki, Werner; Eckert, Jürgen
(10 November 2017). “Giant thermal expansion and α-precipitation
pathways in Ti-alloys”. Nature Communications. 8 (1): 1429.
Bibcode:2017NatCo...8.1429B.
Measurement and Prediction of Solubility of Paracetamol in WaterIsopropanol Solution. Part 2. Prediction H. Hojjati and S. Rohani
Org. Process Res. Dev.; 2006; 10(6) pp 1110–1118; (Article)
Waves and Sound
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CHAPTER 3
WAVES AND SOUND
OBJECTIVES
After reading this chapter, you should be able to:
•
Explain the wave motion / vibrating strings
•
Describe the properties of sound
•
Define the characteristics of sound waves
•
discuss about basic concepts of vibration
•
Focus on vibration measurement
•
Explain the standing waves on strings
•
Understanding to study longitudinal sound waves created in an air column
of variable length
3.1 WAVE MOTION / VIBRATING STRINGS
The physical phenomenon of sound is a disturbance of matter that
is transmitted from its source outward. Hearing is the perception of
sound, just as seeing is the perception of visible light. On the atomic
scale, sound is a disturbance of atoms that is far more ordered than their
thermal motions. In many instances, sound is a periodic wave, and the
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Experimental Physics I
atoms undergo simple harmonic motion. Thus, sound waves can induce
oscillations and resonance effects ((Figure)).
Figure 1: This glass has been shattered by a high-intensity sound wave of
the same frequency as the resonant frequency of the glass.
A speaker produces a sound wave by oscillating a cone, causing
vibrations of air molecules. In (Figure), a speaker vibrates at a constant
frequency and amplitude, producing vibrations in the surrounding air
molecules. As the speaker oscillates back and forth, it transfers energy
to the air, mostly as thermal energy. But a small part of the speaker’s
energy goes into compressing and expanding the surrounding air,
creating slightly higher and lower local pressures. These compressions
(high-pressure regions) and rarefactions (low-pressure regions) move
out as longitudinal pressure waves having the same frequency as the
speaker—they are the disturbance that is a sound wave. (Sound waves
in air and most fluids are longitudinal, because fluids have almost
no shear strength. In solids, sound waves can be both transverse and
longitudinal.)
(Figure 2) (a) shows the compressions and rarefactions, and also
shows a graph of gauge pressure versus distance from a speaker. As
the speaker moves in the positive x-direction, it pushes air molecules,
displacing them from their equilibrium positions. As the speaker moves
in the negative x-direction, the air molecules move back toward their
equilibrium positions due to a restoring force. The air molecules oscillate
in simple harmonic motion about their equilibrium positions, as shown
in part (b). Note that sound waves in air are longitudinal, and in the
figure, the wave propagates in the positive x-direction and the molecules
oscillate parallel to the direction in which the wave propagates.
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101
Figure 2: (a) A vibrating cone of a speaker, moving in the positive x-direction, compresses the air in front of it and expands the air behind it. As the
speaker oscillates, it creates another compression and rarefaction as those
on the right move away from the speaker. After many vibrations, a series of
compressions and rarefactions moves out from the speaker as a sound wave.
The red graph shows the gauge pressure of the air versus the distance from
the speaker. Pressures vary only slightly from atmospheric pressure for
ordinary sounds.
3.1.1 Types of Sound
There are many different types of sound
including, audible, inaudible, unpleasant,
pleasant, soft, loud, noise and music. You’re
likely to find the sounds produced by a piano
player soft, audible, and musical. And while
the sound of road construction early on
Saturday morning is also audible, it certainly
isn’t pleasant or soft. Other sounds, such as a
dog whistle, are inaudible to the human ear.
This is because dog whistles produce sound
waves that are below the human hearing
range of 20 Hz to 20,000 Hz. Waves below 20
Hz are called infrasonic waves (infrasound),
while higher frequencies above 20,000 Hz
are known as ultrasonic waves (ultrasound).
Infrasonic Waves (Infrasound)
Infrasonic waves have frequencies
below 20 Hz, which makes them inaudible to
the human ear. Scientists use infrasound to
detect earthquakes and volcanic eruptions,
to map rock and petroleum formations
Note
Gauge pressure is
modeled with a sine
function, where the
crests of the function line up with the
compressions and
the troughs line up
with the rarefactions.
Sound waves can also
be modeled using the
displacement of the
air molecules. The
blue graph shows the
displacement of the
air molecules versus
the position from the
speaker and is modeled with a cosine
function. Notice that
the displacement is
zero for the molecules
in their equilibrium position and are centered
at the compressions
and rarefactions.
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Experimental Physics I
underground, and to study activity in the human heart. Despite our
inability to hear infrasound, many animals use infrasonic waves to
communicate in nature. Whales, hippos, rhinos, giraffes, elephants, and
alligators all use infrasound to communicate across impressive distances
– sometimes hundreds of miles!
Ultrasonic Waves (Ultrasound)
Sound waves that have frequencies higher than 20,000 Hz produce
ultrasound. Because ultrasound occurs at frequencies outside the human
hearing range, it is inaudible to the human ear. Ultrasound is most often
used by medical specialists who use sonograms to examine their patients’
internal organs. Some lesser-known applications of ultrasound include
navigation, imaging, sample mixing, communication, and testing. In
nature, bats emit ultrasonic waves to locate prey and avoid obstacles.
How is Sound Produced?
Sound is produced when an object vibrates, creating a pressure
wave. This pressure wave causes particles in the surrounding medium
(air, water, or solid) to have vibrational motion. As the particles vibrate,
they move nearby particles, transmitting the sound further through
the medium. The human ear detects sound waves when vibrating air
particles vibrate small parts within the ear. In many ways, sound waves
are similar to light waves. They both originate from a definite source
and can be distributed or scattered using various means. Unlike light,
sound waves can only travel through a medium, such as air, glass, or
metal. This means there’s no sound in space!
Figure 3: Sound waves.
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103
How Does Sound Travel?
Mediums
Before we discuss how sound travels, it’s important to understand
what a medium is and how it affects sound. We know that sound can
travel through gases, liquids, and solids. But how do these affect its
movement? Sound moves most quickly through solids, because its
molecules are densely packed together. This enables sound waves to
rapidly transfer vibrations from one molecule to another. Sound moves
similarly through water, but its velocity is over four times faster than it
is in air. The velocity of sound waves moving through air can be further
reduced by high wind speeds that dissipate the sound wave’s energy.
Mediums and the Speed of Sound
The speed of sound is dependent on the type of medium the sound
waves travel through. In dry air at 20°C, the speed of sound is 343 m/s!
In room temperature seawater, sound waves travel at about 1531 m/s!
When physicists observe a disturbance that expands faster than the local
speed of sound, it’s called a shockwave. When supersonic aircraft fly
overhead, a local shockwave can be observed! Generally, sound waves
travel faster in warmer conditions. As the ocean warms from global
climate, how do you think this will affect the speed of sound waves in
the ocean?
3.1.2 Propagation of Sound Waves
When an object vibrates, it creates kinetic energy that is transmitted
by molecules in the medium. As the vibrating sound wave comes in
contact with air particles passes its kinetic energy to nearby molecules.
As these energized molecules begin to move, they energize other
molecules that repeat the process.
Imagine a slinky moving down a staircase. When falling down a
stair, the slinky’s motion begins by expanding. As the first ring expands
forward, it pulls the rings behind it forward, causing a compression
wave. This push and pull chain reaction causes each ring of the slinky’s
coil to be displaced from its original position, gradually transporting
the original energy from the first coil to the last. The compressions and
rarefactions of sound waves are similar to the slinky’s pushing and
pulling of its coils.
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Experimental Physics I
Compression & Rarefaction
Sound waves are composed of compression and rarefaction patterns.
Compression happens when molecules are densely packed together.
Alternatively, rarefaction happens when molecules are distanced from
one another. As sound travels through a medium, its energy causes the
molecules to move, creating an alternating compression and rarefaction
pattern. It is important to realize that molecules do not move with the
sound wave. As the wave passes, the molecules become energized
and move from their original positions. After a molecule passes its
energy to nearby molecules, the molecule’s motion diminishes until it
is affected by another passing wave. The wave’s energy transfer is what
causes compression and rarefaction. During compression there is high
pressure, and during rarefaction there is low pressure. Different sounds
produce different patterns of high- and low-pressure changes, which
allows them to be identified. The wavelength of a sound wave is made
up of one compression and one rarefaction.
Figure 4: Different sounds produce different patterns of high- and low-pressure changes.
Sound waves lose energy as they travel through a medium, which
explains why you cannot hear people talking far away, but you can hear
them whispering nearby. As sound waves move through space, they
are reflected by mediums, such as walls, pillars, and rocks. This sound
reflection is better known as an echo. If you’ve ever been inside a cave
or canyon, you’ve probably heard your echo carry much farther than
usual. This is due to the large rock walls reflecting your sound off one
another.
3.1.3 Types of Waves
So, what type of wave is sound? Sound waves fall into three
categories: longitudinal waves, mechanical waves, and pressure waves.
Keep reading to find out what qualifies them as such.
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105
Longitudinal Sound Waves
A longitudinal wave is a wave in which the motion of the medium’s
particles is parallel to the direction of the energy transport. Sound
waves in air and fluids are longitudinal waves, because the particles that
transport the sound vibrate parallel to the direction of the sound wave’s
travel. If you push a slinky back and forth, the coils move in a parallel
fashion (back and forth). Similarly, when a tuning fork is struck, the
direction of the sound wave is parallel to the motion of the air particles.
Mechanical Sound Waves
A mechanical wave is a wave that depends on the oscillation of
matter, meaning that it transfers energy through a medium to propagate.
These waves require an initial energy input that then travels through the
medium until the initial energy is effectively transferred. Examples of
mechanical waves in nature include water waves, sound waves, seismic
waves and internal water waves, which occur due to density differences
in a body of water. There are three types of mechanical waves: transverse
waves, longitudinal waves, and surface waves.
Why is sound a mechanical wave? Sound waves move through air
by displacing air particles in a chain reaction. As one particle is displaced
from its equilibrium position, it pushes or pulls on neighboring
molecules, causing them to be displaced from their equilibrium. As
particles continue to displace one another with mechanical vibrations,
the disturbance is transported throughout the medium. These particleto-particle, mechanical vibrations of sound conductance qualify sound
waves as mechanical waves. Sound energy, or energy associated with
the vibrations created by a vibrating source, requires a medium to travel,
which makes sound energy a mechanical wave.
Wireless Sound Sensor
The Wireless Sound Sensor features two key sensors in one portable
package: a sound wave sensor for measuring relative changes in sound
pressure and a sound level sensor with both dBA- and dBC-weighted
scales. With live data reporting and a wide range of displays (FFT,
scope, digits), the Wireless Sound Sensor’s simple design makes it easy
to use for introductory sound explorations, while its onboard memory
and robust software features support higher-level investigations into
the science of sound.
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Experimental Physics I
Pressure Sound Waves
A pressure wave, or compression wave, has a regular pattern
of high- and low-pressure regions. Because sound waves consist of
compressions and rarefactions, their regions fluctuate between low and
high-pressure patterns. For this reason, sound waves are considered
to be pressure waves. For example, as the human ear receives sound
waves from the surrounding environment, it detects rarefactions as lowpressure periods and compressions as high-pressure periods.
Transverse Waves
Transverse waves move with oscillations that are perpendicular
to the direction of the wave. Sound waves are not transverse waves
because their oscillations are parallel to the direction of the energy
transport; however sound waves can become transverse waves under
very specific circumstances. Transverse waves, or shear waves, travel
at slower speeds than longitudinal waves, and transverse sound waves
can only be created in solids. Ocean waves are the most common
example of transverse waves in nature. A more tangible example can
be demonstrated by wiggling one side of a string up and down, while
the other end is anchored (see standing waves video below). Still a
little confused? Check out the visual comparison of transverse and
longitudinal waves below.
Figure 5: Visual comparison of longitudinal and transverse waves.
How to Create Standing Waves
With PASCO’s String Vibrator, Sine Wave Generator, and Strobe
System, students can create, manipulate and measure standing waves in
real time. The Sine Wave Generator and String Vibrator work together to
propagate a sine wave through the rope, while the Strobe System can be
used to “freeze” waves in time. Create clearly defined nodes, illuminate
Waves and Sound
107
standing waves, and investigate the quantum nature of waves in realtime with this modern investigative approach.
3.2 PROPERTIES OF SOUND
What makes music different from noise? A bird’s call is more
melodic than a car alarm. And, we can usually tell the difference between
ambulance and police sirens - but how do we do this? We use the four
properties of sound: pitch, dynamics (loudness or softness), timbre (tone
color), and duration.
3.2.1 Frequency (Pitch)
Pitch is the quality that enables us to judge sounds as being “higher”
and “lower. It provides a method for organizing sounds based on a
frequency-based scale. Pitch can be interpreted as the musical term for
frequency, though they are not exactly the same. A high-pitched sound
causes molecules to rapidly oscillate, while a low-pitched sound causes
slower oscillation. Pitch can only be determined when a sound has a
frequency that is clear and consistent enough to differentiate it from
noise. Because pitch is primarily based on a listener’s perception, it is
not an objective physical property of sound.
3.2.2 Amplitude (Dynamics)
The amplitude of a sound wave determines it relative loudness.
In music, the loudness of a note is called its dynamic level. In physics,
we measure the amplitude of sound waves in decibels (dB), which do
not correspond with dynamic levels. Higher amplitudes correspond
with louder sounds, while shorter amplitudes correspond with quieter
sounds. Despite this, studies have shown that humans perceive sounds
at very low and very high frequencies to be softer than sounds in the
middle frequencies, even when they have the same amplitude.
3.2.3 Timbre (Tone Color)
Timbre refers to the tone color, or “feel” of the sound. Sounds
with various timbres produce different wave shapes, which affect our
interpretation of the sound. The sound produced by a piano has a
different tone color than the sound from a guitar. In physics, we refer to
this as the timbre of a sound. It’s what allows humans to quickly identify
sounds (e.g. a cat’s meow, running water, the sound of a friend’s voice).
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Experimental Physics I
3.2.4 Duration (Tempo/Rhythm)
In music, duration is the amount of time that a pitch, or tone, lasts.
They can be described as long, short, or as taking some amount of time.
The duration of a note or tone influences the timbre and rhythm of a
sound. A classical piano piece will tend to have notes with a longer
duration than the notes played by a keyboardist at a pop concert. In
physics, the duration of a sound or tone begins once the sound registers
and ends after it cannot be detected.
