Fizika I
Hárs, György
Dobos, Gábor
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Fizika I
írta Hárs, György és Dobos, Gábor
Publication date 2014
Szerzői jog © 2014 Hárs György, Dobos Gábor
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Tartalom
Fizika I ................................................................................................................................................ 1
1. Introduction - György Hárs ................................................................................................... 1
2. 1 Kinematics of a particle - György Hárs .............................................................................. 1
2.1. 1.1 Rectilinear motion .............................................................................................. 1
2.1.1. 1.1.1 Uniform Rectilinear Motion ................................................................ 2
2.1.2. 1.1.2 Uniformly Accelerated Rectilinear Motion ......................................... 2
2.1.3. 1.1.3 Harmonic oscillatory motion ............................................................... 3
2.2. 1.2 Curvilinear motion .............................................................................................. 4
2.2.1. 1.2.1 Projectile motion ................................................................................. 5
2.2.2. 1.2.2 Circular motion ................................................................................... 6
2.2.3. 1.2.3 Areal velocity ...................................................................................... 9
3. 2 Dynamics of a Particle - György Hárs ............................................................................... 9
3.1. 2.1 Inertial system ................................................................................................... 10
3.2. 2.2 The mass ........................................................................................................... 10
3.3. 2.3 Linear momentum p .......................................................................................... 10
3.4. 2.4 Equation of motion: .......................................................................................... 12
3.5. 2.5 The concept of weight ...................................................................................... 14
3.6. 2.6 The concept of work in physics ........................................................................ 15
3.7. 2.7 Power ................................................................................................................ 16
3.8. 2.8 Theorem of Work (Kinetic energy) .................................................................. 17
3.9. 2.9 Potential energy ................................................................................................ 19
3.10. 2.10 Conservation of the mechanical energy ........................................................ 21
3.11. 2.11 Energy relations at harmonic oscillatory motion .......................................... 22
3.12. 2.12 Angular momentum ...................................................................................... 23
3.13. 2.13 Torque ........................................................................................................... 23
3.14. 2.14 Central force field ......................................................................................... 24
4. 3 Dynamics of system of particles - György Hárs .............................................................. 26
4.1. 3.1 Momentum in system of particles ..................................................................... 26
4.1.1. 3.1.1 Collisions .......................................................................................... 28
4.1.2. 3.1.2 Missile motion ................................................................................... 32
4.2. 3.2 Angular momentum in system of particles ....................................................... 33
4.2.1. 3.2.1 The skew rotator ................................................................................ 35
4.2.2. 3.2.2 The pirouette dancer (The symmetrical rotator) ................................ 36
4.3. 3.3 Discussion of the total kinetic energy in the system of particles ..................... 38
5. 4 Dynamics of rigid body - György Hárs ............................................................................ 40
5.1. 4.1 Moment of inertia ............................................................................................. 40
5.2. 4.2 Equation of motion of the rigid body: ............................................................... 45
5.2.1. 4.2.1 Demonstration example 1. ................................................................. 46
5.2.2. 4.2.2 Demonstration example 2. ................................................................. 47
5.3. 4.3 Kinetic energy of the rigid body ....................................................................... 49
6. 5 Non-inertial (accelerating) reference frames - György Hárs ............................................ 49
6.1. 5.1 Coordinate system with translational acceleration ............................................ 50
6.2. 5.2 Coordinate system in uniform rotation ............................................................. 52
6.2.1. 5.2.1 Earth as a rotating coordinate system ................................................ 54
7. 6 Oscillatory Motion - Gábor Dobos ................................................................................... 62
7.1. 6.1 The simple harmonic oscillator ......................................................................... 63
7.1.1. 6.1.1 Complex representation of oscillatory motion .................................. 64
7.1.2. 6.1.2 Velocity and acceleration in oscillatory motion ................................ 65
7.2. 6.2 Motion of a body attached to a spring .............................................................. 66
7.3. 6.3 Simple pendulum .............................................................................................. 68
7.4. 6.4 Energy in simple harmonic motion ................................................................... 70
7.5. 6.5 Damped oscillator ............................................................................................. 71
7.6. 6.6 Forced oscillations ............................................................................................ 73
7.7. 6.7 Superposition of simple harmonic oscillations ................................................. 76
7.7.1. 6.7.1 Same frequency, same direction ........................................................ 76
7.7.2. 6.7.2 Different frequency, same direction .................................................. 77
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7.7.3. 6.7.3 Lissajous figures ................................................................................ 78
7.7.4. 6.7.4 Fourier analysis ................................................................................. 79
8. 7 Waves - Gábor Dobos ...................................................................................................... 81
8.1. 7.1 Sine wave .......................................................................................................... 82
8.2. 7.2 Transverse wave on a string ............................................................................. 83
8.3. 7.3 Energy transport by mechanical waves ............................................................. 84
8.4. 7.4 Group velocity .................................................................................................. 86
8.5. 7.5 Wave packets .................................................................................................... 87
8.6. 7.6 Standing waves ................................................................................................. 89
8.7. 7.7 The Doppler Effect ........................................................................................... 92
9. 8 First law of thermodynamics and related subjects - György Hárs .................................... 93
9.1. 8.1 Ideal gas equation ............................................................................................. 94
9.2. 8.2 The internal energy of the gas U ....................................................................... 94
9.3. 8.3 The p-V diagram ............................................................................................... 97
9.4. 8.4 Expansion work of the gas ................................................................................ 97
9.5. 8.5 First law of thermodynamics ............................................................................ 97
9.5.1. 8.5.1 Isochoric process ............................................................................... 98
9.5.2. 8.5.2 Isobaric process ................................................................................. 99
9.5.3. 8.5.3 Isothermal process ........................................................................... 100
9.5.4. 8.5.4 Adiabatic process ............................................................................ 101
9.6. 8.6 Summary of the molar heat capacitances ........................................................ 103
9.7. 8.7 The Carnot cycle ............................................................................................. 103
10. 9 The entropy and the second law of thermodynamics - György Hárs ........................... 106
10.1. 9.1 The entropy ................................................................................................... 106
10.2. 9.2 The isentropic process .................................................................................. 109
10.3. 9.3 The microphysical meaning of entropy ........................................................ 110
10.4. 9.4 Gay-Lussac experiment ................................................................................ 111
10.4.1. 9.4.1 Phenomenological approach ......................................................... 111
10.4.2. 9.4.2 Statistical approach ....................................................................... 112
10.5. 9.5 The Boltzmann equation ............................................................................... 114
10.6. 9.6 Approximate formula
a sketch of proof: ................... 114
10.7. 9.7 Equalization process ..................................................................................... 115
10.7.1. 9.7.1 Equalization between gaseous components ................................... 115
10.7.2. 9.7.2 Equalization of non-gaseous materials without phase transition ... 120
10.7.3. 9.7.3 Ice cubes in the water .................................................................... 121
10.8. 9.8 The second law of thermodynamics ............................................................. 123
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1. Introduction - György Hárs
Present work is the summary of the lectures held by the author at Budapest University of Technology and
Economics. Long verbal explanations are not involved in the text, only some hints which make the reader to
recall the lecture. Refer here to the book: Alonso/Finn Fundamental University Physics, Volume I where more
details can be found.
Physical quantities are product of a measuring number and the physical unit. In contrast to mathematics, the
accuracy or in other words the precision is always a secondary parameter of each physical quantity. Accuracy is
determined by the number of valuable digits of the measuring number. Because of this 1500 m and 1.5 km are
not equivalent in terms of accuracy. They have 1 m and 100 m absolute errors respectively. The often used term
relative error is the ratio of the absolute error over the nominal value. The smaller is the relative error the higher
the accuracy of the measurement. When making operations with physical quantities, remember that the result
may not be more accurate than the worst of the factors involved. For instance, when dividing 3.2165 m with 2.1
s to find the speed of some particle, the result 1.5316667 m/s is physically incorrect. Correctly it may contain
only two valuable digits, just like the time data, so the correct result is 1.5 m/s.
The physical quantities are classified as fundamental quantities and derived quantities. The fundamental
quantities and their units are defined by standard or in other words etalon. The etalons are stored in relevant
institute in Paris. The fundamental quantities are the length, the time and the mass. The corresponding units are
meter (m), second (s) and kilogram (kg) respectively. These three fundamental quantities are sufficient to build
up the mechanics. The derived quantities are all other quantities which are the result of some kind of
mathematical operations. To describe electric phenomena the fourth fundamental quantity has been introduced.
This is ampere (A) the unit of electric current. This will be used extensively in Physics 2, when dealing with
electricity.
2. 1 Kinematics of a particle - György Hárs
Kinematics deals with the description of motion, without any respect to the cause of the motion. Strictly
speaking there is no mass involved in the theory, so force and related quantities do not show up. The
fundamental quantities involved are the length and the time only.
To describe the motion one needs a reference frame. Practically it is the Cartesian coordinate system with x, y, z
coordinates, and corresponding i, j, k unit vectors.
The particle is a physical model. This is a point like mass, so it lacks of any extension.
2.1. 1.1 Rectilinear motion
(Egyenes vonalú mozgás)
The motion of the particle takes place in a straight line in rectilinear motion. This means that the best
mathematical description is one of the axes of the Cartesian coordinate system. So the position of the particle is
described by
function.
The velocity of the particle is the first derivative of the position function. The everyday concept of speed is the
absolute value of the velocity vector. Therefore the speed is always a nonnegative number, while the velocity
can also be a negative number.
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The opposite direction operation recovers the position time function from the velocity vs. time function. Here
is the initial value of the position in
0 moment, denotes the integration parameter from zero to
time.
The acceleration of the particle is the first derivative of the velocity vs. time function, thus it is the second
derivative of the position vs. time function.
The opposite direction operation recovers the velocity time function from the acceleration vs. time function.
Here
is the initial value of the position in
0 moment, denotes the integration parameter from zero to
time.
2.1.1. 1.1.1 Uniform Rectilinear Motion
Here the acceleration of the particle is zero. The above formulas transform to the following special cases.
0, v
vt.
2.1.2. 1.1.2 Uniformly Accelerated Rectilinear Motion
Here the acceleration of the particle is constant. The above formulas transform to the following special cases. a
const,
Typical example is the free fall, where the acceleration is a
g
9.81 m/s .
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2.1.3. 1.1.3 Harmonic oscillatory motion
The trajectory of the harmonic oscillation is straight line, so this is a special rectilinear motion. First let us
consider a particle in uniform circular motion.
The two coordinates in the Cartesian coordinate system are as follows:
If the uniform circular motion is projected to one of its coordinates, the motion of the projected point is
"harmonic oscillatory motion". We choose the x coordinate.
The displacement at oscillatory motion is called excursion. The sum in the parenthesis is called the "phase". The
multiplier of time is called angular frequency, and additive
constant is the initial phase. The multiplier in
front is called the "amplitude". The velocity of the oscillation is the derivative of the displacement function.
The multiplier of the trigonometric term is called the "velocity amplitude" (
The acceleration is the derivative of the velocity:
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If one compares the displacement and the acceleration functions the relation below can readily found:
Accordingly, the acceleration is always opposite phase position relative to the displacement.
In the kinematics of the harmonic oscillations it is very much helpful to go back to the origin of the oscillatory
motion and contemplate the phenomena as projected component of a uniform circular motion. This way one gets
rid of the trigonometric formalism and the original problem could have a far easier geometric interpretation.