Creating Music with the 4 Properties of Sound
Musicians manipulate the four properties of sound to make
repeating patterns that form a song. Duration is the length of time a
musical sound lasts. When you strum a guitar, the duration of the sound
is stopped when you quiet the strings. Pitch is the relative highness or
lowness that is heard in a sound and is determined by the frequency of
sound vibrations. Faster vibrations produce a higher pitch than slower
vibrations. The thicker strings of the guitar produce slower vibrations,
creating a deeper pitch, while the thinner strings produce faster
vibrations and a higher pitch. A sound with a definite pitch, or specific
frequency, is called a tone. Tones have specific frequencies that reach
the ear at equal time intervals, such as 320 cycles per second. When two
tones have different pitches, they sound dissimilar, and the difference
between their pitches is called an interval. Musicians frequently use an
interval called an octave, which allows two tones of varying pitches to
share a similar sound. Dynamics refers to a sound’s degree of loudness
or softness and is related to the amplitude of the vibration that produces
the sound. The harder a guitar string is plucked, the louder the sound
will be. Tone color, or timbre, describes the overall feel of an instrument’s
produced sound. If we were to describe a trumpet’s tone color, we
may refer to it as bright or brilliant. When we consider a cello, we may
say it has a rich tone color. Each instrument offers its own tone color,
and new tone colors can be created by layering instruments together.
Furthermore, modern music styles like EDM have introduced new tone
styles, which were unavailable prior to digital music creation.
What Makes Sound Music or Noise?
Acousticians, or scientists who study sound acoustics, have studied
how different sound types, primarily noise and music, affect humans.
Randomized, unpleasant sound waves are often referred to as noise.
Alternatively, constructed patterns of sound waves are known as music.
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Studies have shown that the human body responds differently to noise
and music, which may explain why road construction on a Saturday
morning makes us more tense than a pianist’s song.
Acoustics
Acoustics is an interdisciplinary science that studies mechanical
waves, including vibration, sound, infrasound and ultrasound in
various environments, such as solids, liquids and gases. Professionals
in acoustics can range from acoustical engineers, who investigate new
applications for sound in technology, to audio engineers, who focus on
recording and manipulating sound, to acousticians, who are scientists
concerned with the science of sound.
3.3 CHARACTERISTICS OF SOUND WAVES
There are five main characteristics of sound waves: wavelength,
amplitude, frequency, time period, and velocity. The wavelength of a
sound wave indicates the distance that wave travels before it repeats itself.
The wavelength itself is a longitudinal wave that shows the compressions
and rarefactions of the sound wave. The amplitude of a wave defines the
maximum displacement of the particles disturbed by the sound wave as
it passes through a medium. A large amplitude indicates a large sound
wave. The frequency of a sound wave indicates the number of sound
waves produced each second. Low-frequency sounds produce sound
waves less often than high-frequency sounds. The time period of a sound
wave is the amount of time required to create a complete wave cycle. Each
vibration from the sound source produces a wave’s worth of sound. Each
complete wave cycle begins with a trough and ends at the start of the next
trough. Lastly, the velocity of a sound wave tells us how fast the wave is
moving and is expressed as meters per second.
Figure 6: A wave cycle occurs between two troughs.
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Experimental Physics I
3.3.1 Units of Sound
When we measure sound, there are four different measurement
units available to us. The first unit is called the decibel (dB). The decibel
is a logarithmic ratio of the sound pressure compared to a reference
pressure. The next most frequently used unit is the hertz (Hz). The hertz
is a measure of sound frequency. Hertz and decibels are widely used to
describe and measure sounds, but phon and sone are also used. A sone
is the perceived loudness of a sound and a phon is the unit of loudness
for pure tones. Additionally, the phon refers to subjective loudness,
while the sone is the perceived loudness.
3.3.2 Sound Wave Graphs Explained
Sound waves can be described by graphing either displacement or
density. Displacement-time graphs represent how far the particles are
from their original places and indicates which direction they’ve moved.
Particles that show up on the zero line in a particle displacement graph
didn’t move at all from their normal position. These seemingly motionless
particles experience more compressions and rarefactions than other
particles. Since pressure and density are related, a pressure versus time
graph will display the same information as a density versus time graph.
These graphs indicate where the particles are compressed and where
they are very expanded. Unlike displacement graphs, particles along the
zero line in a density graph are never squished or pulled apart. Instead,
they are the particles that move back and forth the most.
Figure 7: Sound waves can be described by graphing either displacement or
density.
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3.3.3 Sound Pressure
Sound pressure describes the local pressure deviation from the
ambient atmospheric pressure as a sound wave travels. It’s important
to recognize that sound pressure and air pressure are not the same
concept. Overall, the speed of sound is not influenced by air pressure.
As sound waves pass from the sound source through the air, they alter
the pressure experienced by air nearby particles.
Sound Level
Sound level is a comparison of the sound wave’s pressure relative
to the reference point. Sound level is measured in decibels, with higher
decibels correlating to higher sound levels. Some sound instruments
measure sound level in dBc, which is the power ratio (decibels) of a
signal to its carrier signal. Other sound instruments measure the relative
loudness of sounds as perceived by the human ear using a-weighted
decibels, known as dBa. When dBa is used, sounds at low frequencies
have their decibel values reduced and compared to unweighted decibels.
Figure 8: Sound Level is a comparison of the sound wave’s pressure relative to the reference point. A dBc meter measures high and low frequencies,
while a dBa meter measures mid-level frequencies.
3.3.4 Sound Intensity
Sound intensity is the power per unit area carried by a sound wave.
The more intense the sound is, the larger the amplitude oscillations
will be. As sound intensity increases, the pressure exerted by the sound
waves on nearby objects also increases. Decibels are used to measure the
ratio of a given intensity (I) to the threshold of hearing intensity, which
typically has a value of 1000 Hz for the human ear.
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Experimental Physics I
Figure 9: Sound Intensity is the power per unit area carried by a sound
wave. The more intense the sound is, the larger the amplitude oscillations
will be. As sound intensity increases, the pressure exerted by the sound
waves on nearby objects also increases.
3.3.5 Sound Intensity in an Air Column
An air column is a large, hollow tube that is open on one side
and closed on the other. The conditions created by an air column are
especially useful for investigating sound characteristics such as intensity
and resonance. Check out the video below to see how air columns can be
used to investigate nodes, antinodes and resonance.
3.4 BASIC CONCEPTS OF VIBRATION
Any motion that repeats itself after an interval of time is called
vibration or oscillation. The swinging of a pendulum and the motion
of a plucked string are typical examples of vibration. The theory of
vibration deals with the study of oscillatory motions of bodies and the
forces associated with them.
A vibratory system, in general, includes a means for storing potential
energy (spring or elasticity), a means for storing kinetic energy (mass
or inertia), and a means by which energy is gradually lost (damper).
The vibration of a system involves the transfer of its potential energy to
kinetic energy and of kinetic energy to potential energy, alternately. If the
system is damped, some energy is dissipated in each cycle of vibration
and must be replaced by an external source if a state of steady vibration
is to be maintained. As an example, consider the vibration of the simple
pendulum shown in Figure 10. Let the bob of mass m be released after
being given an angular displacement. At position 1 the velocity of the
bob and hence its kinetic energy is zero. But it has a potential energy of
magnitude mgl(1 − cos θ) with respect to the datum position 2.
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Since the gravitational force mg induces a torque about the point
O, the bob starts swinging to the left from position 1. This gives the bob
certain angular acceleration in the clockwise direction, and by the time
it reaches position 2, all of its potential energy will be converted into
kinetic energy.
Figure 10: A simple pendulum.
Hence the bob will not stop in position
2 but will continue to swing to position 3.
However, as it passes the mean position 2,
a counterclockwise torque due to gravity
starts acting on the bob and causes the bob
to decelerate. The velocity of the bob reduces
to zero at the left extreme position. By this
time, all the kinetic energy of the bob will
be converted to potential energy. Again due
to the gravity torque, the bob continues to
attain a counterclockwise velocity. Hence the
bob starts swinging back with progressively
increasing velocity and passes the mean
position again. This process keeps repeating,
and the pendulum will have oscillatory
motion.
Note
The magnitude of
oscillation gradually decreases and
the pendulum
ultimately stops
due to the resistance
(damping) offered
by the surrounding
medium (air). This
means that some energy is dissipated in
each cycle of vibration due to damping
by the air.
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Experimental Physics I
3.4.1 Time Response
Vibrations can be analyzed either in the time domain or in the
frequency domain. Both free and forced vibrations may have to be
analyzed. Free and natural vibrations occur in systems because of the
presence of two forms of energy storage that are interchangeable. When
the stored energy is repeatedly exchanged between these two forms,
the resulting time response of the system is oscillatory. In a mechanical
system, natural vibrations can occur because kinetic energy, which is
manifested as velocities of mass (inertia) elements, may be converted
into potential energy (which can be classified into two basic types-elastic potential energy resulting from the deformation of spring-like
elements and gravitation potential energy resulting from the elevation
of mass elements under the Earth’s gravitational pull) and back to kinetic
energy, repetitively, during motion.
An oscillatory excitation (forcing function) is able to make a
“dynamic” system respond with an oscillatory motion (at the same
frequency as the forcing excitation) even in the absence of the two forms
of energy storage. Such motions are forced responses rather than natural
or free responses. Nevertheless, clear analogies exist with electrical
and fluid systems as well as mixed systems such as electromechanical
systems. Natural oscillations of electrical signals occur in circuits
because of the presence of electrostatic energy (electric charge storage
in capacitor-like elements) and electromagnetic energy (magnetic fields
in inductor-like elements). Fluid systems can exhibit natural oscillatory
responses as they possess two forms of energy. However, purely
thermal systems do not produce natural oscillations because they, as far
as we know, have only one type of energy. Mechanical vibrations can
occur as both free (natural) and forced responses in numerous practical
situations. Some of these vibrations are desirable and useful, and some
others are undesirable and should be avoided or suppressed.
The sound that is generated when a string of a guitar is plucked is a
free vibration whereas the sound of a violin is a mixture of both free and
forced vibrations. These sounds are generally pleasant and desirable.
The response of an automobile as it hits a road bump is an undesirable
free vibration. The vibrations that are felt while operating a concrete
drill are required for the drilling process itself, but are undesirable
forced vibrations for the human who operates the drill. In the design
and development of a mechanical system, regardless of whether it is
intended for generating desirable vibrations or for operating without
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115
vibrations, an analytical model of the system can serve a very useful
purpose. The model will represent the dynamic system, and may be
analyzed and modified more quickly and
cost effectively than one could build and test
a physical prototype. Similarly, in the control
or suppression of vibrations, it is possible
Note
to design, develop, and evaluate vibration
isolators and control schemes through
A frequencyanalytical means before they are physically
domain model is a
implemented. It follows that analytical
set of input-output
models are useful in the analysis, control, and
transfer functions
evaluation of vibrations in dynamic systems,
with respect to the
and also in the design and development of
independent variable frequency.
dynamic systems for achieving the desired
performance when operating in vibratory
environments.
An analytical model of a mechanical system is a set of equations,
and may be developed either by the Newtonian approach, where
Newton’s second law is explicitly applied to each inertia element, or
by the Lagrangian or Hamiltonian approach, which is based on the
concepts of energy (kinetic and potential). The time response describes
how the system moves (responds) as a function of time. The frequency
response describes the way the system moves when excited by a
harmonic (sinusoidal) forcing input, and is a function of the frequency
of excitation.
3.4.2 Classification of Vibration
Vibration can be classified in several ways. Some of the important
classifications are as follows.
Free and Forced Vibrations
•
Free Vibration. If a system, after an initial disturbance, is left
to vibrate on its own, the ensuing vibration is known as free
vibration. No external force acts on the system. The oscillation
of a simple pendulum is an example of free vibration. Free
vibrations are oscillations where the total energy stays
the same over time. This means that the amplitude of the
vibration stays the same. This is a theoretical idea because in
real systems the energy is dissipated to the surroundings over
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Experimental Physics I
•
time and the amplitude decays away to zero, this dissipation
of energy is called damping.
Forced Vibration. If a system is subjected to an external
force (often, a repeating type of force), the resulting vibration
is known as forced vibration. The oscillation that arises
in machines such as diesel engines is an example of forced
vibration. If the frequency of the external force coincides with
one of the natural frequencies of the system, a condition known
as resonance occurs, and the system undergoes dangerously
large oscillations. Failures of such structures as buildings,
bridges, turbines, and airplane wings have been associated
with the occurrence of resonance.
Free vibrations can be defined as oscillations about a system’s
equilibrium position that occur in the absence of an external excitation.
Examples of free vibrations are oscillations of a pendulum about a vertical
equilibrium position and a motion of a vehicle suspension system after
the vehicle encounters a pothole. The oscillations in free vibrations will
eventually die out over time. Let’s consider a mass attached to a spring,
k and a damper, c. The mass is pulled to the right in horizontal direction
and then released. Oscillations occur about its equilibrium position until
it stops.
Figure 11: Free vibration: free body diagram.
Using Newton’s Second Law, force equilibrium in horizontal
direction (x-direction) is given by:
 + cx + kx =
mx
0 (1)
Forced vibrations occur when work is being done on the system
while the vibrations/oscillations occur. Examples of free vibrations
include a motion caused by an unbalanced rotating component, a motion
of reciprocating pistons in engine. Diagram of forced vibration can be
represented by a spring-mass-damper system with external force F(t) =
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117
F0 sinωt. Oscillations occur about its equilibrium position. Oscillations
do not stop because of applied external force.
Figure 12: Forced vibration: free body diagram.
Force equilibrium in horizontal direction (x-direction) is given by:
mx + cx + kx= F0 sin ωt (2)
Comparing the equations of free vibration and forced vibrations,
free vibrations have no external force term, while forced vibrations have
external force term, F0 sinωt. Forced vibrations are commonly observed
in washing machines due to unbalanced mass, fans due to broken blade,
compressors due to reciprocating action of piston etc.
Undamped and Damped Vibration
If no energy is lost or dissipated in friction or other resistance
during oscillation, the vibration is known as undamped vibration. If
any energy is lost in this way, however, it is called damped vibration. In
many physical systems, the amount of damping is so small that it can be
disregarded for most engineering purposes. However, consideration of
damping becomes extremely important in analyzing vibratory systems
near resonance.
Damped and undamped vibration refer to two different types
of vibrations. The main difference between damped and undamped
vibration is that undamped vibration refer to vibrations where energy
of the vibrating object does not get dissipated to surroundings over
time, whereas damped vibration refers to vibrations where the vibrating
object loses its energy to the surroundings.
Undamped Vibration
In undamped vibrations, no resistive force acts on the vibrating
object. As the object oscillates, the energy in the object is continuously
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Experimental Physics I
transformed from kinetic energy to potential energy and back again,
and the sum of kinetic and potential energy remains a constant value.
In practice, it’s extremely difficult to find undamped vibrations. For
instance, even an object vibrating in air would lose energy over time
due to air resistance.
Let us consider an object undergoing simple harmonic motion.
Here, the objet experiences a restoring force towards the equilibrium
point, and the size of this force is proportional to displacement. If the
displacement of the object is given by x, then for an object with mass m
in simple harmonic motion, we can write:
m
d2 x
= −kx
(3)
dt 2
This is a differential equation. A solution to this equation can be
written in the form:
=
x A cos(ωt + φ)
Here, ω =
k
m
(4)
If vibration is undamped, the object continues to oscillate
sinusoidally.