Best example for that if we want to find out the resultant oscillation of two identical frequency harmonic
oscillations with different amplitudes and different initial phases. In pure trigonometry approach this is a tedious
work, while in the circle diagram this is a simple geometry problem, actually a cosine theorem application in the
most ordinary case.
2.2. 1.2 Curvilinear motion
(Görbervonalú mozgás)
The motion of the particle is described by an arbitrary r
vectors of the coordinate system.
vector scalar function, where i, j, k are the unit
The velocity of the particle is the first derivative of the position function.
The velocity vector is tangential to the trajectory of the particle always.
The vector of acceleration is the derivative of the velocity vector. The vector of acceleration can be decomposed
as parallel and normal direction to the velocity.
The parallel component of the acceleration (called tangential acceleration) is the consequence of the variation in
the absolute value of the velocity. In other words this is caused by the variation of the speed. The normal
component of the acceleration (called centripetal acceleration) is the consequence of the change in the direction
of the velocity vector.
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If one drives a car on the road, speeding up or slowing down causes the tangential acceleration to be directed
parallel or opposite with the velocity, respectively. By turning the steering wheel, centripetal acceleration will
emerge. The direction of the centripetal acceleration points in the direction of the virtual center of the bend.
2.2.1. 1.2.1 Projectile motion
(Hajítás)
In the model of the description the following conditions will be used:
Projectile is a particle,
Gravity field is homogeneous,
Rotation of the Earth, does not take part,
No drag due to air friction will be considered.
In real artillery situation the phenomenon is much more complex. This is far beyond the present scope.
The projectile is fired from the origin of the Cartesian coordinate system. The motion is characterized by the
initial velocity
and the angle of the velocity
relative to the horizontal direction. The motion will take place
in the vertical plane, which contains the velocity vector. The motion is the superposition of a uniform horizontal
rectilinear motion, a uniform vertical rectilinear motion and a free fall. Thus the velocity components are as
follows:
The corresponding position coordinates are the integrated formulas with zero initial condition.
Two critical parameters are needed to find out. These are the height of the trajectory
and the horizontal
flight distance
. First, the rise time should be calculated. The rise time
is the time when the vertical
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velocity component vanishes. Accordingly
results:
0 condition should be met. From the equation the following
The height of the trajectory shows up as a vertical coordinate just in rise time moment.
By substituting the formula of
into the equation above, the height of the trajectory results:
Accordingly:
The rise and the fall part of the motion last the same duration, due to the symmetry of the motion. Because of
this, the total flight time of the motion is twice longer than the rise time alone. The horizontal flight distance
can be calculated as the horizontal x coordinate at double rise time moment.
By using elementary trigonometry, the final formula of horizontal flight distance results:
This clearly shows that the projectile flies the furthest if the angle of the shot is 45 degrees.
2.2.2. 1.2.2 Circular motion
In circular motion, the particle moves on a circular plane trajectory. To describe the position of the particle polar
coordinates are used. The origin of the polar coordinate system is the center of the motion. The only variable
parameter is the angular position
since the radial position is constant.
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The derivative of the angular position is the angular velocity
.
Up to this moment it looks as if the angular velocity were a scalar number. But this is not the case. The angular
velocity is a vector in fact, because it should contain the information about the rotational axis as well. By
definition, the angular velocity vector
is as follows: The absolute value of the
is the derivative of the
angular position as written above. The direction of the
is perpendicular, or in other words, normal to the
plane of the rotation, and the direction results as a right hand screw rotation. This latter means that by turning a
usual right hand screw in the direction of the circular motion, the screw will proceed in the direction of the
vector. Just an example: If the circular motion takes place in the plane of this paper and the rotation is going
clockwise, the
will be directed into the paper. Counter clockwise rotation will obviously result in a
vector
pointing upward, away from the paper.
With the help of
vector number of calculation will be much easier to carry out. For example finding out the
velocity vector of the particle is as easy as that:
This velocity vector is sometimes called "circumferential velocity" however this notation is redundant, since the
velocity vector is always tangential to the trajectory. The cross product of vectors in mathematics has a clear
definition. By turning the first factor ( ) into the second one (r) the corresponding turning direction defines the
direction of the velocity vector by the right hand screw rule. The absolute value of the velocity is the product of
the individual absolute values, multiplied with the sine of the angle between the vectors.
Before going into further details, let us state three mathematical statements. Let a(t) and b(t) are two time
dependent vectors and (t) a time dependent scalar. Then the following differentiation rules apply:
These formulas make it possible to use the same differentiation rules among the vector products, just like among
the ordinary product functions. End this is true both the cross product and the dot product operations. The proof
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of these rules, are quite straightforward. The vectors should be written by components, and the match of the two
sides should be verified.
Using the
vector is a powerful means. This way the acceleration vector of the particle can be determined with
a relative ease.
The derivative of
vector is called the vector of angular acceleration . This is the result of the variation in
the angular velocity either due to spinning faster or slower or by changing the axis of the rotation.
Last term is the derivative of the position vector. This is the velocity, which can be written as above wit the help
of
vector. So ultimately the acceleration vector can be summarized.
The above formula consists of two major terms. The first term is called tangential acceleration. In case of plane
motion, this is parallel or opposite to the velocity and it is the consequence of speeding up or slowing down, as
explained in the earlier part of this chapter. The second term is called the centripetal or normal acceleration.
This component points toward the center of the rotation. The centripetal acceleration is the consequence of the
direction variation of the velocity vector. The absolute values of these components can readily be expressed.
There are two special kinds of circular motion, the uniform and the uniformly accelerating circular motion.
2.2.2.1. 1.2.2.1 Uniform circular motion:
In here the angular velocity is constant. The angle or rotation can be expressed accordingly:
Since the angular acceleration is zero, no tangential acceleration will emerge. However there will be a constant
magnitude centripetal acceleration, with an ever changing direction, pointing always to the center.
2.2.2.2. 1.2.2.2 Uniformly accelerating circular motion
In here the angular acceleration is constant. The corresponding formulas are analogous to that of uniformly
accelerating rectilinear motion, explained earlier in this chapter.
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The magnitude of the tangential acceleration is constant and parallel with the velocity vector.
The magnitude of the centripetal component shows quadratic dependence in time.
2.2.3. 1.2.3 Areal velocity
(Területi sebesség)
Let us consider particle travelling on its trajectory. If one draws a line between the origin of the coordinate
system and the particle, this line is called the "radius vector". The vector of areal velocity is the ratio of the area
swept by the radius vector over time. The crosshatched triangle on the figure above is the absolute value of the
infinitesimal variation ( A) of the swept area vector.
Areal velocity will be used in the study of planetary motion later in this book.
3. 2 Dynamics of a Particle - György Hárs
(Tömegpont dinamikája)
Dynamics deals with the cause of motion. So in dynamics a new major quantity shows up. This is the mass of
the particle (m). The concept of force and other related quantities will be treated as well. In this chapter only one
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piece of particle will be the subject of the discussion, in the next chapter however the system of particles will be
treated.
3.1. 2.1 Inertial system
In kinematics any kind of coordinate system could be used, there was no restriction in this respect. In dynamics
however, a dedicated special coordinate system is used mostly. This is called inertial system. The inertial system
is defined as a coordinate system in which the law of inertia is true. The law of inertia or Newton's first law says
that the motion state of a free particle is constant. This means that if it was standstill it stayed standstill, if it was
moving with a certain velocity vector, it continues its motion with the same velocity. So the major role of
Newton's first law is the definition of the inertial system. Other Newton's laws use the inertial system as a frame
of reference further on. The best approximation of the inertial system is a free falling coordinate system. In
practice this can be a space craft orbiting the Earth, since the orbiting space craft is in constant free fall.
The inertial systems are local. This means that the point of the experimentation and its relative proximity
belongs to a dedicated inertial system. An example explains this statement: Imagine that we are on a huge
spacecraft circularly orbiting the Earth, so we are in inertial system. Now a small shuttle craft is ejected
mechanically from the spacecraft without any rocket engine operation. The shuttle craft also orbits the Earth on
a different trajectory and departs relatively far from the mother ship. Observing the events from the inertial
system of the mother ship the shuttle supposed to keep its original ejection velocity and supposed to depart
uniformly to the infinity. Much rather instead the shuttle craft also orbits the Earth and after a half circle it
returns to the mother ship on its own. So the law of inertia is true in the close proximity of the experiment only.
If one goes too far the law of inertia looses validity.
On the surface of the Earth we are not in inertial system. Partly because we experience weight, which is the
gravity force attracting the objects toward the center, partly because the Earth is rotating, which rotation causes
numerous other effects. Even though in most cases phenomena on the face of our planet can be described in
inertial system, by ignoring the rotation related effects, and by considering the gravity a separate interaction.
3.2. 2.2 The mass
Mass is a dual face quantity. Mass plays role in the interaction with the gravity field. This type of mass called
gravitational mass and this is something like gravitational charge in the Newton's gravitational law.
Here the m and m are the gravitational masses, is the distance between the objects,
is the resulting force
and
is the gravitational constant (6.67x10
m kg s
. When somebody measures the body weight
with a bathroom scale he actually measures the gravitational mass.
Other major feature is that the mass shows resistance against the accelerating effect. This resistance is
characterized by the inertial mass. It has been discovered later that these fundamentally different features can be
related to the same origin, and so the two types of mass are equivalent. Therefore the distinction between them
became unnecessary.
This equivalency makes the free falling objects drop with the same acceleration. The gravity force is
proportional with the gravitational mass, which force should be equal with the acceleration times the inertial
mass. So if the ratio of these masses were different, then the free fall would happen with different acceleration
for different materials. This is harshly against the experience, so mass will be referred without any attribute later
in this book.
3.3. 2.3 Linear momentum p
(Impulzus, mozgásmennyiség, lendület)
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By definition the linear momentum is the product of the mass and the velocity. Therefore linear momentum is a
vector quantity.
*Newton's second law:
This law is the definition of force (F).
The force exerted to a particle is equal to the time derivative of the linear momentum. The unit of force is
Newton (N).
Conclusion 1.
If the force equals to zero, then the linear momentum is constant. This is in agreement with the law of inertia.
However it is worth mentioning, that it is only true in inertial system. Which means that on an accelerating train
or in a spinning centrifuge it is not valid.
Conclusion 2.
According to the fundamental theorem of calculus, the time integral of force results in the variation of the linear
momentum:
The right hand side is called impulse (erőlökés).
Conclusion 3.
The well-known form of the Newton second law can be readily expressed:
Or briefly:
*Newton third law:
(Action reaction principle)
When two particles interact, the force on one particle is equal value and opposite direction to the force of the
other particle.
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3.4. 2.4 Equation of motion:
The particle is affected by numerous forces. The sum of these forces, cause the acceleration of the particle. This
leads to a second order ordinary differential equation. This is called the equation of motion:
In principle the forces may be the function of position, time and velocity.
*Example 1 for the equation of motion:
Attenuated oscillation:
(csillapodó rezgés)
A particle is hanging on a spring in water in vertical position. The particle is deflected to a higher position, and
left alone to oscillate. Describe the motion by solving the equation of motion. Ignore the buoyant force. The
motion will take place in the vertical line. The position is denoted
which is positive upside direction.
The forces affecting the particle are as follows:
Here
is the direction constant of the spring in
.Accordingly the equation of motion can be written:
units,
is the drag coefficient and
Ordering it to the form of a differential equation:
Let us introduce
for the attenuation coefficient with the following definition:
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The mathematical method for solving this differential equation is beyond the scope of this chapter. The solution
below can be verified by substitution:
Here
is the original value of the deflection,
is called the Thomson angular frequency and
angular frequency of the attenuated oscillation with the following definitions:
*Example 2 for the equation of motion:
Conical pendulum
(kúpinga)
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The conical pendulum circulates in horizontal plane with
angular frequency. The angle of the rope
relative to the vertical direction is the unknown parameter to be determined. The coordinate system is an inertial
system with horizontal and vertical axes, with the particle in the origin. There are two forces affecting the
particle, gravity force (mg) and the tension of the rope (
. The equation of the motion is a vector equation in
two dimensions so two scalar equations are used.
In addition the centripetal acceleration can be expressed readily:
After substitution
results.
By means of this result the cosine of the angular position is determined:
3.5. 2.5 The concept of weight
Let us place a bathroom scale on the floor of an elevator. The normal force (
transferred to the object.
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The positive reference direction is pointing down. The following equation of motion can be written:
Here the acceleration of the elevator is denoted (
. Let us express the normal force indicated by the scale:
If the elevator does not accelerate (in most cases it is standstill) the scale shows the force which is considered
the weight of the object in general. (
mg). This force is just enough to compensate the gravity force, so the
object does not accelerate. However, when the elevator accelerates up or down, the indicated value is increased
or decreased, respectively. This also explains that in a freefalling coordinate system, where
the weight
vanishes. Similarly zero gravity shows up on the orbiting spacecraft, which is also in constant freefall.
3.6. 2.6 The concept of work in physics
The concept of work in general is very broad. Besides physics, it is used in economy, also used as "spiritual
work". Concerning the physical concept, the amount of work is not too much related, how much tiredness is
suffered by the person who actually made this work. For example, if somebody is standing with fifty kilogram
sack on his back for an hour without any motion, surely becomes very tired. Furthermore if this person walks on
a horizontal surface during this time, he gets tired even more. Physical work has not been done in either case.
In high school the following definition was learnt. "The work equals the product of force and the projected
displacement". This is obviously true, but only for homogeneous force field and straight finite displacement. In
equation:
. Here we used the mathematical concept of dot product, which results in a scalar
number, and the product of the two absolute values is multiplied with the cosine of the angle.
In general case when the related force field F(r) is not homogeneous and the displacement is not straight, the
above finite concept is not applicable. We have to introduce the infinitesimal contribution of work (dW F(r)
r). The amount of work made between two positions is the sum or in other words integral of dW contributions.
The physical unit is Newton meter (Nm) which is called Joule ( .
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There is a special case when the force is the function of one variable
with the x direction. The above definition simplifies to the following:
only, and its direction is parallel
In this special case the work done between two positions is displayed by the area under the
curve.
3.7. 2.7 Power
(Teljesítmény)
The power (
is associated with the time needed to carry out a certain amount of work. In mathematics, this is
the time derivative of the work done. The physical unit is Joule per second which is called Watt (
.
Provided the force does not depend directly on time, the above formula can be transformed:
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So the instantaneous power is the dot product of the force and the actual velocity vector.
3.8. 2.8 Theorem of Work (Kinetic energy)
Munkatétel (Mozgási energia)
Kinetic energy is the kind of energy which is associated with the mechanical motion of some object. In high
school the following simplified argument was presented to calculate it:
A particle with mass (
is affected by constant force. Initially the particle is standstill. The acceleration is
constant, thus the
graph is a sloppy line through the origin. After
time passed, the displacement
shows up as the area under the
curve. Its shape is a right angle triangle.
The acceleration is the slope of the v(t) line.
Let us multiply the above equation with the mass of the particle:
The left hand side equals the force affecting the particle.
We also know that the work done in this simple case is:
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So let us substitute the related formulas. Time cancels out:
This is the work done on the particle which generated the kinetic energy.
The above argument is not general enough, due to the simplified conditions used. The general argument is
presented below:
Let us start with Newton's second law:
The work done in general is as follows:
Substitute first to the second formula:
Switch the limits of the integration to the related time moments t and t .
Take a closer look at the formulas in the parenthesis. In here the product of the first and the second derivative of
some function are present.
The following rule is known in mathematics:
Using this formula for the last expression of work:
By integrating the variations of the v , the total variation will be the result:
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Thus: The work done on a particle equals the variation of the kinetic energy. This is the theorem of work.
Note there is no any restriction to the kind of force. So the force is not required to be conservative, which
concept will be presented later in this chapter. This can be even sliding friction, drag or whatever other type of
force.
The kinetic energy is accordingly:
3.9. 2.9 Potential energy
(Helyzeti energia)
Potential energy is the kind of energy which is associated with the position of some object in a force field. Force
field is a vector-vector function in which the force vector F depends on the position vector r. In terms of
mathematics the force field F(r) is described as follows:
where i, j, k are the unit vectors of the coordinate system.
Take a particle and move it slowly in the F(r) force field from position 1 to position 2 on two alternative paths.
Let us calculate the amount of work done on each path. The force exerted to the particle by my hand is just
opposite of the force field -F(r). If it was not the case, the particle would accelerate. The moving is thought to
happen quasi-statically without acceleration.
Let us calculate my work for the two alternate paths:
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In general case W and W are not equal. However, in some special cases they may be equal for any two paths.
Imagine that our force field is such, that W and W are equal. In this case a closed loop path can be made
which starts with path 1 and returns to the starting point on path 2. Since the opposite direction passage turns W
to its negative, ultimately the closed loop path will result in zero value. That special force field where the
integral is zero for any closed loop is considered CONSERVATIVE force field. In formula:
At conservative force field, one has to choose a reference point. All other destination points can be characterized
with the amount of the work done against the force field to reach the destination point. This work is considered
the potential energy (
of the point relative to the reference point:
The reference point can be chosen arbitrarily, however it is worth considering the practical aspects of the
problem.
Due to the fact that the reference point is arbitrary, the value of the potential energy is also indefinite since direct
physical meaning can only be associated to the variation of the potential energy. In other words, the individual
potential energy values of any two points can be altered by changing the reference point, but the difference of
the potential energy values does not change.
Now the work done against the forces of the field between r and r points can be expressed:
The last two integrals are the potential energies of r and r points respectively.
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3.10. 2.10 Conservation of the mechanical energy
(Mechanikai energia megmaradása)
Mechanical energy consists of kinetic and potential energy by definition. Earlier in this chapter the theorem of
work was stated. Work done on a particle equals the variation of its kinetic energy. In addition F(r) could be any
kind of force.
Later the potential energy has been treated.
Let us switch the sign of the above equation:
At potential energy however conservative force field is required. This means that the so called dissipative
interactions are excluded, such as the sliding friction and the drag. Let us make the right hand sides of the
relevant equations equal.
Ordering the equation:
Using the conservation of mechanical energy requires conservative force, because this is the more stringent
condition.
Ultimately let us declare again clearly the conservation of mechanical energy: In conservative system the sum of
the kinetic and potential energy is constant in time. Accordingly, these two types of energy transform to each
other during the motion, but the overall value is unchanged. In contrast to this when dissipative interaction
emerges in the system, the total mechanical energy gradually decreases by heat loss.
In this chapter the concept of work end energy have been used extensively. To improve clarity, the following
statement needs to be declared: Work is associated to some kind of process or action. Energy on the other hand
is associated to some kind of state of a system, when not necessarily happens anything, but the capacitance to
generate action is present.
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3.11. 2.11 Energy relations at harmonic oscillatory motion
The equation of motion of the harmonic oscillation is as follows:
Here
is direction coefficient of the spring on which a particle with mass
oscillates.
In the chapter of kinematics the harmonic oscillatory motion has been introduced, and the basic formulae have
all been derived. The following relation was recovered:
Let us multiply it with mass:
The left hand side of the equation is the force affecting the particle.
By comparing the two expressions of the force one can conclude as follows:
The harmonic oscillatory motion is a conservative process. This means that the total mechanical energy (the sum
of kinetic and the potential energy) should be constant.
Let us verify the above statement with the concrete formulas of displacement and velocity:
Now we can proceed on two alternate tracks by substituting the direction coefficient into the equation and using
the most basic trigonometric relation:
Or alternatively:
By using the velocity amplitude (
defined in the chapter of kinematics one can conclude as follows:
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Ultimately we found two alternate formulae for the total mechanical energy. These formulae prove that the
process is truly conservative, and the total energy may show up either as potential or kinetic energy. In
amplitude position the total energy is stored in the spring as potential (elastic) energy, at zero excursion position
the total energy is kinetic energy.
In the figure above the energy relations are displayed. The motion takes place under the solid horizontal line of
total energy.
3.12. 2.12 Angular momentum
(Impulzus nyomaték, perdület)
By definition the angular momentum of the particle is the cross product of the position vector and the linear
momentum.
3.13. 2.13 Torque
(Forgató nyomaték)
By definition the torque (M) is the cross product of the position vector and the force affecting the particle.
Let us consider the situation when r and p and F are in the plane of the sheet. According to the definition, both
the angular momentum and the torque are normal to the sheet.
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If the vectors depend on time, one can determine the derivative of the product:
Since
and
the above equation can be transformed:
The first term on the right cancels out because v and p vectors are parallel. Therefore:
The product on the right hand side is the torque. Ultimately one can conclude:
In words: The time derivative of the angular momentum of some particle equals the torque affecting this
particle. (Obviously the reference point of both L and M must be the same.)
This formula is analogous to that of Newton's second law, expressed with the linear momentum. By means of
the fundamental theorem of calculus, this formula can be integrated.
In words:
The variation of the angular momentums is the time integral of the torque affecting the particle. This integral is
called the angular impulse. (Nyomaték lökés)
3.14. 2.14 Central force field
(Centrális erőtér)
If the force is collinear with the position vector and the magnitude depends on the distance alone, then the force
field is considered central force field:
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here k is a scalar number which may depend only on the distance from the center.
As it has already been calculated:
Let us substitute the central force field:
The cross product is zero because of the collinear arrangement:
Accordingly, in central force field the angular momentum is constant in time: (L const) It has the important
conclusion. Planets, moons or spacecrafts which orbit their central body in the space also move in the central
force field of gravity. Therefore the angular momentum referred to the central body is constant.
In the chapter of kinematics the concept of areal velocity was introduced in general. Accordingly:
On the other hand, the angular velocity is:
By combining these two last equations:
Ultimately the areal velocity is constant in the central force field.
Planetary motion: A meteorite is orbiting the sun on an ellipse trajectory. The ellipse trajectory is the
consequence of the Newton's gravitational law. The constant areal velocity will make the meteorite travel faster
when close to the sun and slower when it is far away. The crosshatched areas in the figure below are equal. So,
the motion is far not uniform.
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4. 3 Dynamics of system of particles - György Hárs
(Tömegpont rendszer dinamikája)
4.1. 3.1 Momentum in system of particles
The subject of analysis will be the system of particles. The system of particles in practice may consist of several
particles (mass points). Each of the particles may travel arbitrarily in 3D space. The particles may exert force to
each other (internal force) and may be affected by forces originating in the environment (external force).