Damped Vibration
In damped vibrations, external resistive forces act on the vibrating
object. The object loses energy due to resistance and as a result, the
amplitude of vibrations decreases exponentially. We can model the
damping force to be directly proportional to the speed of the object at
the time. If the constant of proportionality for the damping force is ,
then we can write:
m
d2 x
dx
=
−kx − b
2
dt (5)
dt
The solution to this differential equation can be given in the form:
−b
t
=
x A e 2m cos(ω′t + φ) (6)
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119
2
ω
Here, the=
k  b 
−
 m  2m 
(7)
We can write this as:
2
ω
=
 b 
1−

 2 mk  (8)
Writing the equation in this form is useful because the quantity
 b 

 can be used to determine the nature of a particular oscillation.
 2 mk 
Often, this quantity is called the damping coefficient ζ ,i.e. ζ =
b .

 2 mk 
If ζ =1, then we have critical damping. Under this condition, the
oscillating object returns to its equilibrium position as soon as possible
without completing any more oscillations. When ζ <1, we have
underdamping. In this case, the object continues to oscillate, but with an
ever-reducing amplitude. For ζ >1 the resistive forces are very strong.
The object would not oscillate again, but the object is slowed down so
much, that it goes towards the equilibrium much more slowly compared
to an object that is critically damped. Over damping is the name given to
this type of scenario. When ζ =0, there is no resistive force and the object
is undamped. Theoretically, the object continues to carry out simple
harmonic motion without any reduction in amplitude.
The graph below illustrates how the displacement of the object
changes under these three different conditions:
Figure 13: Damping under resistive forces with different damping constants.
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Experimental Physics I
Linear and Nonlinear Vibration
If all the basic components of a vibratory system the spring, the mass,
and the damper behave linearly, the resulting vibration is known as linear
vibration. If, however, any of the basic components behave nonlinearly,
the vibration is called nonlinear vibration. The differential equations that
govern the behavior of linear and nonlinear vibratory systems are linear
and nonlinear, respectively. If the vibration
is linear, the principle of superposition
holds, and the mathematical techniques of
analysis are well developed. For nonlinear
Important
vibration, the superposition principle is not
If the value
valid, and techniques of analysis are less
or magnitude of the
well known. Since all vibratory systems
excitation (force or
tend to behave nonlinearly with increasing
motion) acting on a
amplitude of oscillation, a knowledge of
vibratory system is
nonlinear vibration is desirable in dealing
known at any given
with practical vibratory systems.
time, the excitation
Deterministic and Random Vibration
is called deterministic. The resulting
vibration is known
as deterministic
vibration.
In some cases, the excitation is
nondeterministic or random; the value of the
excitation at a given time cannot be predicted.
In these cases, a large collection of records
of the excitation may exhibit some statistical
regularity. It is possible to estimate averages such as the mean and mean
square values of the excitation. Examples of random excitations are wind
velocity, road roughness, and ground motion during earthquakes. If the
excitation is random, the resulting vibration is called random vibration.
In this case the vibratory response of the system is also random; it can
be described only in terms of statistical quantities. Figure 14 shows
examples of deterministic and random excitations.
Figure 14: Deterministic and random excitations.
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3.4.3 Vibration Analysis Procedure
A vibratory system is a dynamic one for which the variables such
as the excitations (inputs) and responses (outputs) are time dependent.
The response of a vibrating system generally depends on the initial
conditions as well as the external excitations. Most practical vibrating
systems are very complex, and it is impossible to consider all the details
for a mathematical analysis. Only the most important features are
considered in the analysis to predict the behavior of the system under
specified input conditions. Often the overall behavior of the system
can be determined by considering even a simple model of the complex
physical system. Thus, the analysis of a vibrating system usually involves
mathematical modeling, derivation of the governing equations, solution
of the equations, and interpretation of the results.
Step 1: Mathematical Modeling. The purpose of mathematical
modeling is to represent all the important features of the system for the
purpose of deriving the mathematical (or analytical) equations governing
the system s behavior. The mathematical model should include enough
details to allow describing the system in terms of equations without
making it too complex. The mathematical model may be linear or
nonlinear, depending on the behavior of the system s components.
Linear models permit quick solutions and are simple to handle;
however, nonlinear models sometimes reveal certain characteristics of
the system that cannot be predicted using linear models. Thus a great
deal of engineering judgment is needed to come up with a suitable
mathematical model of a vibrating system.
(a)
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Experimental Physics I
(b)
(c)
Figure 15: Modeling of a forging hammer.
Sometimes the mathematical model is gradually improved to
obtain more accurate results. In this approach, first a very crude or
elementary model is used to get a quick insight into the overall behavior
of the system. Subsequently, the model is refined by including more
components and/or details so that the behavior of the system can be
observed more closely. To illustrate the procedure of refinement used in
mathematical modeling, consider the forging hammer shown in Figure
15 (a). It consists of a frame, a falling weight known as the tup, an anvil,
and a foundation block.
The anvil is a massive steel block on which material is forged into
desired shape by the repeated blows of the tup. The anvil is usually
mounted on an elastic pad to reduce the transmission of vibration to
the foundation block and the frame. For a first approximation, the
frame, anvil, elastic pad, foundation block, and soil are modeled as a
single degree of freedom system as shown in Figure 15 (b). For a refined
approximation, the weights of the frame and anvil and the foundation
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123
block are represented separately with a two-degree-of-freedom model
as shown in Figure 15 (c). Further refinement of the model can be made
by considering eccentric impacts of the tup, which cause each of the
masses shown in Figure 15 (c) to have both vertical and rocking (rotation)
motions in the plane of the paper.
Step 2: Derivation of Governing Equations. Once the mathematical
model is available, we use the principles of dynamics and derive the
equations that describe the vibration of the system. The equations of
motion can be derived conveniently by drawing the free-body diagrams
of all the masses involved. The free-body diagram of a mass can be
obtained by isolating the mass and indicating all externally applied forces,
the reactive forces, and the inertia forces. The equations of motion of a
vibrating system are usually in the form of a set of ordinary differential
equations for a discrete system and partial differential equations for a
continuous system. The equations may be linear or nonlinear, depending
on the behavior of the components of the system. Several approaches
are commonly used to derive the governing equations. Among them
are Newton s second law of motion, D Alembert s principle, and the
principle of conservation of energy.
Step 3: Solution of the Governing Equations. The equations of
motion must be solved to find the response of the vibrating system.
Depending on the nature of the problem, we can use one of the following
techniques for finding the solution: standard methods of solving
differential equations, Laplace transform methods, matrix methods,
and numerical methods. If the governing equations are nonlinear,
they can seldom be solved in closed form. Furthermore, the solution of
partial differential equations is far more involved than that of ordinary
differential equations. Numerical methods involving computers can be
used to solve the equations. However, it will be difficult to draw general
conclusions about the behavior of the system using computer results.
Step 4: Interpretation of the Results. The solution of the governing
equations gives the displacements, velocities, and accelerations of the
various masses of the system. These results must be interpreted with
a clear view of the purpose of the analysis and the possible design
implications of the results.
3.5 VIBRATION MEASUREMENT
Transducers are available for the direct measurement of
instantaneous acceleration, velocity, displacement and surface strain.
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Experimental Physics I
In noise-control applications, the most commonly measured quantity is
acceleration, as this is often the most convenient to measure. However,
the quantity that is most useful is vibration velocity, as its square is
related directly to the structural vibration energy, which in turn is often
related directly to the radiated sound power. Also, most machines and
radiating surfaces have a flatter velocity spectrum than acceleration
spectrum, which means that the use of velocity signals is an advantage in
frequency analysis as it allows the maximum amount of information to
be obtained using an octave or third-octave filter, or spectrum analyzer
with a limited dynamic range.
For single frequencies or narrow bands of noise, the displacement,
d, velocity, ν, and acceleration, a, are related by the frequency, ω(rad/s),
as dω2=νω=a. In terms of phase angle, velocity leads displacement by 90º
and acceleration leads velocity by 90°. For narrow band or broadband
signals, velocity can also be derived from acceleration measurements
using electronic integrating circuits. On the other hand, deriving
velocity and acceleration signals by differentiating displacement signals
is generally not practical due primarily to the limited dynamic range of
displacement transducers and secondarily to the cost of differentiating
electronics.
One alternative, which is rarely used in noise control, is to bond
strain gauges to the surface to measure vibration levels. However, this
technique will not be discussed further here.
3.5.1 Acceleration Transducers
Vibratory motion for noise-control purposes is most commonly
measured with an accelerometer attached to the vibrating surface. The
accelerometer most generally used consists of a small piezoelectric crystal,
loaded with a small weight and designed to have a natural resonance
frequency well above the anticipated excitation frequency range. Where
this condition may not be satisfied and consequently a problem may
exist involving excitation of the accelerometer resonance, mechanical
filters are available which, when placed between the accelerometer base
and the measurement surface, minimize the effect of the accelerometer
resonance at the expense of the high-frequency response. This results in
loss of accuracy at lower frequencies, effectively shifting the ±3 dB error
point down in frequency by a factor of five. However, the transverse
sensitivity at higher frequencies is also much reduced by use of a
mechanical filter, which in some cases is a significant benefit. Sometimes
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125
it may also be possible to filter out the accelerometer resonance response
using an electrical filter on the output of the amplifier, but this could
effectively reduce the dynamic range of the measurements due to the
limited dynamic range of the amplifier.
Figure 16: Single degree of freedom: (a) forced mass, rigid bas; (b) vibrating
base.
The mass-loaded piezoelectric crystal accelerometer may be thought
of as a one degree- of-freedom system driven at the base, such as that of
case (b) of Figure 16. The crystal, which may be loaded in compression
or shear, provides the stiffness and system damping as well as a small
contribution to the inertial mass, while the load provides the major part
of the system inertial mass. As may readily be shown, the response of
such a system driven well below resonance is controlled by the system
(crystal) stiffness. Within the frequency design range, motion of the
base of the accelerometer imparts in-phase motion to the weight on the
crystal, resulting in small stresses in the crystal. The latter stresses are
detected as induced charge on the crystal by means of some very highimpedance voltage detection circuit, like that provided by an ordinary
sound level meter or a charge amplifier.
Let the displacement, d, of the motion to be measured be:
d=D0ejωt(9)
Referring to Figure 16 (b), the amplitude, D, of displacement
response of the accelerometer mass is as follows:
D=D0X2/Z
(10)
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Experimental Physics I
where
|Z|=[(1−X2)2+(2Xζ)2]1/2, and X=ƒ/ƒ0(11)
In the above equations, X is the ratio of the driving frequency to
the resonance frequency of the accelerometer, ζ is the damping ratio of
the accelerometer and Z is the modulus of the impedance seen by the
accelerometer mass, which represents the reciprocal of a magnification
factor. The voltage generated by the accelerometer will be proportional
to D and, as shown by Equation (10), to the acceleration D0X2 divided by
the modulus of the impedance, |Z|.
If a vibratory motion is periodic it will generally have overtones.
Alternatively, if it is not periodic, the response may be thought of as
a continuum of overtones. In any case, if distortion in the measured
acceleration is to be minimal, then it is necessary that the magnification
factor be essentially constant over the frequency range of interest. In
this case the accelerometer mass displacement amplitude, D (and the
associated voltage which is proportional to the mass displacement),
associated with any component of acceleration will be proportional to
the amplitude of the component, and there will then be no distortion.
However, as the magnification factor, 1/|Z|, is a function of frequency
ratio, X, it can only be approximately constant by design over some
prescribed range and some distortion will always be present. The
percent amplitude distortion is defined as:
Amplitude distortion=[(1/|Z|)−1)]×100%
(12)
To minimize distortion, the accelerometer should have a damping
ratio of between 0.6 and 0.7, giving a useful frequency range of 0<X<0.6.
Where voltage amplification is used, the sensitivity of an accelerometer
is dependent upon the length of cable between the accelerometer and
its amplifier. Any motion of the connecting cable can result in spurious
signals. The voltage amplifier must have a very high input impedance
to measure low frequency vibration and not significantly load the
accelerometer electrically because the amplifier decreases the electrical
time constant of the accelerometer and effectively reduces its sensitivity.
Commercially available high impedance voltage amplifiers allow
accurate measurement down to about 20 Hz, but are rarely used due to
the above-mentioned problems.
Alternatively, charge amplifiers (which, unfortunately, are
relatively expensive) are usually preferred, as they have a very high
Waves and Sound
127
input impedance and thus do not load the accelerometer output; they
allow measurement of acceleration down to frequencies of 0.2 Hz;
they are insensitive to cable lengths up to 500 m and they are relatively
insensitive to cable movement. Many charge amplifiers also have
the capability of integrating acceleration signals to produce signals
proportional to velocity or displacement. This facility should be used
with care, particularly at low frequencies, as phase errors and high levels
of electronic noise may be present, especially if double integration is
used to obtain a displacement signal. Some accelerometers have in-built
charge amplifiers and thus have a low impedance voltage output, which
is easily amplified using standard low impedance voltage amplifiers.
Figure 17: Basic configuration of strain Gage Acceleration Transducer.
The minimum vibration level that can be measured by an
accelerometer is dependent upon its sensitivity and can be as low as 10−4
m/s2. The maximum level is dependent upon size and can be as high as
106 m/s2 for small shock accelerometers. Most commercially available
accelerometers at least cover the range 10−2 to 5×104 m/s2. This range is
then extended at one end or the other, depending upon accelerometer
type. The transverse sensitivity of an accelerometer is its maximum
sensitivity to motion in a direction at right angles to its main axis. The
maximum value is usually quoted on calibration charts and should be
less than 5% of the axial sensitivity. Clearly, readings can be significantly
affected if the transverse vibration amplitude at the measurement
location is an order of magnitude larger than the axial amplitude.
The frequency response of an accelerometer is regarded as
essentially flat over the frequency range for which its electrical output
is proportional to within ±5% of its mechanical input. The upper limit
is generally just less than one third of the resonance frequency. The
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Experimental Physics I
resonance frequency is dependent upon accelerometer size and may be
as low as 2,500 Hz or as high as 180 kHz.
In general, accelerometers with higher resonance frequencies are
smaller in size and less sensitive. When choosing an accelerometer,
some compromise must always be made between its weight and
sensitivity. Small accelerometers are more convenient to use; they
can measure higher frequencies and are less likely to mass load a test
structure and affect its vibration characteristics. However, they have low
sensitivity, which puts a lower limit on the acceleration amplitude that
can be measured. Accelerometers range in weight from miniature 0.65
grams for high-level vibration amplitude (up to a frequency of 18 kHz)
on light weight structures, to 500 grams for low level ground vibration
measurement (up to a frequency of 700 Hz). Thus, prior to choosing
an accelerometer, it is necessary to know approximately the range of
vibration amplitudes and frequencies to be expected as well as detailed
accelerometer characteristics, including the effect of various types of
amplifiers.