In mathematical calculations however it is worth reducing the number of particles to two particles. This way,
calculations become much easier without loosing generality. The physical meaning behind the equations
becomes even more apparent. At the end of the argument the result will be stated in full generality for any
number of particles.
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The center of mass is the weighted average of the position vectors.
Its time derivative is the velocity of the center of mass.
The numerator is the total momentum of the system of particles. So the total momentum can be expressed as the
product of the velocity of the center of mass multiplied by the total mass.
Let us make one more time derivation:
Accordingly:
Now consider the Newton equation for m and m :
Internal forces show up with double subscript. By substituting the forces to the above equation:
Here we have to take into account the fact, that the internal forces show up in pairs and they are opposite of each
other. F
-F . So they cancel out and only the external forces remain.
In words: The sum of the external forces accelerates the center of mass. Internal forces do not affect the
acceleration of the center of mass. This is the theorem of momentum.
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If on the other hand the sum of the external forces is zero, the acceleration of the center of mass becomes also
zero, or in other words, the velocity of the center of mass is constant. If the velocity of the center of mass is
constant, then the total momentum of the system of particles will also be constant.
So all together, let us state the conservation of momentum: In an isolated mechanical system (in here the sum of
the external forces is zero) the total momentum of the system of particles is constant.
This law can also be used in coordinate components. So if the system of particles is mounted on a little rail cart,
and external force parallel with the rail does not affect the system, then that component of the total momentum
will be constant which is parallel with the rail. In terms of other directions no any law applies.
4.1.1. 3.1.1 Collisions
(Ütközések)
At commonly happening collisions the conservation of momentum is valid because the system of the two
colliding particles represents an isolated mechanical system. There are two specific types of collisions, the
inelastic and elastic collision. The distinction is based on the kinetic energy variation during the process.
4.1.1.1. 3.1.1.1 Inelastic collisions:
(Rugalmatlan ütközés)
The two colliding particles get stuck together. The kinetic energy of the system is partly dissipated. Substantial
amount of heat can be generated. Let us write the conservation of momentum:
The velocity after collision (u) results:
The "lost" mechanical energy, which has been dissipated to heat, is the difference of the total kinetic energy
before and after the collision:
4.1.1.2. 3.1.1.2 Elastic collision:
(Rugalmas ütközés)
Word "elastic" means that the mechanical energy is conserved. Thus, both the momentum and the kinetic energy
are conserved. After collision the particles get separated with different velocities. The velocities before and after
the collision are denoted with v v and u u respectively. The conservation of momentum follows:
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The conservation of mechanical energy is also valid. Here the total mechanical energy is kinetic energy since no
potential energy is involved.
Let us group terms with subscript 1 to the left and terms with subscript 2 to the right hand side for two equations
above.
Now factor out m and m from the equations, multiply the kinetic energy equation with two and use the
equivalency for the difference of squares:
Up to this point of discussion the 3D vector equations above are fully valid. Among dot products, division
operation is impossible. This is due to the fact that reverse direction of the operation is ambiguous.
From this point, the mathematical argument is confined to the central collision only. At central collision the
velocities before the collision are parallel with the line between the centers of the particle. This way the collision
process takes place in a single line, and the velocities before and after the collision will all be 1D vectors in the
line of the collision. The 1D vectors are practically plus, minus or zero numbers, and the dot product between
these vectors is basically product between real numbers. So from the above equations the vector notation will be
omitted. Accordingly any division can readily be carried out.
Let us divide the last equation with the former one:
Now group the
terms to the left and
Multiply the equation with
terms to the right hand side:
:
The original equation for momentum conservation is simplified for 1D central collision:
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Let us subtract the former equitation from the last one. Here
Thus
term will cancel out:
can be expressed:
Due to symmetry, formula for
can be easily derived by switching the subscripts 1 and 2.
The above formulas are not simple enough to provide plausible results. For this purpose some special cases will
be treated separately:
*Discussion 1:
What if
is the case.
Basically the masses are equal. Then the final result simplifies to:
and
Accordingly the particles swap their velocities. If on the other hand one of the particle had zero velocity
originally
0), and the other particle slammed into it with velocity. Then :
and
This means that the standing particle will start travelling with the velocity of the moving particle, and the
originally moving particle will stop.
*Discussion 2:
What if
is far larger than
.
and
This is the case when a ball bounces back from the face of the incoming bus. The velocity of bus does not
change (
, and the velocity of ball is reflected plus the speed of the buss is added.
*Discussion 3:
Billiard ball collision:
In this game, the balls are equal in mass, but the collisions are not necessarily central. Consider the situation
when one ball is standing and an equal weight ball collides to it in a skew elastic collision. Let us go back to the
original equation with vectors.
The momentum conservation for the present case:
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The mechanical energy conservation for the present case:
After some obvious mathematical simplifications:
First equation means that the vectors create a closed triangle. The second equation shows that the created
triangle is a right angle triangle, since the Pythagoras theorem is true only then. As a summary, one can say that
the balls travel 90 degree angle relative to each after collision in billiard game.
4.1.1.3. 3.1.1.3 Ballistic pendulum
(Ballisztikus inga)
This is a pendulum with some heavy sand bag on the end of some meter long rope. The rope is hung on a high
fix point, letting the pendulum swing. A simple indicator mechanism shows the highest angular excursion.
The pendulum is left to get quiet and hang vertically. Then the gun is fired, the bullet penetrates into the
sandbag and get stuck in it. The pendulum starts to swing. The first highest angular excursion is detected. From
the above information, the speed of the bullet can be found.
The whole process consists of two steps. In step 1 the bullet collides with the sandbag. Up to this point,
conservation momentum is valid but the mechanical energy is not conserving quantity, due to the inelastic
collision. After collision in step 2, there are no more dissipative effects, so conservation of mechanical energy is
true. The two relevant equations are as follows:
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Here
and
are the mass of the bullet and the sandbag respectively. The and are the speed of the
bullet and the speed of the sandbag respectively. The
is the length of the rope and
is the gravity
acceleration.
By eliminating u from the equations one can readily express the incoming speed of the bullet:
This is an excellent example how careful one must be. If wrongly the whole process is assumed to be
conservative, the resulting bullet speed will be some ten meters per second which is roughly hundred times
smaller than the real result.
4.1.2. 3.1.2 Missile motion
(Rakéta mozgás)
Jet propulsion is the fundamental basis of the missile motion. This is based on the conservation of momentum. If
one tries to hold the garden hose when sprinkling the garden, one will experience a recoil type force, which is
pushing back. This force is called "thrust", and this drives the missiles, aircrafts and jet-skis.
The missile ejects mass in continuous flow with the ejection speed ( relative to the missile. The rate with
which the mass is ejected is denoted (
and measured in kg/s. The infinitesimal ejected momentum will
provide the impulse to the missile:
So the product of the speed and ejection rate determines the thrust (
. During the missile motion the thrust is a
constant force. As the missile progresses the overall mass is continuously reduced by burning the fuel. The
equation of motion is as follows:
Here
is the reducing mass and m is the initial mass:
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The acceleration can be expressed:
In order to find out the velocity time function, the above formula needs to be integrated:
After the integration the velocity function is revealed:
The final formula shows that approaching the
time the speed grows to the infinity. This value can not be
reached since there must be a payload on the missile.
4.2. 3.2 Angular momentum in system of particles
(Tömegpont rendszer impulzus nyomatéka)
Consider the total angular momentum of a system of particles.
This is the sum of the angular momentums of the individual particles:
Check out the time derivative oft this equation:
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In Chapter 2 it has been shown that the derivative of angular momentum is the torque affecting the particle.
Accordingly the above equation is transformed:
Based on the definition of torque following formulas are true:
Let us substitute them to the equation above:
Here we have to use that F
-F
which is the consequence of action reaction principle.
Now regroup the left hand side:
If now one looks at the figure above, the fact is readily apparent that the r -r vector and F force vector are
collinear vectors, thus their cross product is zero. Therefore the torques of the internal forces cancels out.
The left hand side terms are all the torques of the external forces. So all together the generalized statement is as
follows:
In words:
In system of particles, the time derivative of the total angular momentum is the sum of the external torques. This
statement is called the theorem of angular momentum. Therefore the internal torques are ineffective in terms of
total angular momentum.
If on the other hand the total external torque is zero, then the total angular momentum is constant. This is the
conservation of angular momentum.
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In summary: In a system of particles where the total external torque is zero, the total angular momentum is
constant, or in other words it is a conserving quantity.
This law can also be used in coordinate components. So if the system of particles is mounted on a bearing, and
external torque parallel with the axis of the bearing does not affect the system, then that component of the
angular momentum will be constant which is parallel with the axis of the bearing. In terms of other directions no
any law applies.
4.2.1. 3.2.1 The skew rotator
(Ferdeszögű forgás)
Consider the figure below. Two equal masses are placed on the ends of a weightless rod. The center of mass is
mounted on a vertical axis, which is rotating freely in two bearings. The angle of the fixture is intentionally not
ninety degrees, but a skew acute angle. The system rotates with a uniform angular velocity. The job is to find
out the deviational torque which emerges, due to the rotation of asymmetric structure.
The origin of the coordinate system is the center of mass. The coordinate system is not rotating together with the
mechanical structure and it is considered inertial system. Gravity cancels out from the discussion, since the
center of mass is supported by the axis, and the gravity does not affect torque to the system. The mechanical
setup is in the plane of the figure. The two position vectors of the particles are (r) and (-r).
Momentums of the particles are p and p . They are normal to the paper sheet.
The corresponding L and L angular momentums are equal, because of the twice negative multiplication:
So the total angular momentum is the sum of these two:
Now we use the theorem of angular momentum:
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Now take it into consideration that the derivative of the position is the velocity which can be expressed by
means of angular velocity vector:
After substitution:
The first term on the right hand side is zero, because this is a cross product of collinear vectors. The final result
comes up immediately.
After following the numerous cross products in terms of direction one can conclude, that the torque is pointing
out of the paper sheet. The direction of torque is rotating together with the mechanical structure and always
perpendicular to its plane. This is deviational torque and this emerges because the angular momentum vector is
constantly changing, not in absolute value but in direction. This effect is very detrimental to the bearings due to
the load that it generates. There are cases however when such effect does not show up. When the angular
momentum vector is parallel with angular velocity vector no deviational torque will emerge. These are called
principal axes. In general there are three perpendicular directions of principal axes.
A new interesting aspect:
Imagine that this experiment is carried out on a spacecraft orbiting the Earth. Suddenly the bearing and the
mechanical axis disappear. How will the mass-rod-mass structure move after this?
Since no external torque affects the system, the angular momentum will be constant. But now the angular
velocity vector starts to go around on the surface of a virtual cone. The symmetry axis of such virtual cone is
just the angular momentum vector. This kind of motion is called precession.
4.2.2. 3.2.2 The pirouette dancer (The symmetrical rotator)
(A piruett táncos)
In conjunction with the previous section this section could be called as "symmetrical rotator". The setup
fundamentally similar, the major difference is that the mass-rod-mass system is positioned perpendicularly to
the rotation axis.
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Similarly to the previous section total angular momentum is:
In present case however the position vectors and the angular velocity vector are normal to each other. Therefore
the direction of the angular momentum and the angular velocity will be both parallel with the rotation axis.
Since the direction of the vectors is clear, it is enough to deal with the absolute values only.
Notice: The emerging 2mr quantity is the so called the moment of inertia. Find details in Chapter 4.
Let us see what happens when the pirouette dancer pulls his arms in. Then
is the case.
The conserving quantity is the angular momentum.
From here:
Accordingly, the dancer is spinning much faster.
Let us check out how the kinetic energy changed during the pirouette. Clearly the difference between the final
state and the initial state should be calculated.
Now let us substitute
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The kinetic energy increased. This only could happen due to the work done by the dancer. The force of the
dancer is internal force, so the kinetic energy of the system of particles can be changed by internal forces too. In
order to calculate the work done, the first task is to find out the force function. The force against which the
dancer pulls his arm is the centrifugal force. The centrifugal force is an inertial force which emerges in spinning
coordinate system only. See details in chapter 5.
Let us find out the angular velocity as a function of the arbitrary position:
The formula of centrifugal force in present case is:
The work carried out by the dancer is the integral of the dancer's force, which is just opposite of the centrifugal
force.
Now let us study the integral alone. Use the fundamental theorem of calculus:
After substitution:
This result completely matches the growth of the kinetic energy calculated earlier, so the increase of the kinetic
energy is the consequence of the work done by the dancer.
4.3. 3.3 Discussion of the total kinetic energy in the system of
particles
In contrast to the earlier habit the system of particles will be treated in general up to
pieces of particles.
Particle positions are defined by r position vectors relative to the origin of the coordinate system. The center of
mass has the position vector r . Now, we introduce the center of mass as a new coordinate origin. In this new
coordinate system the corresponding position vectors will be denoted .
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Therefore the following equation is true:
After derivation similar rule is found among the velocities:
Here v and v are the velocity of i-th particle and the center of mass respectively in the original coordinate
system. . The
is the velocity of the i-th particle in the center of mass coordinate system.
Take a look at the total kinetic energy of the system in the laboratory coordinate system:
Let us discuss the individual terms separately:
In the first term the velocity of center of mass can be factored out:
The first energy term is associated with the velocity of the center of mass in the laboratory coordinate system.
The second term is associated with the velocities relative to the center of mass.
The third term gives zero result. The proof is as follows: From the third term v can be factored out:
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The right hand side summa is the total momentum in the center of mass coordinate system. The total momentum
is the product of the total mass multiplied with the velocity of the center of mass. But in the center of mass
coordinate system the velocity of center of mass is obviously zero. QED.
Accordingly the total kinetic energy of the system of particles can be divided into two parts: Kinetic energy
associated to the velocity of the center of mass in the laboratory coordinate system, and kinetic energy
associated to the velocities in the center of mass coordinate system. In formulas:
The first term can be changed external force only, due to the theorem of momentum. The second term can be
changed by any type of forces.
5. 4 Dynamics of rigid body - György Hárs
(Merev testek dinamikája)
5.1. 4.1 Moment of inertia
(Tehetetlenségi nyomaték)
Up to this point only the motion of particles was discussed. The rigid bodies however are extended objects. The
kinematics of extended objects contains major distinctions in terms of motion. Translation means that the points
of the object travel the same trajectory except for a shift, by which all the trajectories cover each other. The
rotation however contains circular trajectories with different radii.
Rigid body is a special system of particles. Here the positions of the particles are fixed relative to each other. In
addition the geometrical shape of the body is constant, and it is independent of the mechanical load.
In present discussion a majorly simplified theory will be treated. The simplifications are as follows:
• The origin of the coordinate system will be the center of mass of the rigid body.
• The rotation will take place around principal axis, thus the angular momentum and the angular velocity
vectors are parallel.
Let us consider the mass-rod-mass structure in chapter 3. The angular momentum in general case can be
expressed as follows:
In present case however the position vectors and the angular velocity vector are normal to each other. Therefore
the direction of the angular momentum and the angular velocity will be parallel. If two vectors are parallel then
a scalar multiplier can be found between them. (In the general case tensor describes the relation.)
The scalar multiplier is called the "moment of inertia". This is denoted with the Greek letter
symmetrical pair of particles the moment of inertia is:
.
For one single piece of particle this value is:
.
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. So for a
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Let us calculate the moment of inertia for a large diameter thin wheel with the radius
.
Since the radius is constant this can be factored out.
Here
means the total mass of the wheel.
Now consider an arbitrary rotationally symmetric object. Imagine as this object consisted of several wheels,
each of them with dm mass.
If the density of the object is constant then
. With this:
By means of the above formula the moment of inertia for several symmetrical objects can be calculated:
• Tube: Here
Since
Const.
is constant this can be factored out.
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• Cylinder or disc:
Let us put together the cylinder from several tubes with increasing radii. Here
is the length of the cylinder.
On the other hand the mass of the cylinder is the product of its volume and density.
Divide them:
From here the moment of inertia is found:
• Spherical shell. (Hollow sphere)
The radius is denoted
, the thickness of the shell is
.
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is the case, where
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According to the general expression:
The function to be integrated is:
The dV volume here is as follows:
Let us deal with the integral for a while separately. The antiderivative is as follows:
By means of the integral value the moment of inertia can be expressed:
On the other hand the total mass is the product of the density and the volume.
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Divide them.
The result is clear.
• The bulky sphere
This can be composed of spherical shells with increasing radii. The infinitesimal volume is the product of the
surface of the inflating sphere and the infinitesimal thickness (dr):
On the other hand the mass of the sphere is the product of its volume and the density:
Divide them:
Thus the result is:
• Stick or rod.
The stick is put together from several particles with increasing radii. Early in this chapter it was shown that in
case of particles the moment of inertia is as follows:
equal to
.
. In addition the infinitesimal volume is
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The mass of the rod can also be expressed:
Accordingly:
Now, we should find out how much the moment of inertia is through the center of mass. A symmetrically
rotating rod can be composed of two half size rods. Therefore half length and half mass rod will be used and
finally doubled:
• Steiner's theorem
If one calculates the difference between the moment of inertia of the rod with different axes, an interesting
relation can be revealed:
The difference is the total mass multiplied with the square of the distance between the axes. The calculated
result is a special case of a far more general law which is called the Steiner's theorem. This general law states
that moment of inertia is smallest through the axis of center of mass. In addition the moment of inertia for any
parallel axes can be calculated as follows:
where
and
are the mass and the distance between the axes respectively.
5.2. 4.2 Equation of motion of the rigid body:
(Merev test mozgásegyenlete)
Let us take a look at the equation of motion of an extended rigid body. Two laws will be used, such as the
theorem of momentum
and the theorem of angular momentum.
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In case of rigid body the theorem of momentum is written with the relevant quantities:
The theorem of angular momentum is written by means of the moment of inertia. Subscript c means that the
moment of inertia is calculated to a principal axis through the center of mass.
So all together the equation of motion consists of two vector equations:
They represent in principle six scalar equations, but in the most cases the system of equations contains only
three equations. This is due to the fact that mechanical problems are mostly 2D plane problems. This case the
theorem of momentum contains two variables x and y components in the plane of the sheet, and the theorem of
angular momentum contains z component normal to the sheet.
The time derivative of angular velocity is the angular acceleration
acceleration is always parallel with the angular velocity.
. In present simplified situation the angular
5.2.1. 4.2.1 Demonstration example 1.
*The yoyo
Find the acceleration and the tension of the rope (rope force). The mass is 3 kg, g
the radii are
6 cm and r
3cm
The equations of motion:
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(1,0)300
In the second equation the first factor is the moment of inertia for cylinder, the second factor is the angular
acceleration. From the second equation
cancels out. The mathematical steps are as follows:
comes out. This and the first equation are added so
The rope force results as the acceleration is substituted to the expression of
.
5.2.2. 4.2.2 Demonstration example 2.
*Cylinder on the table
Find the accelerations and the tension of the rope. Determine the smallest possible static friction coefficient,
which provides sliding free rolling for the cylinder. m
3 kg, M
8 kg.
Equation of motion for the block:
I.
Equations of motion for the cylinder:
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II.
III.
IV.
In addition there is a condition for the static friction force:
V.
(1,0)300
IV.
IV.
III.
IV.
III.
I.
IV.
III.
I.
IV.
III.
I.
IV.
III.
Let us find the tension of the rope:
IV.
III.
Now determine the friction force:
-IV.
IV.
(IV.
III.
III).-IV.
The static friction coefficient is the last to deal with:
V.
II.
II.
After substitution:
V.
V.
V.
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The summary of the solution is the following:
The acceleration of the block is
5 m/s , The acceleration of the cylinder is a/2
2,5 m/s . The tension of
the rope is
15 N, the static friction force is
5 N. The minimum required static friction coefficient is
1/16
0.0625.
5.3. 4.3 Kinetic energy of the rigid body
(Merev test mozgási energiája)
The motion of the rigid body represents kinetic energy. In chapter 3 the kinetic energy of system of particles
was discussed. Let us extend the validity of this discussion to the rigid bodies.
The first term on the right hand side represents the kinetic energy of the rigid body due to the speed of the center
of mass. The second term is associated with the motion relative to the center of mass. This is typically the
rotation in case of rigid bodies, therefore
. Since they are normal to each other the absolute value
of the velocity is
. Let us substitute to the formula of the kinetic energy.
The last summa term on the right hand side is the moment of inertia of the rigid body. The total kinetic energy
of a rigid body can be written as follows:
So the total kinetic energy consists of two terms. One of them is in conjunction with the translation of the center
of mass, the other one is related to the rotation around the center of mass.
sectionCorrespondence between translation and rotation in the framework of the simplified model
6. 5 Non-inertial (accelerating) reference frames György Hárs
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(Gyorsuló vonatkoztatási rendszerek)
So far dynamics has been treated in inertial systems. The inertial systems are equivalent, which means that the
equations of physical laws show up in the same shape in all inertial systems. There are practical aspects however
which make the application of the non-inertial reference frames necessary. In here the shape of equation of
motion changes relative to that in inertial system. New terms, the inertial forces will appear in the equation of
motion in addition to the real forces. The real forces are associated with real interactions with other objects.
6.1. 5.1 Coordinate system with translational acceleration
There are two coordinate systems. One of them is an inertial system, the other one is a coordinate system with
translational acceleration. The position vector of the origin of the accelerating system is r . The position, the
velocity and the acceleration in the two systems are generated by consecutive derivations. The vectors in the
accelerating system are denoted with the subscript "rel" standing for relative.
Let us multiply the last equation with the mass
.
The left hand side of the equation is the sum of the real forces. Regroup the terms.
The second term is called inertial force.
So altogether the sum of the real forces should be completed with the inertial force. This way the equation of
motion can formally be handled just like in the inertial system.
*Example:
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A simple pendulum is hanging in a uniformly accelerating train. Find the angle of the rope in stationary state
relative to the vertical direction. The rope force is denoted
the angle is
Inertial system approach: In practice the observer is standing on the ground next to the train.
The angle can be determined accordingly:
Accelerating system approach: In practice the observer is on the train.
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The result is obviously the same as earlier. The point is that by choosing accelerating system, the equation of
motion becomes equilibrium equation, which easier to handle in most cases.