Sources of Measurement Error
Temperatures above 100ºC can result in small reversible changes in
accelerometer sensitivity up to 12% at 200ºC. If the accelerometer base
temperature is kept low using a heat sink and mica washer with forced
air cooling, then the sensitivity will change by less than 12% when
mounted on surfaces having temperatures up to 400ºC. Accelerometers
cannot generally be used on surfaces characterized by temperatures in
excess of 400ºC.
Strain variation in the base structure on which an accelerometer
is mounted may generate spurious signals. Such effects are reduced
using a shear type accelerometer and are virtually negligible for piezoresistive accelerometers.
Magnetic fields have a negligible effect on an accelerometer output,
but intense electric fields can have a strong effect. The effect of intense
electric fields can be minimized by using a differential pre-amplifier
with two outputs from the same accelerometer (one from each side of the
piezoelectric crystal with the accelerometer casing as a common earth)
in such a way that voltages common to the two outputs are cancelled.
This arrangement is generally necessary when using accelerometers
near large generators or alternators.
Waves and Sound
129
If the test object is connected to ground, the accelerometer must
be electrically isolated from it, otherwise an earth loop may result, and
producing a high level 50 Hz hum in the resulting acceleration signal.
Sources of Error in the Measurement of Transients
If the accelerometer charge amplifier lower limiting frequency
is insufficiently low for a particular transient or very low frequency
acceleration waveform, then the phenomenon of leakage will occur. This
results in the waveform output by the charge amplifier not being the
same as the acceleration waveform and errors in the peak measurement
of the waveform will occur. To avoid this problem, the lower limiting
frequency of the preamplifier should be less than 0.008/T for a square
wave transient and less than 0.05/T for a half-sine transient, where T is
the period of the transient in seconds. Thus, for a square wave type of
pulse of duration 100 ms, the lower limiting frequency set on the charge
amplifier should be 0.1 Hz.
Another phenomenon, called zero shift, that can occur when any
type of pulse is measured is that the charge amplifier output at the
end of the pulse could be negative or positive, but not zero and can
take a considerable time longer (up to 1000 times longer than the pulse
duration) to decay to zero. Thus, large errors can occur if integration
networks are used in these cases. The problem is worst when the
accelerometers are being used to measure transient acceleration levels
close to their maximum capability. A mechanical filter placed between
the accelerometer and the structure on which it is mounted can reduce
the effects of zero shift.
The phenomenon of ringing can occur when the transient
acceleration that is being measured contains frequencies above the
useful measurement range of the accelerometer and its mounting
configuration. The accelerometer mounted resonance frequency should
not be less than 10/T, where T is the length of the transient in seconds.
The effect of ringing is to distort the charge amplifier output waveform
and cause errors in the measurement. The effects of ringing can be
minimized by using a mechanical filter between the accelerometer and
the structure on which it is mounted.
Another phenomenon, called zero shift, that can occur when any
type of pulse is measured is that the charge amplifier output at the
end of the pulse could be negative or positive, but not zero and can
take a considerable time longer (up to 1000 times longer than the pulse
130
Experimental Physics I
duration) to decay to zero. Thus, large errors can occur if integration
networks are used in these cases. The problem is worst when the
accelerometers are being used to measure transient acceleration levels
close to their maximum capability. A mechanical filter placed between
the accelerometer and the structure on which it is mounted can reduce
the effects of zero shift.
The phenomenon of ringing can occur when the transient
acceleration that is being measured contains frequencies above the
useful measurement range of the accelerometer and its mounting
configuration. The accelerometer mounted resonance frequency should
not be less than 10/T, where T is the length of the transient in seconds.
The effect of ringing is to distort the charge amplifier output waveform
and cause errors in the measurement. The effects of ringing can be
minimized by using a mechanical filter between the accelerometer and
the structure on which it is mounted.
Accelerometer Calibration
In normal use, accelerometers may be subjected to violent
treatment, such as dropping, which can alter their characteristics.
Thus, the sensitivity should be periodically checked by mounting the
accelerometer on a shaker table which either produces a known value
of acceleration at some reference frequency or on which a reference
accelerometer of known calibration may be mounted for comparison.
Accelerometer Mounting
Generally, the measurement of acceleration at low to middle
frequencies poses few mechanical attachment problems. Use of a
magnetic base usually limits the upper frequency bound to about 2
kHz. Beeswax may be used on surfaces that are cooler than 30ºC, for
frequencies below 10 kHz. Thus, for the successful measurement of
acceleration at high frequencies, some care is required to ensure (1)
that the accelerometer attachment is firm, and (2) that the mass loading
provided by the accelerometer is negligible. With respect to the former
it is suggested that the manufacturer’s recommendation for attachment
be carefully followed. With respect to the latter the following is offered
as a guide.
Let the mass of the accelerometer be ma grams. When the mass, ma,
satisfies the appropriate equation which follows, the measured vibration
level will be at most 3 dB below the unloaded level due to the mass
loading by the accelerometer. For thin plates: and for massive structures:
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131
ma≤3.7×10−4 (ρcLh2/ƒ) (grams)(13)
ma≤0.013 (ρcL2 Da/ƒ2) (grams)(14)
In the above equations ρ is the material density (kg/m3), h is the
plate thickness (mm), Da is the accelerometer diameter (mm), ƒ is the
frequency (Hz) and cL is the ongitudinal speed of sound (m/s). For the
purposes of Equations (13) and (14) it will be sufficient to approximate
cL as
E
ρ
As a general guide, the accelerometer mass should be less than 10%
of the dynamic mass (or modal mass) of the vibrating structure to which
it is attached. The effect of the accelerometer mass on any resonance
frequency, ƒs, of a structure is given by:
fm = fs
ms
ms + ma
(15)
where ƒm is the resonance frequency with the accelerometer
attached, ma is the mass of the accelerometer and ms is the dynamic
mass of the structure (often approximated as the mass in the vicinity
of the accelerometer). One possible means of accurately determining a
structural resonance frequency would be to measure it with a number
of different weights placed between the accelerometer and the structure,
plot measured resonance frequency versus added mass and extrapolate
linearly to zero added mass. If mass loading is a problem, an alternative
to an accelerometer is to use a laser Doppler velocimeter, especially if
the frequency range of interest is below a few kHz.
Piezo-resistive Accelerometers
An alternative type of accelerometer is the piezo-resistive type, which
relies upon the measurement of resistance change in a piezo-resistive
element (such as a strain gauge) subjected to stress. Piezo-resistive
accelerometers are less common than piezoelectric accelerometers and
generally are less sensitive by an order of magnitude for the same size
and frequency response. Piezo-resistive accelerometers are capable
of measuring down to d.c. (or zero frequency), are easily calibrated,
and can be used effectively with low impedance voltage amplifiers.
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Experimental Physics I
However, they require a stable d.c. power supply to excite the piezoresistive element (or elements).
3.5.2 Velocity Transducers
Measurement of velocity provides an estimate of the energy
associated with structural vibration; thus, a velocity measurement is often
a useful parameter to quantify sound radiation. Velocity transducers
are generally of three types. The least common is the non-contacting
magnetic type consisting of a cylindrical permanent magnetic on which
is wound an insulated coil. As this type of transducer is only suitable
for relative velocity measurements between two surfaces or structures,
its applicability to noise control is limited; thus, it will not be discussed
further.
The most common type of velocity transducer consists of a moving
coil surrounding a permanent magnet. Inductive electromotive force
(EMF) is induced in the coil when it is vibrated. This EMF (or voltage
signal) is proportional to the velocity of the coil with respect to the
permanent magnet. In the 10 Hz to 1 kHz frequency range, for which
the transducers are suitable, the permanent magnet remains virtually
stationary and the resulting voltage is directly proportional to the
velocity of the surface on which it is mounted. Outside this frequency
range the electrical output of the velocity transducer is not proportional
to velocity. This type of velocity transducer is designed to have a low
natural frequency (below its useful frequency range); thus it is generally
quite heavy and can significantly mass load light structures. Some care
is needed in mounting but this is not as critical as for accelerometers
due to the relatively low upper frequency limit characterizing the basic
transducer.
The preceding two types of velocity transducer generally cover the
dynamic range of 1 to 100 mm/s. Some extend down to 0.1 mm/s while
others extend up to 250 mm/s. Sensitivities are generally high, of the
order of 20 mV/mm s−1.
Low impedance, inexpensive voltage amplifiers are suitable for
amplifying the signal. Temperatures during operation or storage should
not exceed 120ºC. A third type of velocity transducer is the laser Doppler
velocimeter. Currently available instrumentation has a dynamic range
typically of 80 dB or more. Instruments can usually be adjusted using
different processing modules so that the minimum and maximum
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133
measurable levels can be varied, while maintaining the same dynamic
range. Instruments are available that can
measure velocities up to 20 m/s and down
to 1 μm/sec (although not with the same
Important
processing electronics).
3.5.3 Instrumentation Systems
Around 1830,
Michael Faraday
established that the
reactions at each of the
two electrode–electrolyte interfaces provide
the “seat of emf” for
the voltaic cell, that is,
these reactions drive
the current and are not
an endless source of
energy as was initially
thought.
The instrumentation system which is
used in conjunction with the transducers
just described, depends upon the level of
sophistication desired. Overall or octave
band vibration levels can be recorded in
the field using a simple vibration meter.
If more detailed analysis is required, a
portable spectrum analyzer can be used.
Alternatively, if it is preferable to do the data
analysis in the laboratory, samples of the
data can be recorded using a high quality FM or DAC tape recorder
and replayed through the spectrum analyzer. This latter method has
the advantage of enabling one to re-analyze data in different ways and
with different frequency resolutions, which is useful when diagnosing a
particular vibration problem.
Units of Vibration
It is often convenient to express vibration amplitudes in decibels.
The International Standards Organization has recommended that
the following units and reference levels be used for acceleration and
velocity. Velocity is measured as a root mean square (r.m.s.) quantity
in millimeters per second and the level reference is one nanometer per
second (10−6 mm/s) The velocity level expression, Lν, is:
Lν=20 log10(ν/νref); νref=10−6 mms−1(16)
Acceleration is measured as an r.m.s. quantity in meters per second2
(m/s) and the level reference is one micrometer per second squared (10−6
m/s). The acceleration level expression, La, is:
La=20 log10(a/aref) aref=10−6ms−2(17)
Although there is no standard for displacement, it is customary to
measure it as a peak to peak quantity in micrometers (μm) and use a
level reference of one picometer (10−6mm).
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Experimental Physics I
The displacement level expression, Ld, is:
Ld=20 log10(d/dref) dref=10−6 μm(18)
When vibratory force is measured in dB, the standard reference
quantity is 1 μN. The force level expression is then:
Lƒ=20 log10(F/Fref) Fref=1 μN(19)
Wave Function
The wave function is the most fundamental concept of quantum
mechanics. It was first introduced into the theory by analogy; the
behavior of microscopic particles likes wave, and thus a wave function
is used to describe them. Schrödinger originally regarded the wave
function as a description of real physical wave. But this view met
serious objections and was soon replaced by Born’s probability
interpretation, which becomes the standard interpretation of the wave
function today. According to this interpretation, the wave function is a
probability amplitude, and the square of its absolute value represents
the probability density for a particle to be measured in certain locations.
However, the standard interpretation is still unsatisfying when applying
to a fundamental theory because of resorting to measurement. In view
of this problem, some alternative realistic interpretations of the wave
function have been proposed and widely studied.
There are in general two possible ways to interpret the wave function
of a single quantum system in a realistic interpretation. One view is to
take the wave function as a physical entity simultaneously distributing
in space such as a field, and it is assumed by de Broglie-Bohm theory,
many-worlds interpretation and dynamical collapse theories etc. For
example, in de Broglie-Bohm theory the wave function is generally taken
as an objective physical field, called Ψ-field, though there are various
views on exactly what field the wave function is. The other view is to
take the wave function as a description of some kind of ergodic motion
of a particle (or corpuscle), and it is assumed by stochastic interpretation
etc. The essential difference between a field and the ergodic motion of a
particle lies in the property of simultaneity. The field exists throughout
space simultaneously, whereas the ergodic motion of a particle exists
throughout space in an essentially local way; the particle is still in one
position at each instant, and it is only during a time interval that the
ergodic motion of the particle spreads throughout space.
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135
It is widely expected that the correct realistic interpretation of the
wave function can only be determined by future precise experiments.
We will argue that the above two interpretations of the wave function
can in fact be tested by analyzing the mass and charge density
distributions of a quantum system, and the former has already been
excluded by experimental observations. Moreover, a further analysis
can also determine which kind of ergodic motion of particles the wave
function describes. The plan of this paper is as follows. We first argue
that a quantum system with mass m and charge Q, which is described
by the wave function ψ (x,t), has effective mass and charge density
distributions
in space respectively. This argument
is strengthened by showing that the result is also a consequence of
protective measurement. We argue that the field explanation of the
wave function entails the existence of an electrostatic self-interaction
for the wave function of a charged quantum system, as the charge
density will be distributed in space simultaneously for a physical
field. This contradicts the predictions of quantum mechanics as well as
experimental observations. Thus we conclude that the wave function
cannot be a description of a physical field. This leads us to the second
view that interprets the wave function as a description of the ergodic
motion of particles.
It is argued that the classical ergodic models, which assume
continuous motion of particles, cannot be consistent with quantum
mechanics, and thus they have been excluded.
Further investigates the possibility that the wave function is a
description of the quantum motion of particles, which is random and
discontinuous in nature. It is shown that this new interpretation of the
wave function provides a natural realistic alternative to the orthodox
interpretation, and its implications for other realistic interpretations of
quantum mechanics.
How do Mass and Charge Distribute for a Single Quantum
System?
The mass and charge of a charged classical system always localize
in a definite position in space at each moment. For a charged quantum
system described by the wave function ψ (x,t), how do its mass and
charge distribute in space then? We can measure the total mass and
charge of the quantum system by gravitational and electromagnetic
interactions and find them in some region of space. Thus the mass and
136
Experimental Physics I
charge of a quantum system must also exist in space with a certain
distributions if assuming a realistic view. Although the mass and charge
distributions of a single quantum system seem meaningless according
to the probability interpretation of the wave function, it should have a
physical meaning in a realistic interpretation of the wave function such
as de Broglie-Bohm theory.
As we think, the Schrödinger equation of a charged quantum
system under an external electromagnetic potential already provides an
important clue. The equation is
(20)
where m and Q is respectively the mass and charge of the system, ϕ
and A are the electromagnetic potential,
is Planck’s constant divided
by 2π , c is the speed of light. The electrostatic interaction term Qϕψ
(x,t) in the equation seems to indicate that the charge of the quantum
system distributes throughout the whole region where its wave
function ψ (x,t) is not zero. If the charge does not distribute in some
regions where the wave function is nonzero, then there will not exist any
electrostatic interaction there. But the term Qϕψ (x,t) implies that there
exists an electrostatic interaction in all regions where the wave function
is nonzero. Thus it seems that the charge of the quantum system should
distribute throughout the whole region where its wave function is not
zero. Furthermore, since the integral
is the total charge of the
system, the charge density distribution in space will be
. Similarly,
the mass density can be obtained from the Schrödinger equation of a
quantum system with mass m under an external gravitational potential
V G:
(21)
The gravitational interaction term
in the equation also
indicates that the (passive gravitational) mass of the quantum system
distributes throughout the whole region where its wave function ψ (x,t)
is not zero, and the mass density distribution in space is
.