6.2. 5.2 Coordinate system in uniform rotation
All planets rotate thus this phenomenon is very frequent in nature. Let us have two coordinate systems with
common origin, the inertial system (
with the unit vectors i, j, k) and the uniformly rotating system (
with the unit vectors e , e , e . The i, j, k vectors are constant in time since they belong to an inertial
system, but the e , e , e vectors rotate together with the spinning system.
Consider an r vector in both coordinate systems:
Let us derivate it:
Regroup the right hand side:
The first three terms contain the derivative in the spinning system. Here it is worth introducing a notation for the
derivation in the spinning system. This will be similar to the usual derivation symbol, but instead of a Greek
letter is used.
In the kinematics chapter it has been shown that the velocity due to spinning can be written as follows:
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So altogether:
Let us factor out the angular velocity:
So the rule of transformation between the derivatives in the inertial and in the spinning system is as follows:
This has a direct consequence to the velocities:
If the above transformation rule is applied to the velocity vector then the relation between the accelerations is
recovered:
The following notations are used:
By means of which the acceleration is expressed:
Regroup the above formula and express the relative acceleration.
Multiply the equation with mass.
This equation has an important message. The equation of motion in a spinning system is very much similar to
that in the inertial system. The major difference is that additional terms appear which are called the inertial
forces. These forces are not exerted by some other object, but they are due to the fact that the coordinate system
is spinning. So when solving problem in a spinning system these inertial force should be added to the real
forces. Take a closer look at these forces.
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The second term is called Coriolis force. This force only appears if the particle is moving relative to the
spinning system. If we are on a spinning contraption in the amusement park Coriolis force can be felt if we
move our hand perpendicular direction to the axis of spinning.
Coriolis force does not make any work because its direction is normal to the relative velocity. This only deflects
the moving objects.
The third term is called the centrifugal force. This force affects also the stationary objects in the spinning system
and shows up in a turning car. Its direction points away from the axis of rotation.
Therefore the absolute value of the centrifugal force depends on the distance from the rotational axis and
proportional with the square of the angular velocity.
In technology it is a major limiting factor at manufacturing turbines and fast spinning motors.
6.2.1. 5.2.1 Earth as a rotating coordinate system
It is well known that planet Earth rotates with one day period. Accordingly inertial forces appear. In most cases
these are ignored, but in some special cases these forces make major qualitative changes in the physical
processes.
6.2.1.1. 5.2.1.1 The effects of the centrifugal force
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The gravity force and the centrifugal force affect the object together. The resulting free fall acceleration will
increase as one goes closer to the poles. At higher latitude we are closer to the rotational axis therefore the
centrifugal force is smaller. The vertical is defined by the resulting direction of both effects. If planet Earth
ceased to rotate strange effect would follow. The water of the oceans would move to the poles, and the ocean
floor close to the equator would be a huge desert.
Let us calculate the effect of the rotation on the equator numerically: The acceleration associated with the
centrifugal effect can be calculated as follows:
The radius of Earth is denoted
which is 6370 km. The duration of a normal day shows up in the denominator
in seconds. If the acceleration value is compared to the nominal value of the free fall acceleration we find that it
is around one third of a percent.
6.2.1.2. 5.2.1.2 The effects of the Coriolis force
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Since the angular velocity is a vector, this vector can be decomposed to two components as shown in the figure.
The two resulting omega vector components are the horizontal and vertical component. On the surface of the
Earth one makes rotation around two axes simultaneously. On the northern hemisphere there is an omega vector
which is lying on the ground horizontally and points to North, and another one which is standing out of the
ground vertically and points upside direction. Each of these rotations has different Coriolis effects.
Let us study them separately.
The first and second terms on the right hand side are denoted as Coriolis 1 and 2 forces.
The effects of Coriolis 1 force
Coriolis 1 force is associated with the vertical omega vector.
• The hurricanes
The first effect to be treated is the whirling motion of the hurricanes, which is a counter clockwise (CCW)
rotation on the northern hemisphere. Imagine that at certain spot the air warms up more than in the surrounding
places. Here the air lifts up and the surrounding air will horizontally move to the place where the airlift
occurred. This will cause an air flow vector field, in which the horizontal velocity vectors will all point to the
place of the airlift in
cylindrically symmetric arrangement. In order to describe the situation quantitatively a simplified mechanical
model is introduced.
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Consider a huge hypothetical frozen ocean on the northern hemisphere which is completely flat and the ice is
free of any friction. We put down a particle on the ice (imagine it weights some kilograms) and pull the particle
with v speed by means of a weightless ideal thread through several kilometers toward the center. The initial
and the final positions of the particle are R and R respectively.
The Coriolis 1 force is as follows:
The associated torque to the center can be written:
Now the following mathematical rule is used:
By applying this rule to the case we find the following outcome:
The first term in the parenthesis cancels out since the factors in the dot product are normal to each other.
Now we use the integral form of the theorem of angular momentum. It declares that the angular impulse equals
the variation of the angular momentum.
For present case:
The integration by time is switched to position integral by using the fact:
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After integration we find the result:
On the other hand it is known that the angular momentum can be expressed as the product of the angular
velocity and the moment of inertia. The angular velocity is denoted as .
The right hand sides are equal
Finally the angular velocity of the rotation can be expressed:
Back to the hypothetical experiment, it can be concluded that the mass on the thread will rotate around the
central position. The ratio of the initial and the final radii will determine the angular velocity of the rotation.
The result above for the angular velocity can also be interpreted in the inertial system. In this approach the law
to be used is the conservation of the angular momentum. The result in the inertial system is very much similar to
the actual formula except for minus one in the parenthesis, which is missing in inertial system approach. In
rotating coordinate system the angular velocity of the rotating coordinate system should be subtracted.
When the speed of the hurricane is of interest this can readily be expressed:
The angular frequency of the Earth's rotation at 45 degree latitude is as follows:
Let us take some realistic numerical values. Assume that R
400 km and R
Substitute these values to find out the wind speed in the hurricane.
The result is realistic though the physical model was the most basic possible.
• The Foucault pendulum
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The second effect to be treated is famous Foucault pendulum experiment. Leon Foucault French physicist made
the experiment in 1851. A 67 meter long pendulum was hung up in the Panthéon in Paris. As the pendulum was
swinging for several hours the plain of the oscillation was gradually turning. The stick at the end of the
pendulum was drawing a rosette shape figure into the sand on the floor. The angular velocity of the turning was
measured to be 11.29 degrees per hour. Which value gave an exact match with the actual vertical angular
velocity (
) in Paris. Since Paris is at the 48.83 degree latitude, the sine of this angle multiplies the total
angular velocity of planet Earth.
Let us discuss the motion of the pendulum in the rotating coordinate system of planet Earth. The problem is
considered to be two dimensional. The pendulum is treated in linear approximation. The equation of motion is
as follows:
At little angular excursions the motion can be approximated with a horizontal plane motion. Therefore
and is the length of the pendulum. On the right hand side the first term is the returning force due to gravity,
the second term is the centrifugal force and the third one is the Coriolis force. After rearranging the equation the
following state is reached:
Here
. This equation is difficult to handle due to the cross product. At this point it is worth to consider
the variable as a complex number rather than a two dimensional vector. The complex calculation provides a
straightforward means to rotate a complex number. Let us switch to z as a complex variable. By this way the
following equation is recovered: Here is the imaginary unit.
This is an ordinary second order differential equation which is widely used. The solution is looked for in the
following form:
. After substitution the equation is as follows:
By solving this equation the roots emerge:
Ultimately there are two linearly independent solutions:
Any linear combination of the above solutions is a valid solution. For the sake of simplicity wee choose the
average of the above:
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The real and the imaginary parts give the solutions:
Finally the two coordinates of the motion can be written:
In inertial system approach the situation is completely different. The plane of the oscillation is constant and only
planet Earth rotates under the experiment with
angular velocity. This makes possible that basically a
kinematical solution is given to this problem.
The inertial coordinate system is characterized by the e , e unit vectors. The rotating coordinate system,
(which rotates together with Earth) is characterized by i, j unit vectors.
The oscillation takes place along the e axis therefore it is written as follows:
Looking from the rotating system the e
Accordingly:
unit vector seems turning to the negative
Let us substitute and take it into consideration that
angle direction.
.
The two coordinates of the motion can be written accordingly:
This perfectly matches the result above, which was calculated in a more tedious way by solving the differential
equation of motion in the coordinate system of planet Earth.
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The effects of Coriolis 2 force
The Coriolis 2 force is associated with the horizontal omega vector component.
Dependent on the direction of the relative velocity two effects emerge.
If the relative velocity points vertically down, the F
force will point to the East. This force will deflect the
freefall from the perfect vertical direction slightly to the East.
Consider a tower with height denoted . A particle is dropped with zero initial velocity. The velocity of free
fall is as follows:
.The Coriolis acceleration in present case is:
. After substitution the
formula results. This acceleration should be integrated twice in order to find out the magnitude
of the deflection.
The displacement is the integral of the above formula:
On the other hand the total duration of the freefall (denoted
this formula
above.
) can be determined as follows:
results. The amount of the deflection emerges by substituting
After arranging it to a decent form:
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. From
formula
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Let us take the constant values numerically and find out the value in parenthesis above. The
and angular velocity at 45 degree latitude is the following:
equals 9,81 m/s
After substitution the final formula shows up:
If the height is 30 meters the deflection is 2.53 millimeters, when the height goes up to 100 meters the deflection
becomes 1,54 centimeters. These values are so small that any air disturbance will causes much higher deflection,
which makes this specific Coriolis effect practically ignorable.
If the relative velocity points horizontally in East or West direction the F
force will point to vertically up or
down respectively. Consequently this effect will cause a virtual decrease or increase in the weight of any object.
Let us calculate the magnitude of the effect. Assume that an aircraft travels 1008 km/h (280m/s) speed in either
direction mentioned. The absolute value of the F
force is easy to calculate due to the perpendicular position
of the vectors in the cross product.
The associated acceleration is
. Substitute the numerical values:
If one compares this to the nominal value of the freefall acceleration the following comes out:
Roughly one third of a percent reduction or increase in the weight of any object seems negligible in most
practical cases. And do not forget that the speed was high. At speeds usual in ground transportation the effect is
far more insignificant.
chapter5
7. 6 Oscillatory Motion - Gábor Dobos
A body is doing oscillatory motion when it is moving periodically around an equilibrium position. In
mathematical terms this can be expressed in the following form:
where r is the position-vector of the body, is the time and
is called the period of the oscillation. (1) means
that if the body is found in position r at a given time, it will return to the same position
time later. Any
kind of motion that can be described by an
function which satisfies (1) is called and oscillation.
Oscillations and vibrations are one of the most common types of motion in nature. A pendulum or a body
attached to a spring is doing oscillatory motion. Musical instruments and our vocal cords also utilise vibrations
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to create sounds. Most modern clocks measure time by counting oscillations. From switching mode power
supplies to radio transmitters there is a wide variety of electronic devices that rely on the periodic motion of
charged particles. Atoms in solids are also vibrating around their equilibrium positions.
To understand all these phenomena, first we have to understand oscillations. We will start our discussion of
oscillatory motion with the simple harmonic oscillator, which is the easiest to describe in mathematical terms.