The above result can be more readily understood when the wave
function is a complete realistic description of a single quantum system
Waves and Sound
137
as in many-worlds interpretation and dynamical collapse theories. If
the mass and charge of a quantum system does not distribute as above
in terms of its wave function ψ (x,t) , then other supplement quantities
will be needed to describe the mass and charge distributions of the
system in space, while this obviously contradicts the premise that the
wave function is a complete description. In fact, the dynamical collapse
theories such as GRW theory already admit the existence of mass density.
In addition, which takes the wave function as an incomplete
description and admits supplement hidden variables, there are also
some arguments for the above mass and charge density explanation.
It was argued that since the Ψ-field depends on the parameters
such as mass and charge, it may be said to be massive and charged.
Brown, argued that properties sometimes attributed to the “particle”
aspect of a neutron, for example. Mass and magnetic moment, cannot
straightforwardly be regarded as localized at the hypothetical position
of the particle in Bohm’s theory. They also argued that it is hard to
understand how the Aharonov-Bohm effect is possible if that the charge
of the electron which couples with the electromagnetic vector-potential
is not co-present in the regions on all sides of the confined magnetic field
accessible to the electron.
One may object that de Broglie-Bohm theory and many-worlds
interpretation seemingly never admit the above mass density
explanation, and no existing interpretation of quantum mechanics
including dynamical collapse theories endows charge density to the
wave function either. As we think, however, protective measurement
provides a more convincing argument for the existence of mass and
charge density distributions. The wave function of a single quantum
system, especially its mass and charge density, can be directly measured
by protective measurement.
3.5.4 Protective Measurement
Different from the conventional measurement, protective
measurement aims at measuring the wave function of a single quantum
system by repeated measurements that do not destroy its state. The
general method is to let the measured system be in a non-degenerate
eigenstate of the whole Hamiltonian using a suitable interaction, and
then make the measurement adiabatically so that the wave function of
the system neither changes nor becomes entangled with the measuring
device appreciably. The suitable interaction is called the protection.
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Experimental Physics I
As a typical example of protective measurement, we consider a
quantum system in a discrete nondegenerate energy eigenstate ψ(x).
The protection is natural for this situation, and no additional protective
interaction is needed.
The interaction Hamiltonian for measuring the value of an
observable A in the state is:
(22)
where P denotes the momentum of the pointer of the measuring
device, which initial state is taken to be a Gaussian wave packet
centered around zero. The time-dependent coupling g(t) is normalized
to
, where T is the total measuring time. In conventional von
Neumann measurements, the interaction HI is of short duration and so
strong that it dominates the rest of the Hamiltonian. As a result, the
time evolution
will lead to an entangled state: eigenstates
of A with eigenvalues αi are entangled with measuring device states
αi
in which the pointer is shifted by these values . Due to the collapse
of the wave function, the measurement result can only be one of the
eigenvalues of observable A , say αi , with a certain probability pi . The
expectation value of A is then obtained as the statistical average of
eigenvalues for an ensemble of identical systems, namely
. By
contrast, protective measurements are extremely slow measurements.
We let g(t) =1/T for most of the time T and assume that g(t) goes to zero
gradually before and after the period T. In the limit T → ∞ , we can obtain
an adiabatic process in which the system cannot make a transition from
one energy eigenstate to another, and the interaction Hamiltonian does
not change the energy eigenstate. As a result, the corresponding time
evolution
shifts the pointer by the expectation value < A
> . This result strongly contrasts with the conventional measurement in
which the pointer shifts by one of the eigenvalues of A.
It should be stressed that T → ∞ is only an ideal situation, and a
protective measurement can never be performed on a single quantum
system with absolute certainty because of the tiny unavoidable
entanglement.
For example, for any given values of P and T, the energy shift of the
above eigenstate, given by first-order perturbation theory, is
Waves and Sound
139
(23)
We can only obtain the exact expectation value < A > with a
probability very close to one, and the measurement result can also be the
expectation value < A >⊥ , with a probability proportional to 1/T2 , where
⊥ refers to the normalized state in the subspace normal to the initial
state ψ(x) as picked out by first-order perturbation theory. Therefore, an
ensemble, which may be considerably small, is still needed for protective
measurements.
Although a protective measurement can never be performed on
a single quantum system with absolute certainty, the measurement is
distinct from the standard one: in no stage of the measurement we obtain
the eigenvalues of the measured variable. Each system in the small
ensemble contributes the shift of the pointer proportional not to one of
the eigenvalues, but to the expectation value. This essential novel point
has been repeatedly stressed by the inventors of protective measurement.
As we know, in the orthodox interpretation of quantum mechanics, the
expectation values of variables are not considered as physical properties
of a single system, as only one of the eigenvalues is observed in the
outcome of the standard measuring procedure and the expectation value
can only be defined as a statistical average of the eigenvalues. However,
for protective measurements, we obtain the expectation value directly
for a single system and not as a statistical average of eigenvalues for
an ensemble. Since the expectation value of a variable can be directly
measured for a single system, it must be a physical characteristic of a
single system, not of an ensemble. This is a definite conclusion we can
reach by the analysis of protective measurement.
In the following we will show that the mass and charge density
can be measured by protective measurement as expectation values of
certain variable for a single quantum system, and thus it is the physical
property of the system. Consider again a quantum system in a discrete
nondegenerate energy eigenstate ψ(x). The interaction Hamiltonian for
measuring the value of an observable An in the state assumes the same
form as equation (24):
(24)
where An is a normalized projection operator on small regions Vn
having volume n vn, which can be written as follows:
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Experimental Physics I
(25)
Then a protective measurement of An will yield the following result:
(26)
It is the average of the density |ψ (x)|2 over the small region Vn .
When vn →0 and after performing measurements in sufficiently many
regions Vn we can find the whole density distribution |ψ (x)|2 . For a
charged system with charge Q the density |ψ (x)|2 times the charge yields
the effective charge density Q |ψ (x)|2. In particular, an appropriate
adiabatic measurement of the Gauss flux out of a certain region will
yield the value of the total charge inside this region, namely the integral
of the effective charge density Q |ψ (x)|2 over this region. Similarly, we
can measure the effective mass density of the system in principle by an
appropriate adiabatic measurement of the flux of its gravitational field.
Therefore, protective measurement shows that the mass and charge of
a single quantum system described by the wave function ψ(x) is indeed
distributed throughout space with effective mass density m |ψ (x)|2
and effective charge density Q |ψ (x)|2 respectively. For instance, in the
double-slit experiment of an electron, a protective measurement of the
charge density will show that there is a charge of e/2 in each of the slits
when the electron is passing the slits.
Although protective measurement strongly suggests a realistic
interpretation of the wave function, it does not directly tell us what the
wave function is. It may describe a physical wave or field, as suggested
by the inventors of protective measurement. It is also possible that the
wave function describes some kind of ergodic motion of particles, though
this view was rejected by Aharonov and Vaidman. Correspondingly, the
mass and charge density may result from a physical field or the ergodic
motion of a particle. These two explanations are essentially different in
that a field exists throughout space simultaneously, whereas the ergodic
motion of a particle exists throughout space in an essentially local way.
Why the Wave Function is not a Physical Field?
If the wave function is a physical field, then its mass and charge
density will simultaneously distribute in space. This has two disaster
Waves and Sound
141
consequences at least. One is that charge will not be quantized; the
total charge inside a very small region can be much smaller than an
elementary charge for a single quantum system. This obviously
contradicts the common expectation that charge should be quantized.
But maybe our expectation needs to be revised. So this consequence is
not fatal for the field explanation of the wave function. The other is that
the wave function will not satisfy the superposition principle.
For example, for the wave function of a single electron, different
spatial parts of the wave function will have gravitational and electrostatic
interactions, as these parts have mass and charge simultaneously.
Let’s analyze the second consequence in more detail. Interestingly,
the so-called Schrödinger-Newton equation, which was proposed for
other purposes, just describes the gravitational self-interaction of the
wave function.
The equation for a single quantum system can be written as:
(27)
where m is the mass of the quantum system, V is an external
potential, and G is Newton’s gravitational constant. Some experimental
schemes have been also proposed to test its physical validity. As we will
see, although such gravitational self-interactions cannot yet be excluded
by experiments the existence of electrostatic self-interaction already
contradicts experimental observations.
(28)
If there is also an electrostatic self-interaction, then the equation
for a free quantum system with mass m and charge Q will be where k
is the Coulomb constant. Note that the gravitational self-interaction is
an attractive force, while the electrostatic self-interaction is a repulsive
force. It has been shown that the measure of the potential strength
of a gravitational self-interaction is
for a free particle with
mass m. This quantity represents the strength of the influence of selfinteraction on the normal evolution of the wave function; when ε2 ≈1
the influence will be significant. Similarly, for a free charged particle
with charge Q, the measure of the potential strength of the electrostatic
142
Experimental Physics I
self-interaction is
. As a typical example, for a free electron with
charge e, the potential strength of the electrostatic self-interaction will
be
. This indicates that the electrostatic self-interaction
will have significant influence on the evolution of the wave function
of a free electron. If such an interaction indeed exists, it should have
been detected by precise experiments on charged microscopic particles.
As another example, consider the electron in the hydrogen atom. Since
the potential of its electrostatic self-interaction is of the same order as
the Coulomb potential produced by the nucleus, the energy levels of
hydrogen atoms will be significantly different from those predicted
by quantum mechanics and confirmed by experimental observations.
Therefore, the electrostatic self-interaction cannot exist for the wave
function of a charged quantum system. Since the field explanation of the
wave function entails the existence of such electrostatic self-interactions,
it cannot be right, i.e. the wave function cannot be a description of a
physical field.
One may object to the above argument with the example of
classical electromagnetic field. Electromagnetic field is a field, but it
has no self-interaction. Thus a field does not require the existence of
self-interaction. However, this is a common misunderstanding. The
crux of the matter is that the non-existence of electromagnetic selfinteraction results from the fact that electromagnetic field itself has no
charge. If the electromagnetic field had charge, then there would also
exist electromagnetic self-interaction due to the nature of field, namely
the simultaneous existence of its properties in space. In fact, although
electromagnetic field has no electromagnetic self-interaction, it does
have gravitational self-interaction; the simultaneous existence of energy
densities in different spatial locations for an electromagnetic field must
generate a gravitational interaction, though the interaction is too weak
to be detected by current technology.
One may further object that the superposition principle in quantum
mechanics already prohibits the existence of the above self-interactions.
But this is just the key point we use to argue against the field explanation
of the wave function. Let’s state the argument more explicitly. If the
wave function of a charged quantum system is a physical field, then
the different spatial parts of this field will have gravitational and
electrostatic interactions. But the superposition principle in quantum
mechanics, which has been verified within astonishing precision, does
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143
not permit the existence of the remarkable electrostatic self-interaction.
Therefore, the field explanation of the wave function is already refuted
by the superposition principle of quantum mechanics.
Why Classical Ergodic Models Fail
If the wave function is not a description of a physical field, then
exactly what does the wave function describe? This naturally leads us
to the second view that takes the wave function as a description of some
sort of ergodic motion of particles. On this view, the effective mass and
charge density are formed by time average of the motion of a charged
particle, and they distribute in different locations at different moments.
At every instant, there is only a localized particle with mass and charge.
Thus there will not exist any self-interaction for the wave function, and
this view can be consistent with quantum mechanics and experimental
observations. In fact, if the mass and charge density does not exist in
different regions simultaneously as the field explanation holds, they
can only be formed by the ergodic motion of a particle. As a result, the
wave function must be a description of some sort of ergodic motion of
particles.
There are indeed some realistic interpretations of quantum mechanics
that attempt to explain the wave function in terms of the ergodic motion
of particles. A well-known example is the stochastic interpretation of
quantum mechanics. Nelson derived the Schrödinger equation from
Newtonian mechanics via the hypothesis that every particle of mass m
is subject to a Brownian motion with diffusion coefficient h / 2m and
no friction. In more technical terms, the quantum mechanical process
is claimed to be equivalent to a classical Markovian diffusion process.
On this interpretation, particles have continuous trajectories but no
velocities, and the wave function is a statistical average description of
their ergodic motion.
However, it has been pointed out that the classical stochastic
interpretations are inconsistent with quantum mechanics argued that
the Schrödinger equation is not equivalent to a Markovian process, and
the various correlation functions used in quantum mechanics do not
have the properties of the correlations of a classical stochastic process.
One must add by hand a quantization condition, as in the old quantum
theory, in order to recover the Schrödinger equation, and thus the
Schrödinger equation and the Madelung hydrodynamic equations are
not equivalent. There is an empirical difference between the predictions
of quantum mechanics and his stochastic mechanics when considering
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Experimental Physics I
quantum entanglement and nonlocality. In addition, it can be generally
argued that the classical ergodic models that assume continuous motion
of particles cannot be consistent with quantum mechanics. In order to
see this let’s examine whether the continuous motion of particles can
generate the charge density
at the level of time average.
Consider an electron in a one-dimensional box in an energy eigenstate
such as the first excited state. Its wave function has a node at the center
of the box, where its charge density is zero. The electron performs a very
fast motion in the box. At a particular time the charge density is either
zero (if the electron is not there) or singular (if the electron is inside the
infinitesimally small region including the space point in question). But
during a time interval, the motion of the electron will generate a charge
density cloud with an effective charge density. The question is whether
this density can assume the same form as e |ψ (x) |2. Since the effective
charge density is proportional to the amount of time the electron spends
in a given position, the electron must be in the left half of the box half
of the time and in the right half of the box half of the time. But it can
spend no time at the center of the box where the effective charge density
is zero; in other words, it must move at infinite velocity at the center.
Certainly, the appearance of infinite velocity or velocity faster than light
may be not a fatal problem, as our discussion is entirely in the context of
non-relativistic quantum mechanics and especially the infinite potential
assumed in the example is also an ideal situation. However, it is hard
to understand why the electron speeds up at the node and where the
infinite energy required for the acceleration comes from. Moreover, the
sudden acceleration of the electron near the node will result in large
radiation, which is inconsistent with both the predictions of quantum
mechanics and experimental observations. Maybe one can also assume
in an ad hoc way that the accelerating electron does not radiate here in
order to make the model be consistent with quantum mechanics and
experimental observations.
Let’s further consider a superposition of two energy eigenstate
respectively limited in two widely-separated boxes. In this example,
even if one assumes that the electron can move with an infinite velocity,
it cannot move from one box to another due to the limitation of box
walls. Therefore, any sort of continuous motion cannot generate the
charge density e |ψ (x) |2 at the level of time average. One may still
object that this is merely an artificial result of the idealization of infinite
potential. But even in this ideal example, the model should also be able
to generate the charge density by means of some sort of ergodic motion
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145
of the electron. In fact, there is a very similar situation in double-slit
experiment. The wave function of a single electron passes through two
channels that are well separated in space. The wave function disappears
outside the channels for all practical purposes, and the electron can
only move inside the channels (otherwise the electron will be detected
outside the channels, which contradicts experimental observations).
Again, a classical ergodic model cannot explain this experiment.
There is a general objection to all classical ergodic models. Any
classical ergodic model will inevitably introduce a finite ergodic time
parameter, which is needed to generate the effective mass and charge
density, because it must take a finite time for the particle to continuously
move throughout all regions where the wave function is not zero.
However, it can be argued that no finite time scale is permitted to exist
for the ergodic motion. First of all, the existence of a finite time scale,
denoted by Tc, is inconsistent with the standard quantum theory, as
there is no such a time constant in the theory. Next, if there exists a time
scale Tc , then when the measuring time T of protective measurement
is shorter than Tc (i.e. T < Tc ), the measurement result will be not the
expectation value of a variable such as charge density, as no whole time
average can be obtained. This also contradicts the prediction of protective
measurement. As an extreme example, consider a spatial superposition
state ψ L +ψ R, where ψ L and ψ R are Gaussian wave packets and their
centers are well separated in space. When T < Tc, the particle has no
enough time to move throughout the whole regions including both L and
R. Then the result of a protective position measurement will be not the
expectation value of ψ L +ψ R, but the eigenvalue corresponding to ψL or
ψR. Moreover, the results distribution is also different from that predicted
by protective measurement. When T < Tc , the distribution of position
measurement results will concentrate near L and R, while according to
protective measurement, the distribution should concentrate near the
midpoint between L and R. In fact, for protective measurement, during
any period of time the pointer of the measuring device always shifts
by an amount proportional to the expectation value of the measured
variable, rather than to one of its eigenvalues. Thus the expectation
value can be associated with any short period of time. Certainly, pointer
shifts during short time intervals are practically unobservable since they
are much smaller than the uncertainty, and only when the total shift
accumulated during the whole period of measurement is much larger
than the width of the initial distribution it becomes observable.
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Experimental Physics I
Therefore, we conclude that the continuous ergodic motion
of particles cannot generate the effective mass and charge density
measurable by protective measurement, and the classical ergodic
models cannot be consistent with quantum mechanics. As a result, the
wave function cannot be a description of the continuous ergodic motion
of particles.
The Wave Function as a Description of Quantum Motion of Particles
The failure of classical ergodic models does not exclude all possible
ergodic motion of particles. We will argue that another different kind
of motion – random discontinuous motion can naturally generate the
effective mass and charge density measurable by protective measurement
and what the wave function describes is probably such quantum motion
of particles, which is essentially discontinuous and random.
If the motion of a particle is not continuous but discontinuous and
random, and the probability density of the particle being in certain
positions is proportional to the square of the absolute value of its wave
function at every instant, then the particle can readily move throughout
all possible regions where the wave function is nonzero during an
arbitrarily short time interval near a given instant.
This will solve the problems plagued by the classical ergodic models.
The discontinuous ergodic motion requires no existence of a finite time
scale. A particle undergoing discontinuous motion can also move from
one region to another spatially separated region, no matter whether
there is an infinite potential wall between them. Besides, discontinuous
motion can also solve the problems of infinite velocity and accelerating
radiation.
The reason is that no classical velocity and acceleration can be defined
for discontinuous motion, and energy and momentum will require new
definition and new understanding as in quantum mechanics. Thus it
seems that the discontinuous ergodic motion of particles can in principle
generate the effective mass and charge density measurable by protective
measurement, and thus the wave function is probably a description of
such random discontinuous motion of particles.
In some sense, the above interpretation of the wave function seems
to be an inevitable consequence of protective measurement. According
to protective measurement, a charged quantum system has effective
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147
mass and charge density distributing in space, proportional to the
square of the absolute value of its wave function. There are two possible
ways to explain the existence of the mass and charge density; one is to
take the wave function as a description of some kind of physical field,
and the mass and charge density of this field exists throughout space
simultaneously, the other is to take the wave function as a description
of some sort of ergodic motion of particles, and the effective mass
and charge density formed by such motion exists throughout space
in an essentially local way. The first view has been rejected because it
entails the existence of a remarkable electrostatic self-interaction that
contradicts experimental observations. Thus the wave function can only
be a description of some sort of ergodic motion of particles. Since the
classical ergodic models have also been excluded, the ergodic motion of
particles cannot be continuous and must be essentially discontinuous.
Besides, when considering the randomness of the results of conventional
quantum measurement, such motion must be also random. Therefore,
what the wave function describes can only be random discontinuous
motion of particles.
The wave function is a (complete) description for the motion of
particles, we can reach the random discontinuous motion in a more
direct way, independent of the analysis of protective measurement. If
the wave function ψ(x,t) is a description of the state of motion for a single
particle, then the quantity |ψ (x,t)|2 dx not only gives the probability
of the particle being found in an infinitesimal space interval dx near
position x at instant t, but also gives the objective probability of the
particle being there. This accords with the well-accepted assumption that
the probability distribution of the measurement outcomes of a property
is the same as the actual distribution of the property in the measured
state. Then at instant t the particle may appear in any location where
the probability density |ψ (x,t)|2 is nonzero, and during an infinitesimal
time interval near instant t , the particle will move throughout the
whole space where the wave function ψ(x,t) spreads, though it is still
in one position at each instant. Moreover, the density distribution of its
positions is equal to the probability density |ψ (x,t)|2. Obviously, this
kind of motion is essentially random and discontinuous.
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Experimental Physics I
Figure 18: The description of RDM of a single particle.
The strict mathematical description of random discontinuous
motion (RDM henceforth) can be obtained by using the measure theory.
Consider the motion state of a single particle in finite intervals Δt and
Δx near a space-time point (ti , xj ) as shown in Fig. 17. For RDM, the
position of the particle forms a random discontinuous point set in the
whole space for the time interval Δt near the instant ti. Accordingly,
there is a local discontinuous point A random discontinuous point set
can be defined as a set of points (t, x) in continuous space and time, for
which the function x(t) is discontinuous and random at all instants. The
definition of a discontinuous function is as follows. Suppose A is an open
set in ℜ (say an interval A = (a,b) , or A = ℜ ), and f : A→ ℜ is a function.
Then f is discontinuous at x ∈ A, if f is not continuous at x. Note that a
function f: A→ ℜ is continuous if and only if for every x ∈ A and every
real number ε > 0 , there set in the space interval Δx near the position
xj. The local discontinuous point set represents the motion state of the
particle in the finite intervals Δt and Δx near the space-time point (ti, xj ).
It is projection in the t-axis, namely the corresponding dense instant set
in the time interval Δt . Let W be the discontinuous trajectory or worldset of the particle and Q be the square region [xj, xj + Δx ] × [ti , ti + Δt ].
The dense instant set can be denoted by π t(W IQ) ⊂ ℜ, where πt is the
projection on the t-axis.
According to the measure theory, we can define the Lebesgue
measure:
(29)
Since the sum of the measures of all such dense instant sets in the
time interval Δt is equal to the length of the continuous time interval Δt,
we have:
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149
(30)
Then we can define the measure density:
(31)
The limit exists for a random discontinuous point set. This provides
a strict mathematical description of the point distribution situation for
the above local discontinuous point set. We call this measure density
position measure density.
Since the local discontinuous point set represents the motion state
of the particle, the position measure density ρ(x,t) will be a descriptive
quantity for the RDM of a single particle. It represents the relative
frequency of the particle appearing in an infinitesimal space interval dx
near position x during an infinitesimal interval dt near instant t. From
equation (31) we can see that ρ(x,t) satisfies the normalization relation,
namely
. Furthermore, we can define the position measure flux
density j(x,t) through the relation j(x,t) = ρ(x,t) ν (x,t) , where ν (x,t) is
the velocity of the local discontinuous point set. It describes the change
of the position measure density with time. Due to the conservation of
measure, ρ(x,t) and j(x,t) satisfy the following equation:
(32)
The position measure density ρ(x,t) and the position measure flux
density j(x,t) provide a complete description for the RDM of a single
particle.
It is very natural to extend the description of the motion of a single
particle to the motion of many particles. For the RDM state of N particles,
we can define a joint position measure density
. This
represents the relative probability of the situation in which particle 1 is
in position x1, particle 2 is in position x2,… , and particle N is in position
xN. In a similar way, we can define the joint position measure flux density
. It satisfies the joint measure conservation equation.
(33)
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Experimental Physics I
When these N particles are independent, the joint position measure
density
can be reduced to the direct product of the position
measure density of each particle, namely
. It is worth
noting that the joint position measure density
and the joint
position measure flux density
are not defined in the threedimensional real space, but defined in the 3N-dimensional configuration
space. As we will see later, these two quantities can further constitute
the wave function, and the many-body wave functions thus defined also
live on the 3N-dimensional configuration space.
With respect to the RDM of a particle, the motion of the particle is
completely discontinuous and random. The probability density for the
particle to appear at position x at instant t is ρ(x,t) . There is no evolution
law for the position state of a particle, and the trajectory function x(t) is
random and discontinuous at every instant. However, the discontinuity
of RDM is absorbed into the motion state of a particle, which is defined
during an infinitesimal time interval, by the descriptive quantities of
position measure density ρ(x,t) and position measure flux density j(x,t).
Therefore, the evolution law for the motion state of a particle will contain
no discontinuities and can be a continuous equation.
By assuming that the nonrelativistic evolution equation of RDM
is the Schrödinger equation, the wave function ψ(x,t) can be uniquely
expressed by the position measure density ρ(x,t) and the position
measure flux density j(x,t):
(34)
where
. Since ρ(x,t) and j(x,t) provide a complete
description of the RDM of a single particle, the wave function ψ(x,t) also
provides a complete description of the RDM of a single particle.
The new interpretation of the wave function in terms of RDM of
particles can be taken as a natural realistic alternative to the orthodox
view. According to the standard probability interpretation of the wave
function, the square of the absolute value of a N-particle wave function,
which can be denoted by
, gives the probability of
particle 1 being found in the infinitesimal interval dx1 near position x1
and particle 2 being found in the infinitesimal interval dx2 near position
x2, …, and particle N being found in the infinitesimal interval dxN near
position xN. By contrast, according to the new interpretation, the square
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151
of the absolute value of the wave function not only gives the probability
of a particle being found in certain locations, but also gives the objective
probability of the particle being there.
For example,
also represents the objective probability
of particle 1 being in the infinitesimal interval dx1 near position x1 and
particle 2 being in the infinitesimal interval dx2 near position 2 x2, … ,
and particle N being in the infinitesimal interval dxN near position xN.
Certainly, the transition process from “being” to “being found”, which
is closely related to the notorious quantum measurement problem, also
needs to be explained.
Properties of the Wave Function
The physical interpretation of the wave function in terms of a
probability implied that a function needs to satisfy some conditions,
before it is an acceptable wave function. We will now look at some more
properties of the solutions to the Schr¨odinger equation. For simplicity
we will restrict our attention to one-dimensional spatial problems,
although most results are applicable to higher dimensional problems.
The wave function is the solution of the eigenvalue equation
where
is the Hamiltonian operator and has the form
Over and above the conditions we will now show that, for finite
dψ
potentials, the first derivative of the wave function, dx should also be
single valued and continuous. To this end we rewrite the Schrödinger
equation in the form
dψ
A discontinuity in dx would mean that the LHS of this expression
would become infinite, which for finite V is unacceptable.
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Experimental Physics I
We will analyze the one-dimensional Schrödinger equation in
slightly greater detail, which will allow us to draw some other general
conclusions about the wave function. Consider a potential of the form
given in the figure below.
We will consider four different regions and will focus our attention
on the Schrödinger equation written in the form given in equation.
E < Vmin
In this region V −E is always positive, so that the second derivative
of the wave function, which represents the concavity, has the same sign
as the wave function. This may be a good time to step aside and remind
ourselves some simple things about how one sketches functions from a
knowledge of the derivatives.
The derivative of a function y = f(x), is the rate at which y changes
with respect to x. It defines the slope of the function’s graph at x and
provides an estimate of how much y changes when x is changed by a
small amount. If a function has a derivative over an interval, then it is
continuous over the interval and its graph over the interval is unbroken.
Even more information about the graph of a differentiable function
can be obtained from a knowledge of where the derivative is positive,
negative, and zero. Suppose that over an interval a function y = f(x) has
derivative at every point x, then f increases in the interval if
f ′(x) > 0
and decreases if ( )
. Moreover, if f ′ changes continuously from
positive to negative as we move from left to right through a point c, then
the value of f at c is a local maximum of f. But, a horizontal tangent does
not imply a maximum or a minimum. Take the function y = x3 which
crosses the horizontal tangent at x = 0 but keeps on rising. If you look
at the function closely, however, we find that the portions of the curves
in the region (−∞, 0) and (0, ∞) turn in different ways. To the left of the
origin the curve turns to the right, while to the right of the origin it turns
to the left. In other words, the left portion “bends downward” while the
right portion “bends upward.” More rigorously we would say that the
curve is concave down on the interval (−∞, 0) and concave up on (0, ∞).
The direction of concavity is determined by the sign of f ′′ being concave
down if f ′′ < 0 and concave up if f ′′ > 0. A point on the curve where the
concavity changes from up to down or vice versa is called a point of
inflection, f ′′ = 0.
f′ x < 0
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153
Figure 19: A typical one-dimensional potential showing a minimum at x0.
The potential has two classical turning points at x1 and x2 when the Vmin < E
< V− and a turning point x3 when V− < E < V+.
Coming back to the problem at hand, from the discussion in the last
paragraph we now know that function will be concave upwards when
ψ > 0 and concave downwards when ψ < 0 as is represented in Fig. 20.
Since V − E is positive for all x, it is clear that a solution which remains
finite everywhere cannot be found.
Figure 20: The curvature of the wavefunction when the energy is below Vmin
is of the same sign as the wavefunction and is hence concave upwards if ψ
is > 0 and concave downwards if ψ is < 0.
In fact, |ψ(x)| grows without limit as x → +∞ or/and as x → −∞. The
best we could do is to select from among the two linearly independent
solutions a function which approaches the x-axis asymptotically either
on the left or on the right, but then this solution will necessarily ‘blow
up’ on the other side. We conclude therefore that there is no acceptable
solution of the Schr¨odinger equation when E < V (x) for all x. classically
also this is not possible since this would correspond to negative kinetic
energy.