Then we will expand this simple model to take into account other effects such as damping and external
excitation. Finally we will discuss how the superposition of oscillations can be used to describe any arbitrary
periodic motion.
7.1. 6.1 The simple harmonic oscillator
A body is doing a simple harmonic motion if its displacement from the equilibrium position can be described by
a sinusoidal function:
where:
•
•
is the position (or displacement) of the body (Measured in meters)
is called the amplitude of the oscillation. This defines the range in which the body is moving. If the
equilibrium point is in the origin, the body is oscillating between
and
. (Measured in
meters)
• The argument of the sinusoidal function (
• At
the phase is equal to
t+
) is called the phase of the oscillation. (Measured in radians)
, thus it is called the initial phase. (Also measured in radians)
• The constant
is called angular frequency (measured in radians/s, or simply 1/s), and it is determined by the
period of the oscillation.
is a periodic function: it takes the same value when the phase is increased or
decreased by 2 . Thus the time required to change the phase by 2 is equal to the period of the oscillation.
In mathematical terms:
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• The number of oscillations in a unit of time is called the frequency of the oscillation. It is usually measured in
Hertz (Hz), which is one oscillation per second. Like the angular frequency, it is determined by the period of
the oscillation:
7.1.1. 6.1.1 Complex representation of oscillatory motion
An interesting feature of harmonic oscillations is their connection to circular motion. Consider a body moving
on a circular trajectory of radius
around the origin. When the position of the body is projected to a straight
line (such as the or axis) it can be described by a sinusoidal function, thus the projection of the body is
doing a simple harmonic oscillation. The angular frequency of this oscillation is the same as the angular velocity
of the circular motion, and its amplitude is the radius of the circle.
This connection can be used to describe harmonic oscillations in complex form. A complex number may be
represented by a vector in the complex plane. The coordinates of this vector can be given either in Cartesian or
in polar coordinates:
where
is the imaginary unit,
(usually marked by
and
and
are called the real and imaginary parts of the complex number
), while
and
are its magnitude and phase, respectively.
Consider a complex function
, whose magnitude is constant (
), while its phase is a linear
function of time (
). The endpoint of the vector representing
in the complex plane is moving on
a circle around the origin. The real and imaginary parts of
can be determined using Euler’s formula:
Using this, the real and imaginary parts of
are:
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Since the real and imaginary parts are basically projections of
to the real and imaginary axes, and are
sinusoidal functions of time. This also means that a simple harmonic oscillation can be described as the real part
of such a “rotating” complex function. Such functions are usually referred to as phase vectors or phasors. The
main advantage of such a representation is that the dependencies on ,
and
can be separated into three
independent factors:
As the first two terms on the right-hand side are time independent, they are usually merged to a single constant,
called complex amplitude (
), which describes both the amplitude and the initial phase of the
oscillation.
7.1.2. 6.1.2 Velocity and acceleration in oscillatory motion
The velocity and the acceleration of the particle can be determined from (2):
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Figure 46 shows the position, velocity and acceleration as a function of time. All of these are sinusoidal
functions, but in different phases. The velocity is late by a quarter period with respect to the displacement, and
the acceleration is also shifted by another quarter period. This means, that in simple harmonic motion the
acceleration is always proportional and opposite to the displacement. This also gives us a hint about what kind
of systems may exhibit simple harmonic oscillations. As the sum of the forces acting on the body is proportional
to its acceleration there must be a retracting force, pulling the body back towards the equilibrium with a strength
proportional to its displacement.
7.2. 6.2 Motion of a body attached to a spring
One of the simplest mechanical systems that exhibit simple harmonic oscillations consists of a body attached to
a fix point by a spring. Since the spring force is proportional to the elongation of the spring, this system satisfies
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the aforementioned criterion. Because of its simplicity it is ideal to demonstrate how harmonic motion can be
described in mathematical terms. Using Newton’s law of motion:
(18) is a homogenous second-order linear differential equation, which can be solved by the ansatz (or trial
function)
equation:
. The values of
can be determined by substituting the trial function into the differential
(21) is usually referred to as the characteristic polynomial. Since (21) has two different roots, (18) has two
independent solutions. (These are the so called fundamental solutions of the differential equation.)
According to the superposition principle any solution of a linear differential equation can be built up as a linear
combination of its fundamental solutions.
where
and
are complex constants, that can be determined from the initial conditions. (The state of the
system – such as the position and velocity at the start of the experiment – is usually referred to as initial
conditions.) But there is a further important point to consider: the displacement of the object is a measurable
quantity, thus it must always be a real number. (There are no imaginary quantities in the real physical world.)
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This means that only those solutions have valid physical meaning, where the imaginary parts of the fundamental
solutions cancel out each other for all moments of time. This can be easily achieved by making the two constant
multipliers the complex conjugates of each other. This way the imaginary parts of the two terms on the righthand side of (25) are always the opposites of each other and their sum is a real number. The general solution of
(18) is:
where
is determined by the initial conditions. Using Euler’s formula (26a) can be transformed to:
Or by using trigonometric identities:
(26a), (26b) and (26c) are equivalent: they all give precisely the same result for all values of t, and they can be
transformed into each other using mathematical identities. Comparing (26c) to (2) shows, that the body is doing
a simple harmonic oscillation, with angular frequency
are determined by the initial conditions. (In fact
and
of , that are also determined by the initial conditions.)
. The amplitude ( ) and initial phase ( )
can be expressed in terms of
and
or in terms
An important feature of this system is that the period of oscillation (
) is independent of the
initial conditions. It is influenced only by the parameters of the system itself (such as the mass of the body or the
spring constant). If the amplitude is increased, the body is going to move proportionally faster, and finish each
cycle in precisely the same amount of time. This means that such systems are ideal for building clocks: as each
cycle takes the same time regardless of the initial conditions, time can be measured by counting cycles.
7.3. 6.3 Simple pendulum
Another mechanical system, that exhibits similar oscillations, is the simple pendulum, which is a bob suspended
on a massless rope from a frictionless pivot. When it is displaced from the vertical (equilibrium) position the
weight of the bob pulls it back towards the equilibrium. As the length of the rope is constant, the bob moves on
a circular trajectory. We can apply Newton’s second law to describe its tangential acceleration:
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The left-hand side of (27) is the tangential component of the weight of the bob. (
) The minus
sign indicates that it points towards the equilibrium position. The right-hand side is the mass of the bob ( )
multiplied by the tangential acceleration (which can be derived from the angular acceleration, by multiplying it
by the length of the rope). Rearranging (27) gives:
Although this is another second order homogenous differential equation, unlike (18) this is not a linear
differential equation, and it has no simple analytical solution. However for small angles
is very close to
, therefore when the amplitude of the oscillation is small we may approximate
by , and (28)
becomes:
(29) is identical to (18), with
replaced by
and
replaced by
. Therefore the solution is also identical:
where
and
and
are determined by the initial conditions. The period of the oscillation (
)
depends only on the length of the pendulum ( ), and the gravitational acceleration ( ). As both of these
parameters can be kept reasonably constant, pendulum clocks were considered to be the most precise clocks
from their invention in 1565 by Christiaan Huygens until the invention of quartz clocks in 1927.
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Note, that the calculation above is not an exact solution of (27), merely an approximation. It is true only for
small angles. For higher amplitudes the difference between
and increases, and the approximation
becomes less and less precise. It can be shown, that at high amplitudes the behaviour of the pendulum becomes
significantly different from that of a harmonic oscillator, and the period of the oscillation is not completely
independent of the amplitude.
In fact the calculation above is never completly accurate. Even at small angles
and are not exactly the
same, which means that the formulas above are never completly precise. But for small angles the deviations are
so small, that they are indistinguishable from other errors.1 For example the length of the pendulum can be
measured only with a limited precision, and it may even change with temperature. The gravitational acceleration
may be different at different locations, the pivot is not completely frictionless, and the viscosity of air may also
influence the movement of the pendulum.
These disturbing effects may be decreased by careful design, but they can never be completely eliminated.
Physical measurements always have some level of error. If the error which is caused by our approximations is
considerably smaller than measurement errors, it becomes undetectable: it is hidden by other disturbing effects.
The important lesson of this sub-chapter is that in many cases mathematical description of physical systems
becomes too complicated, and no analytical solution can be found. In such cases one may attempt to find an
approximate solution by simplifying the problem, like we did in case of the simple pendulum. But it is important
to keep it in mind that these kinds of calculations are not completely precise. They are applicable only under
certain circumstances, when the error of the approximation becomes negligible relative to other errors.
7.4. 6.4 Energy in simple harmonic motion
Equation (14) gives the acceleration of a body doing a simple harmonic oscillation. Using Newton’s law of
motion we can calculate the force which is acting on the body during harmonic oscillations:
This is a conservative force, which means there is a potential associated with it. The potential energy of the body
can be obtained by integrating (32) .
An indefinite integral is only defined up to an additive constant. We may choose the value of this constant, so
that the potential energy is zero in the origin:
Besides its potential energy the body also has kinetic energy:
It must be noted however, that approximations cause so called systematic errors, while some measurement errors are random. When the
experiment is repeated many times sytematic errors remain the same, while random errors are different each and every time. This means that
random errors tend to average out when calculating the mean of a large number of measurement values, while systematic errors cause a bias
in the mean value, no matter how many measurements are made.
1
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According to the Pythagoren theorem
, thus:
The total energy of the system is:
which is a constant quantity. This is not surprising since there are no dissipative forces in the system. As the
body is moving, energy is transformed from one form to the other: at the equilibrium the potential energy is
zero, while the kinetic energy is maximal. At the extrema the velocity and kinetic energy of the body becomes
zero, and all of the system’s energy is stored in the spring as potential energy.
7.5. 6.5 Damped oscillator
So far we have ignored friction and the drag force of the medium in which the oscillator is moving. But in
practice some level of dissipation is inevitable. Therefore it is necessary to expand our model of the simple
harmonic oscillator to take this into account. (17) can be modified by adding a new term representing a drag
force, which is proportional to the velocity of the body, but points to the opposite direction:
We may use the same trial function as before:
Substituting it into (40) gives:
We already know that
is the angular frequency of the undamped oscillator, which is usually
referred to as the natural angular frequency. Introducing the new constant
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Depending on wether
•
is less than, equal or greater than
we have three distinct cases:
(underdamped case):
introducing the new constant
, (47) becomes:
In the underdamped case oscillations occur due to the imaginary component of
. It must be noted however
that the angular frequency of these oscillations
is lower than the natural angular frequency
( ). The other main difference between the undamped and the underdamped oscillator is the result of the
real part of , which is responsible for the
factor. Damped oscillators lose energy due to the dissipative
drag force, thus the amplitude of the oscillation is not constant but decreases exponentially with time.
•
(critical damping): Increasing the viscosity of the medium further decreases the angular frequency of
the oscillations. The angular frequency reaches zero when
becomes equal to
. In this case the second
term on the right-hand side of (45) is zero, and both roots are the same. (The solution is degenerated.) But a
second order linear differential equation must have two independent fundamental solutions. The second
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fundamental solution can be found using the ansatz
combination of the two fundamental solutions:
. The general solution is the linear
At this point the frequency of the oscillation decreases to zero, thus even a single cycle would take an
infinitely long time. This is the so called critically damped oscillator.