Vmin < E < V−
This region can be broken up into three parts: left of the classical
turning point x1, between the two turning points, and to the right of the
turning point x2. The classical turning points are those points at which
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Experimental Physics I
E = V, that is points where the kinetic energy becomes zero. Classically
at these points the particle would stop and turn around and no classical
motion is possible to the left of x1 and to the right of x2. In the first and
the third parts, V − E > 0, and hence the curvature of the wavefunction
is similar to that in region 0 and is represented by Fig. 21. This implies
that we cannot have oscillatory solutions in these two parts. Between
the two turning points V − E < 0 and hence the curvature is opposite to
the sign of the wavefunction. The function is concave downwards when
ψ > 0 and concave upwards when ψ < 0. The curvature in this region is
represented in Fig. 21. Thus in this part the wavefunction will exhibit
oscillatory behavior. For some values of the energy, say E’, the oscillatory
solution between the two turning points will connect smoothly with the
asymptotic solutions to the left of x1 and to the right of x2 as in the middle
part of Figure 21. For values of the energy either greater or lesser than
E’ the solutions are diverged to +∞ or −∞ when x → ∞. It is important
to realize that this smooth connection to give physically acceptable
solutions called eigenfunctions only happens for some values of energy,
which are called the eigenvalues. The number of such energies for
which reasonable solutions exist will depend on the characteristics of
the potential. Finally, we observe that we have obtained a fundamental
result: the quantization of the bound state energies, the determination
of which appears in Schrödinger’s mechanics as an eigenvalue problem.
Figure 21: For values of the energy greater or less than the eigenvalue E’ the
solutions of the Schrodinger equation give physically unreasonable results.
Note that there may be a number of E’s for which physically acceptable
results are obtained.
V − < E < V+
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155
In this energy range we note there is only one classical turning
point, at x = x3. No classical motion is allowed for x > x3 or else it would
be reflected at x3. The classical particle is unbound since it can move in
an infinite region of the x-axis. Quantum mechanically, we see that for x
< x3 there are two linearly independent solutions (since we are solving a
second order differential equation) both of which are oscillatory (reasons
evident by looking at Fig. 21), one corresponding to the particle moving
to the left and the other to it moving to the right. For values of x > x3
also there will be two solutions, one which tends to zero as x → ∞ and
the other which is unbounded as x approaches ∞. Obviously the only
physically acceptable solution is the former. The continuity conditions
of ψ and dψ/dx at x3 yields two conditions which determine a unique
linear combination of the two independent oscillatory solutions in the
region x < x3. It is important to note that in this region ’smooth matching’
at x = x3 is possible for every energy. We conclude that in this interval the
allowed energies form a continuum.
E > V+
In this region there are no classical turning points because V − E is
negative for all values of x. For reasons discussed with regard to Fig. 21 we
expect the solutions to be oscillatory. There are two linearly independent
solutions corresponding to the two directions that the particle could be
moving for every value of the energy. Thus in this region the energy
values are continuous and doubly degenerate.
Infinite Potential Energy
So far our discussions have been restricted to the case where the
potential energy was finite. We would like to look finally at the situation
when the potential energy becomes infinite. Indeed, because a particle
cannot have infinite potential energy, it cannot penetrate into a region of
space where V = ∞. We focus our attention on the Schr¨odinger equation
written in the form given in equation 1. Both E and ψ on the RHS are
finite while the LHS has one term that is infinite. This equality can only
hold when ψ is identically zero, which implies that the wavefunction
goes to zero when the potential becomes infinite. We also note that
since V exhibits a second order discontinuity (infinite jump) across the
border, d2ψ/dx2 will also exhibit this discontinuity, so that dψ/dx will be
discontinuous there.
One last thing that we will look at is the relation that exists between
the energy of a state and the number of nodes. Even without carrying
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Experimental Physics I
out detailed calculations, it is often possible to make general conclusions
concerning the ordering of the states by considering their nodal structure.
We consider a one-dimensional potential, one-particle system with the
Hamiltonian
where
and V (x) is some function of x depending only
upon the spatial coordinates. If the potential supports the existence of
bound states we can show that:
•
•
The ground state wave function does not change sign (has no
nodes)
The bound state with n sign changes has a lower energy than
the state with n + 1 sign changes.
Consider first the functions ψ0 and ψ1, which are the Eigen states
of
, with zero and one node respectively and energies E0 and E1
respectively. Let the boundaries of the system be a and b and let ψ1
undergo a zero-crossing at c. Consider the region a < x < c so that both ψ0
and ψ1 are positive. From the eigenvalue equation we then have
Taking the difference of these two expressions we get
The Hermiticity of
implies that the term in parenthesis is zero
(the integrand is generally not zero). To estimate the sign of E1 − E0,
multiply by ψ1ψ0, then integrate from a to c.
We then obtain:
where
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157
Consider the first term in B. Integrating this by parts we get
Combining these results we obtain
Because
we obtain
that is,
Thus the node less wavefunction has a lower energy than the wave
function with one node. A similar proof shows that ψ0 has a lower energy
than any wave function with more than one node. Hence the ground
state of the system has no nodal points.
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Experimental Physics I
Similarly, the above proof can be applied to the comparison of
En and En+1, that is, the energies for wave functions having n and n +
1 nodes, respectively. The result is that and hence the eigenstates of a
system have energies increasing in the same sequence as the number of
nodes.
Formal Properties of Wave Functions
It is very useful to have a broader, more precise framework for
analyzing quantum properties. This is done in terms of Hilbert spaces.
Hilbert Space of Linear Vectors
Hilbert space is a generalization of Euclidian space to finite
or infinite number of dimensions and real or complex vectors. It is
employed for the mathematical description of quantum mechanics.
The description of quantum states as vectors in Hilbert space gives the
model an algebraic structure with means to measure distances, lengths
and angles. Furthermore, it guarantees the use of the techniques of
calculus on functions in H.
Every Hilbert space H has an orthonormal basis. This means that
any vector a in an n-dimensional Hilbert space can be constructed from
n linear independent basis vectors ei with a normalized absolute value of
1. The basis is required to be complete with no points in space missing.
There exists an inner product in H, which associates a pair of vectors
a and b with a real or complex number.
The inner product is linear in its first argument and turns into its
complex conjugate when the argument order is reversed. The inner
product of a vector a with itself is always larger than 0 and vanishes if a
is the null vector.
A norm is defined for a Hilbert space, which allows for the
measurement of a vector’s length, the distance between points, and
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159
the angle between two vectors. Hilbert space is therefore both an inner
product space and a complete metric space.
For real numbers the inner product is the multiplication of two
numbers, for complex numbers the inner product is the multiplication
of one number z1 with the complex conjugate of the second number, z2*.
In Euclidian space this operation is the dot product.
In Euclidian geometry, the three Cartesian coordinates ex, ey and
ez span a three dimensional Hilbert space. In quantum mechanics the
n eigenstates of a quantum mechanical system span an n-dimensional
Hilbert space, where the dimensionality of the Hilbert space is equal
to the number of eigenstates. n may be infinite. For isolated spins the
dimensionality of the Hilbert space is finite, 2 in the case of single
spins 1/2 and 4 in the case of two coupled spins. Interactions with
the surroundings, i.e. in spin relaxation, increases the number of
possible states. Relaxation is therefore often treated in a semi-classical
approximation. Each normalized basis vector corresponds to a pure
eigenstate, which is fully populated.
If the system is in a superposition state, the vector components are
the coefficients in a linear combination of eigenstates. The absolute value
of a vector must always be unity to fulfil the requirements of quantum
mechanics that the described object exists.
Coherences are superposition’s, in which the coherent states are
equally populated and running in phase. The phase factor is generally
a complex number.
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Experimental Physics I
The complex vector components of mixed states do not necessarily
have a straightforward, graphical interpretation, but fulfil the eigenvalue
equations for the respective cartesian operators.
The metric of Hilbert space makes it possible to measure observables
of the quantum mechanical state, as well as the probabilities with which
they are measured.
Liouville Space
Since they work on vectors in a complex metric space, each operator
has a representation as a square, hermitian matrix. Operators that work
on vectors in Hilbert space can in turn be subject to transformations.
The density operator for example, which gives the probability density
for a state in an ensemble, evolves over time due to precession, coupling
and relaxation. Transformations of the operators of Hilbert space form
an analogous algebra called Liouville space with super operators
working on the operators in H. The super operators also have a matrix
representation that can be found by turning the n x n operator matrices
column-wise into a single column vector. The super operators are n2 x
n2 matrices. The application of a super operator on an operator vector
in Liouville space corresponds to a two-sided matrix multiplication in
Hilbert space.
The Hamiltonian in Liouville space can be constructed by taking
the commutator of the Hamiltonian in Hilbert space with the identity
matrix.
Super operators are denoted by a double hat. Vectors in Liouville
space are written in analogy to the bra–ket notation with round brackets.
The equation of motion in super operator space is called the Liouvillevon Neumann equation.
Waves and Sound
161
Hermitian Operators
Most operators in quantum mechanics are of a special kind called
Hermitian. An operator is called Hermitian when it can always be
flipped over to the other side if it appears in an inner product:
f Ag = Af g always iff A is Hermitian
That is the definition, but Hermitian operators have the following
additional special properties:
•
•
•
They always have real eigenvalues, not involving i= −1 .
(But the eigenfunctions, or eigenvectors if the operator is a
matrix, might be complex.) Physical values such as position,
momentum, and energy are ordinary real numbers since they
are eigenvalues of Hermitian operators {N.3}.
Their eigenfunctions can always be chosen so that they
are normalized and mutually orthogonal, in other words,
an orthonormal set. This tends to simplify the various
mathematics a lot.
Their eigenfunctions form a complete set. This means that
any function can be written as some linear combination of
the eigenfunctions. (There is a proof in derivation {D.8} for
an important example. But see also {N.4}.) In practical terms,
it means that you only need to look at the eigenfunctions to
completely understand what the operator does.
In the linear algebra of real matrices, Hermitian operators are simply
symmetric matrices. A basic example is the inertia matrix of a solid body
in Newtonian dynamics. The orthonormal eigenvectors of the inertia
matrix give the directions of the principal axes of inertia of the body.
An orthonormal complete set of eigenvectors or eigenfunctions is
an example of a so-called “basis.” In general, a basis is a minimal set of
vectors or functions that you can write all other vectors or functions in
terms of.
ˆˆ
For example, the unit vectors i, j, and k̂ are a basis for normal threedimensional space. Every three-dimensional vector can be written as a
linear combination of the three.
The following properties of inner products involving Hermitian
operators are often needed, so they are listed here:
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Experimental Physics I
∗
If A is Hermitian : g Af = f Ag , f Af is real.
The first says that you can swap f and g if you take the complex
conjugate. (It is simply a reflection of the fact that if you change the sides
in an inner product, you turn it into its complex conjugate. Normally,
that puts the operator at the other side, but for a Hermitian operator, it
does not make a difference.) The second is important because ordinary
real numbers typically occupy a special place in the grand scheme of
things. (The fact that the inner product is real merely reflects the fact that
if a number is equal to its complex conjugate, it must be real; if there was
an i in it, the number would change by a complex conjugate.)
3.6 STANDING WAVES ON STRINGS
To observe standing waves on a stretched string and to verify the
formula relating the wave speed to the tension and mass per length of
the string.
Figure 22: String vibrator, string, set of slotted weights, weight holder, pulley and clamp, meter stick, analytical balance.
3.6.1 Theory
Standing waves are produced by the interference of two waves of
the same wavelength, speed of propagation and amplitude, travelling
in opposite directions through the same medium. If one end of a light,
flexible string is attached to a vibrator and the other end passes over a
fixed pulley to a weight holder, the waves travel down the string to the
pulley and are then reflected, producing a reflected wave moving in the
opposite direction. The vibration of the string is then a composite motion
resulting from the combined effect of the two oppositely directed waves.
If the proper relationship exists between the frequency, the length and
the tension, a standing wave is produced and when the conditions are
such as to make the amplitude of the standing wave a maximum, the
Waves and Sound
163
system is said to be in resonance. A standing wave has points of zero
displacement (due to destructive interference) and points of maximum
displacement (due to constructive interference). The positions of no
vibration are called nodes (N) and the positions of maximum vibration
are called antinodes (A). The segment between two nodes is called a
loop. Standing waves with one, two, three and four loops are given
below.
The solid line represents the form of the string at an instant of
maximum displacement and the dotted line represents the configuration
one half-period later when the displacements are reversed. In each case
λ = 2l, where λ is the wavelength and l are the distance between two
nearby nodes. For a string with both ends fixed the allowed wavelengths
for standing waves can only take fixed values related to the length L of
the string, as can be seen in the figure above. If one changes the tension
in a vibrating string, the number of loops between the ends of the string
change. As a result the distance between neighboring nodes changes,
thus producing a change in wavelength. The speed of the wave can be
obtained if the frequency f is known and the wave length λ has been
measured:
v = λf.
The frequency is fixed by the string vibrator; the wavelength can
only take on fixed values related to the length of the string, as shown
in the figure above. Thus, standing waves can only exist for particular
values of v that is controlled in this experiment by the tension of the
string. The velocity of the wave is given by the Mercenne’s law:
(35)
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Experimental Physics I
where m is the mass per length of the string and T is the tension.
The tension of the string (in newtons) equals the total hanging mass M
times the gravitational acceleration g = 9.8 m/s2 , that is, T = Mg. The
main objective of the work is to compare the experimental value of the
wave speed given by Eq. (1) and its theoretical value of Eq. (35).
Experimental Setup
A string is attached to a vibrator made of steel and then passed
over a small pulley. The coil producing the alternating magnetic field
acting on the vibrator is being fed by the standard ac current with a
fixed frequency of 60 Hz. The attraction force exerted on the vibrator is
proportional to the square of the magnetic field and thus of the electric
current in the cirquit. As the result, the vibrator is vibrating at the
double frequency, 120 Hz. The weight on the hanger has to be adjusted
to achieve the tension T and thus the wave speed v at which a standing
wave is clearly visible. As the range of the tension is limited, not all
kinds of standing waves can be observed without changing the string
length L. For instance, to observe lower overtones, the tension T can
be increased or L can be decreased. The best way to measure the wave
length λ is to measure the distance d between the end of the string at the
pulley (where there is a node) and the node closest to the vibrator and
count the number n of loops within this region. Then
Keep in mind that there is no node directly at the vibrator, thus
using L to obtain λ as shown in the figure above will result in errors.
Procedure
1. Measure the length of the loose string (not the one attached to
the vibrator) and then measure its mass using the analytical
balance to the nearest milligram. Calculate the mass per unit
lengh m for the string. 2. Suspend a light weight holder from
the string and adjust the load until the string vibrates with
maximum amplitude. Record the load in kilograms, including
the weight of the holder. Measure λ as explained above and
record it in the table together with the number of loops you
observe. 3. Repeat the observations with load (and maybe the
string length) adjusted to give other numbers of loops. Take
measurements for at least three different numbers of loops.
Waves and Sound
165
Data
f= (frequency)
Length= m Mass= kg m=mass/length= kg/m (length of loose string)
(mass of loose string) (mass per length of string)
Analysis
1) Calculate the mass per unit length and express it in kg/m. 2)
Calculate the velocity of the wave on the string using equations (1) and
(2) and compare the results. Calculate the percent discrepancy. Fill all of
this in the data table. 3) In your conclusion discuss your results. 4) If the
frequency of the vibrator were 240 Hz, calculate theoretically how much
tension is necessary to produce a standing wave of two loops. Consider
that the string is fixed at its two ends, has a length of 1 meter and has the
same mass per length as determined before.
3.7 TO STUDY LONGITUDINAL SOUND WAVES
CREATED IN AN AIR COLUMN OF VARIABLE
LENGTH
Sound is a mechanical wave that results from the back-and-forth
vibration of the particles of the medium through which the sound wave
is moving. If a sound wave is moving from left to right through air,
then particles of air will be displaced both rightward and leftward as the
energy of the sound wave passes through it. The motion of the particles
is parallel (and anti-parallel) to the direction of the energy transport.
This is what characterizes sound waves in air as longitudinal waves.
3.7.1 Compressions and Rarefactions
A vibrating tuning fork is capable of creating such a longitudinal wave.
As the tines of the fork vibrate back and forth, they push on neighboring
166
Experimental Physics I
air particles. The forward motion of a tine pushes air molecules horizontally to the right and the backward retraction of the tine creates a lowpressure area allowing the air particles to move back to the left.
Because of the longitudinal motion of the air particles, there are
regions in the air where the air particles are compressed together and
other regions where the air particles are spread apart. These regions are
known as compressions and rarefactions respectively. The compressions
are regions of high air pressure while the rarefactions are regions of
low air pressure. The diagram below depicts a sound wave created
by a tuning fork and propagated through the air in an open tube. The
compressions and rarefactions are labeled.
labeled.
The wavelength of a wave is merely the distance that a disturbance
travels along the medium in one complete wave cycle. Since a wave repeats
its pattern once every wave cycle, the wavelength is sometimes referred
to as the length of the repeating patterns - the length of one complete
wave. For a transverse wave, this length is commonly measured from
one wave crest to the next adjacent wave crest or from one wave trough
to the next adjacent wave trough. Since a longitudinal wave does not
contain crests and troughs, its wavelength must be measured differently.
A longitudinal wave consists of a repeating pattern of compressions
and rarefactions. Thus, the wavelength is commonly measured as the
distance from one compression to the next adjacent compression or the
distance from one rarefaction to the next adjacent rarefaction.
Waves and Sound
167
What is a Pressure Wave?
Since a sound wave consists of a repeating pattern of high-pressure
and low-pressure regions moving through a medium, it is sometimes
referred to as a pressure wave. If a detector, whether it is the human
ear or a man-made instrument, were used to detect a sound wave, it
would detect fluctuations in pressure as the sound wave impinges upon
the detecting device. At one instant in time, the detector would detect
a high pressure; this would correspond to the arrival of a compression
at the detector site. At the next instant in time, the detector might detect
normal pressure. And then finally a low pressure would be detected,
corresponding to the arrival of a rarefaction at the detector site. The
fluctuations in pressure as detected by the detector occur at periodic
and regular time intervals. In fact, a plot of pressure versus time would
appear as a sine curve. The peak points of the sine curve correspond
to compressions; the low points correspond to rarefactions; and the
“zero points” correspond to the pressure that the air would have if there
were no disturbance moving through it. The diagram below depicts the
correspondence between the longitudinal nature of a sound wave in
air and the pressure-time fluctuations that it creates at a fixed detector
location.
The above diagram can be somewhat misleading if you are not
careful. The representation of sound by a sine wave is merely an attempt
to illustrate the sinusoidal nature of the pressure-time fluctuations. Do
not conclude that sound is a transverse wave that has crests and troughs.
Sound waves traveling through air are indeed longitudinal waves with
compressions and rarefactions. As sound passes through air (or any
fluid medium), the particles of air do not vibrate in a transverse manner.
Do not be misled - sound waves traveling through air are longitudinal
waves.
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Experimental Physics I
3.7.2 Open-End Air Columns
Many musical instruments consist of an air column enclosed inside
of a hollow metal tube. Though the metal tube may be more than a
meter in length, it is often curved upon itself one or more times in order
to conserve space. If the end of the tube is uncovered such that the air at
the end of the tube can freely vibrate when the sound wave reaches it,
then the end is referred to as an open end. If both ends of the tube are
uncovered or open, the musical instrument is said to contain an openend air column. A variety of instruments operate on the basis of openend air columns; examples include the flute and the recorder. Even
some organ pipes serve as open-end air columns.
3.7.3 Standing Wave Patterns for the Harmonics
As has already been mentioned, a musical instrument has a set
of natural frequencies at which it vibrates at when a disturbance is
introduced into it. These natural frequencies are known as the harmonics
of the instrument; each harmonic is associated with a standing wave
pattern. In Lesson 4 of Unit 10, a standing wave pattern was defined
as a vibrational pattern created within a medium when the vibrational
frequency of the source causes reflected waves from one end of the
medium to interfere with incident waves from the source in such a
manner that specific points along the medium appear to be standing still.
In the case of stringed instruments (discussed earlier), standing wave
patterns were drawn to depict the amount of movement of the string
at various locations along its length. Such patterns show nodes - points
of no displacement or movement - at the two fixed ends of the string.
In the case of air columns, a closed end in a column of air is analogous
to the fixed end on a vibrating string. That is, at the closed end of an
air column, air is not free to undergo movement and thus is forced into
assuming the nodal positions of the standing wave pattern. Conversely,
air is free to undergo its back-and-forth longitudinal motion at the open
end of an air column; and as such, the standing wave patterns will depict
antinodes at the open ends of air columns.
So the basis for drawing the standing wave patterns for air columns
is that vibrational antinodes will be present at any open end and
vibrational nodes will be present at any closed end. If this principle is
applied to open-end air columns, then the pattern for the fundamental
frequency (the lowest frequency and longest wavelength pattern) will
have antinodes at the two open ends and a single node in between. For
Waves and Sound
169
this reason, the standing wave pattern for the fundamental frequency
(or first harmonic) for an open-end air column looks like the diagram
below.
The distance between antinodes on a standing wave pattern is
equivalent to one-half of a wavelength. A careful analysis of the diagram
above shows that adjacent antinodes are positioned at the two ends of
the air column. Thus, the length of the air column is equal to one-half of
the wavelength for the first harmonic.
The standing wave pattern for the second harmonic of an openend air column could be produced if another antinode and node was
added to the pattern. This would result in a total of three antinodes and
two nodes. This pattern is shown in the diagram below. Observe in the
pattern that there is one full wave in the length of the air column. One
full wave is twice the number of waves that were present in the first
harmonic. For this reason, the frequency of the second harmonic is two
times the frequency of the first harmonic.
And finally, the standing wave pattern for the third harmonic of
an open-end air column could be produced if still another antinode
and node were added to the pattern. This would result in a total of four
antinodes and three nodes. This pattern is shown in the diagram below.
Observe in the pattern that there are one and one-half waves present in
the length of the air column. One and one-half waves is three times the
number of waves that were present in the first harmonic. For this reason,
the frequency of the third harmonic is three times the frequency of the
first harmonic.
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Experimental Physics I
3.7.4 Length-Wavelength Relationships
The process of adding another antinode and node to each consecutive
harmonic in order to determine the pattern and the resulting length-wavelength
relationship could be continued. If doing so, it is important to keep antinodes
on the open ends of the air column and to maintain an alternating pattern of
nodes and antinodes. When finished, the results should be consistent with the
information in the table below. The relationships between the standing wave
pattern for a given harmonic and the length-wavelength relationships for open
end air columns are summarized in the table below.
Harm.
#
# of
Waves in
Air Column
# of
Nodes
# of
Antinodes
LengthWavelength
Relationship
1
1/2
1
2
Wavelength = (2/1)*L
2
1 or 2/2
2
3
Wavelength = (2/2)*L
3
3/2
3
4
Wavelength = (2/3)*L
4
2 or 4/2
4
5
Wavelength = (2/4)*L
5
5/2
5
6
Wavelength = (2/5)*L
Problem-Solving Scheme
Now the aim of the above discussion is to internalize the mathematical
relationships for open-end air columns in order to perform calculations
predicting the length of air column required to produce a given natural
frequency. And conversely, calculations can be performed to predict the
natural frequencies produced by a known length of air column. Each
of these calculations requires knowledge of the speed of a wave in air
(which is approximately 340 m/s at room temperatures). The graphic
below depicts the relationships between the key variables in such
calculations. These relationships will be used to assist in the solution to
problems involving standing waves in musical instruments.
To demonstrate the use of the above problem-solving scheme,
consider the following example problem and its detailed solution.
Waves and Sound
171
EXERCISE
Answer the following questions
1.
Explain the wave motion.
2.
Discuss about vibrating strings.
3.
Describe the properties of sound.
4.
Define the characteristics of sound waves.
5.
Discuss about basic concepts of vibration.
6.
Focus on vibration measurement.
7.
Understanding an air column of variable length.
MULTIPLE CHOICE QUESTIONS
Tick the correct answer:
1. ________ is a wave motion of altering pressure.
a.
Compression
b.
Sound
c.
Vibration
d.
None is correct
2. Low frequencies produce bass tone.
a.
True
b.
False
3. High frequencies produce ________ tone.
a.
bass
b.
treble
c.
complex
d.
pure
4. A point of maximum pressure in a sound wave is called
compression.
a.
True
b.
False
5. The minimum pressure in a sound wave is called ________.
a.
fraction
b.
rarefaction
c.
vibration
d.
wavelength
172
Experimental Physics I
6. Loudness is an intensity of a soundwave.
a.
True
b.
False
7. The formula to calculate the wavelength of sound wave is
__________.
a.
λ=F/V
b.
λ = FV
c.
λ=V/F
d.
all are correct
8. The human ears detects sound waves by eardrum.
a.
True
b.
False
9. The velocity of sound wave in the air is __________.
a.
1130 feet per seconds
b.
344.3 feet per seconds
c.
1280 feet per seconds
d.
a & b are correct
10. The unit of frequency wave is Hertz or cycles/sec.
a.
True
b.
False
ANSWERS
1. (b)
2. (a)
3. (b)
4. (a)
5. (b)
6. (a)
7. (c)
8. (a)
9. (d)
10. (a)
REFERENCES
1.
2.
3.
Acoustical Society of America. “PACS 2010 Regular Edition—
Acoustics Appendix”. Archived from the original on 14 May 2013.
Retrieved 22 May 2013.
“The Propagation of sound”. Archived from the original on 30
April 2015. Retrieved 26 June 2015.
“What Does Sound Look Like?”. NPR. YouTube. Archived from the
original on 10 April 2014. Retrieved 9 April 2014.
Waves and Sound
4.
5.
6.
7.
8.
9.
173
Handel, S. (1995). Timbre perception and auditory object
identification Archived 2020-01-10 at the Wayback Machine.
Hearing, 425–461.
“Scientists find upper limit for the speed of sound”. Archived from
the original on 2020-10-09. Retrieved 2020-10-09.
Trachenko, K.; Monserrat, B.; Pickard, C. J.; Brazhkin, V. V.
(2020). “Speed of sound from fundamental physical constants”.
Science Advances. 6 (41): eabc8662. arXiv:2004.04818.
Bibcode:2020SciA....6.8662T. doi:10.1126/sciadv.abc8662. PMC
7546695. PMID 33036979.
“The American Heritage Dictionary of the English Language”
(Fourth ed.). Houghton Mifflin Company. 2000. Archived from the
original on June 25, 2008. Retrieved May 20, 2010.
Burton, R.L. (2015). The elements of music: what are they, and who
cares? Archived 2020-05-10 at the Wayback Machine In J. Rosevear
& S. Harding. (Eds.), ASME XXth National Conference proceedings.
Paper presented at: Music: Educating for life: ASME XXth National
Conference (pp. 22–28), Parkville, Victoria: The Australian Society
for Music Education Inc.
Nemiroff, R.; Bonnell, J., eds. (19 August 2007). “A Sonic Boom”.
Astronomy Picture of the Day. NASA. Retrieved 26 June 2015.
INDEX
A
Acousticians 108
Amplitude (Dynamics) 107
Angular acceleration 113
C
Calorimeter 52, 53, 54, 55, 57, 58,
59, 68, 69, 96
Characteristics of Sound Waves
109
Chlamydomonas 3, 4, 5, 8, 16, 17,
36
Chromoplast 12
compressions and rarefactions
100, 101, 103, 106, 109, 110,
166, 167
Constant Flux Calorimeter 54
Convective heat transfer 79, 80,
81, 82, 95, 96
Convective heat transfer coefficients 80
D
Damped vibration 117
Diffusion dominates 78
Displacement 112, 118, 119, 123,
124, 125, 126, 127, 133, 134
disturbance of matter 99
Double-membrane-bounded mitochondria 3
E
Electromotive force (EMF) 132
equilibrium positions 100
F
Flagellar membrane 7, 8, 10
Flagellum 4, 5, 6, 7, 9, 10, 11, 16,
17, 18
Fluid system 114
Forced convection 93, 95
Forced vibration 116
Free Convection 79, 80
Free vibration 114, 115, 117
G
Gauge pressure 101
Green alga 3, 4, 5, 7, 16
H
Heat Balance Calorimeter 54
Heat capacity 55
Heat Flow Calorimeter 53
I
Infrasonic waves 101
Isotropic material 45
Isotropic materials 43, 44
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Experimental Physics I
K
Kinesin proteins 5
Kinetic energy 112, 113, 114, 118
L
Latent heat 40, 60, 61, 97
Linear expansion 40, 41, 42, 43,
48, 49, 50, 96
longitudinal. 100
low-pressure regions 100, 106,
167
M
measure sound 110, 111
Mechanical Sound Waves 105
N
Negative thermal expansion 46
O
oscillating 100, 119
oscillation 105, 107, 112, 113, 115,
116, 117, 119, 120
P
Particle-particle distance 41
Phaeophyceae 12, 28, 29
Phycology 1, 2, 35
Piezoelectric 124, 125, 128, 131
Plasma membrane 3, 4, 5, 8, 13,
16, 17, 30, 32
positive x-direction 100, 101
Power Compensation 54
Pressure Sound Waves 106
Prokaryotic cells 2
Propagation of Sound Waves 103
Properties of Sound 107, 108
Q
Quantum mechanics 134, 135,
137, 139, 142, 143, 144, 146,
158, 159, 161
Quantum system 134, 135, 136,
137, 138, 139, 140, 141, 142,
146
S
Schrödinger equation 136, 143,
150
simple harmonic motion 100, 118,
119
Sound Intensity 111, 112
sound travels 103, 104
sound waves 99, 100, 101, 102,
103, 104, 105, 106, 107, 108,
109, 111, 112, 165, 167, 171,
172
Sound waves 100, 101, 102, 104,
105, 106, 110, 167
Surface shear stress 84
T
Thermal expansion 40, 41
Thermal expansion coefficient 43,
44, 45, 96
Thermal linear expansion 40
Thermal stress 51
Thylakoids 3, 12, 13, 14, 15, 23, 24
transverse 100, 105, 106, 124, 127,
166, 167
U
ultrasonic waves 101, 102
Index
177
V
W
vibrating air particles 102
Vibratory system 112, 120, 121
Volumetric coefficient 44, 45
Volumetric thermal expansion 44,
45
Wave function 134, 135, 136, 137,
138, 140, 141, 142, 143, 144,
145, 146, 147, 150, 151