•
(overdamped case): In the criticaly damped case, and at higher viscosities
has no imaginary part,
thus no oscillations can occur. The system simply crawls back toward the equilibrium position (without ever
reaching it):
7.6. 6.6 Forced oscillations
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In most practical cases losses are inevitable, and vibrations continuously lose energy. For an oscillator to be able
to operate for an extended period of time these losses have to be replenished. In other words a periodic external
force has to act on the system to avoid the decay of its vibration. This can be taken into account by expanding
(39):
where the right hand side represent the external force. Dividing both sides by
and
again:
, and using the constants
This is a second order inhomogeneous linear differential equation. The solution of such equations is the sum of
the solution of the homogeneous differential equation and a particular solution of the inhomogeneous equation:
where
is one of (49), (50) or (51), depending on the values of
following form:
This means that
and
. We may look for
in the
is the superposition of two oscillatory motions. In the beginning (transient state) the
system is oscillating like a damped oscillator with
angular frequency. But the amplitude of this
oscillation decreases exponentially due to damping. On the other hand the system is forced to start vibrating
with the frequency of the external driving force. While the damped oscillation quickly dies out, the amplitude of
the forced oscillation increases until the power lost to damping becomes equal to the power supplied to the
system by the external driving force. The amplitude and phase of the oscillation (both included in the complex
amplitude) in this stationary state can be determined by substituting (55) into (53) :
The magnitude of the complex amplitude is the amplitude of the oscillation:
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The phase of the complex amplitude represents the phase difference between the oscillation and the periodic
external driving force:
As the system is linear, the amplitude of the oscillation in stationary state is proportional to
. A more
interesting fact is that both the amplitude and phase-shift depends not only on the parameters of the oscillator
(such as the mass off the body, the spring constant and the strength of damping), but also on the frequency of the
external driving force. Figure 51 shows the amplitude of the oscillation as a function of the driving frequency
for different dampings. When the damping is strong, the amplitude decreases with increasing frequency. But as
the damping decreases a pronounced peak appears on the graph: the amplitude is maximal, when the
denominator on the right-hand side of (59) is minimal. This occurs when
reaches
This is called the angular frequency of amplitude resonance. The smaller the damping, the more pronounced the
resonance becomes and its frequency shifts towards the natural frequency of the oscillator.
The velocity of the oscillator can be calculated by differentiating the displacement with respect to time. In
stationary state the velocity amplitude is:
The velocity amplitude also depends on the frequency of the external driving force.
is maximal, when the
denominator on the right-hand side of (63) is minimal. This occurs when the quantity within the parenthesis
becomes zero.
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The velocity of the oscillator becomes maximal, when the frequency of the external driving force is equal to the
natural frequency of the oscillator. As the kinetic energy also reaches its maximum at this frequency, the
phenomenon is called energy resonance.
In forced oscillators both the amplitude and the kinetic energy increases until losses due to damping become
equal to the power supplied to the oscillator by the external driving force. When the damping is weak this may
result extremely strong vibrations, giving the false impression that a well-tuned oscillator can generate energy.
Many con-men try to utilise this false impression and claim that they are able to build perpetum mobiles based
on oscillators. Of course, this is not possible. It is important to understand, that oscillators cannot generate
energy, only accumulate the power of the external driving. (In fact, they are continuously losing energy to
damping.) Although the amount of accumulated energy can be very high, it is not generated by the oscillator.
Without the external driving force, vibrations decay exponentially.
7.7. 6.7 Superposition of simple harmonic oscillations
As we have seen in in the previous section, different oscillations may get superimposed on each other. In the
following we will discuss some special cases:
7.7.1. 6.7.1 Same frequency, same direction
Let us consider two oscillations in the
different initial phases:
direction with the same frequency, but with different amplitudes and
The superposition of the two oscillations is:
where
. In other words the superposition of the two oscillations is another harmonic oscillation of
the same frequency, with a complex amplitude, that is the sum of the complex amplitudes of the two
oscillations. The amplitude of this oscillation (which is the magnitude of the complex amplitude) can be
determined by using the law of cosines:
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The initial phase is:
7.7.2. 6.7.2 Different frequency, same direction
Let us consider two oscillations in the x direction with the same amplitude, but with different frequencies:
The superposition of the two oscillations is:
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The superposition of the two oscillations is an oscillation (represented by the
term in (75))
whose frequency is the average of the frequencies of the two oscillations, with an amplitude modulated by a
sinusoidal function (the
term in (75)). For example when two musical instruments emit
similar sounds, we cannot detect them individually. Instead we hear a single tone with a fluctuating intensity.
The phenomenon is called beat.
The amplitude, of the oscillation will be maximal when the cos() term equals 1 or -1. This means, that in each
cycle of the modulation function we can detect two fluctuations. Even though the frequency of the modulation
function is half of the difference of the two original frequencies, the frequency of the beat is the difference of the
two frequencies.
7.7.3. 6.7.3 Lissajous figures
It is also possible to describe the superposition of oscillations in different directions. The simplest case is when
two oscillations are perpendicular to each other. In this case we may choose the and axes of our coordinate
system to point in the direction of the two oscillations. The trajectory of the oscillating particle (the so called
Lissajous figure) depends both on the frequencies of the two oscillations, and also on the difference of their
initial phases. (We will assume, that both oscillations have the same amplitude. If this is not true, the Lissajous
figure will be scaled in the x and y directions proportionally to the respective amplitudes.)
Figure 54 represents typical Lissajous figures. In the first four cases the frequencies of the two oscillations are
identical. When the initial phases are equal to each other, the x and y coordinate of the oscillating particle
changes in unison and the particle moves on a straight line. Since
, the particle also
moves on a straight line, when the phase difference is
particle are:
. When the phase difference is
, the coordinates of the
These are the horizontal and vertical components of circular motion, and consequently the Lissajous figure
becomes a circle. For any other arbitrary phase difference the trajectory of the oscillating particle is an ellipse,
with an aspect ratio depending on the phase difference.
When the ratio of the frequencies in the two directions is n:m, the particle finishes n cycles in one direction
while it does m cycles in the other direction. This makes the Lissajous figures considerably more complicated.
When the frequencies are the same, the particle repeats the same trajectory in every cycle. When the horizontal
and vertical frequencies are different, the number of cycles it takes to close the Lissajous figure and start
repeating the same motion again is the least common multiple of n and m.
This also shows that Lissajous figures appear only when the ratio of frequencies is a rational number. When the
ratio is an irrational number there is no common multiple other than infinity, which means that the trajectory is
not a closed curve: although the horizontal and vertical components of the motion are both periodic, their
superposition is not.
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The ratio of frequencies can be determined by counting how many times the figure touches the horizontal and
vertical edges of the square that can be drawn around the figure. Since the particle finishes n cycles in one
direction while it does m cycles in the other, it touches the edges perpendicular to the first direction 2n times,
while it touches the other edges 2m times. Consequently the ratio of the horizontal and vertical frequencies is
the inverse of the ratio of the number of cross-sections with the horizontal and vertical edges. This technique can
be used to measure unknown frequencies with a two channel oscilloscope (that can plot the two signals against
each other in the so called X-Y mode) and a reference oscillator of known frequency.
7.7.4. 6.7.4 Fourier analysis
During this chapter we have focused on oscillatory motions that can be described by sinusoidal functions. But
the definition of oscillatory motion is more general: it does not require the motion to be harmonic, only to be
periodic. Therefore it is necessary to formulate a mathematical method to describe anharmonic oscillations. The
problem is that not all kinds of motion can be expressed in closed-form formulas. The best we can do is to create
a method that enables us to approximate any periodic motion at arbitrary precision.
Our task is slightly similar to what we had to do to describe the simple pendulum. Even though we cannot give a
completely precise mathematical description, we may be able to give an approximation that is so close to the
actual behaviour of the system that any deviations are unnoticable. Since we already have a basic understanding
of harmonic motion it is a sensible idea to approximate arbitrary periodic motions by the superposition of
sinusoidal functions.
Consider a periodic function
, with a period of . In the zero order approximation we may try to calculate
the time-average of
. Naturally, this is a very poor approximation since a constant cannot describe how
changes with time. We may improve the approximation by adding a sinusoidal function with the same
period as
to the average value, and adjusting its phase and amplitude, to achieve the best possible fit. Since
is not a sinusoidal function we may not achieve a perfect fit, but the difference between the arbitrary
function and this simple harmonic approximation is much smaller than the difference between
and its
average value.
If adding a single sinusoidal function has improved the approximation, we may try to further improve it, by
adding even more sinusoidal functions. Adding a second sinusoidal component of the same frequency is
pointless, since the sum of two sinusoidal functions of equal frequency is another sinusoidal function of the
same frequency with different phase and amplitude. As we have already optimised the phase and amplitude of
the first sinusoidal component, adding another one with the same frequency will not improve the approximation.
Therefore the second sinusoidal component needs to have a different frequency. But we also have to make sure,
that the approximation has the same period as
. This can be achieved by choosing the frequencies of further
sinusoidal components to be integer times the frequency of the first component. This way each sinusoidal
component will finish an integer number of cycles in
time, and their sum will have the same period as
.
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Adding each new sinusoidal component will slightly increase the quality of the approximation. By adding
together a suitably high number of sinusoidal components we may create an approximation of arbitrary
precision.
Ususally it is more efficient to represent all sinusoidal components in complex form:
To create an approximation of any arbitrary periodic function, all what we have to do, is to determine the
optimal values of , , , … to achieve the best possible fit. These values may be calculated by the formulas
deduced by Joseph Fourier in the nineteenth century:
Using these formulas we may express any arbitrary periodic function as a sum of sinusoidal functions. It must
be noted however that this is merely an approximation. The more components included in the sum, the more
similar it becomes to the original function, but to reach a perfect fit an infinite number of components would be
required, which is not possible in practice. When choosing the parameters of the approximation, two criteria
must be observed:
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• The frequency of the first sinusoidal component is determined by the period of the oscillation. The longer the
period, the lower the frequency needs to be.
• Low frequency sinusoidal functions change very slowly. Therefore if the
function changes rapidly, high
frequency components are required to follow these rapid changes. Since we have already chosen the
frequency of the first sinusoidal component to set the period, this can be achieved only by increasing the
number of sinusoidal components. As a thumb rule the period of the highest frequency component should not
be longer than the double of the required time resolution. Any changes faster than that are going to be lost in
the approximation.
8. 7 Waves - Gábor Dobos
Most people imagine waves as some sort of periodic disturbance which is moving in a medium, such as the
waves on the surface of a pond after a pebble is dropped into it. It must be noted however that in physics the
term is used in a much broader sense. Electromagnetic waves can propagate in perfect vacuum without the
presence of any medium, shock waves are non-periodic, and standing waves are not moving in space. The most
general definition of a wave that includes the above examples is a disturbance which can transport energy and
momentum without any long range movement of matter.
Waves on the surface of water as well as sound waves, or waves in a string of a musical instrument are called
mechanical waves as they mechanically distort an elastic medium. Electromagnetic waves consist of oscillations
of electrical and magnetic fields that can travel in space. In a plasma these fields can influence the movement of
charged particles forming different kinds of magnetohydrodynamic waves such as Alfvén waves or
magnetosonic waves. In some cases elementary particles also behave as waves. According to Einstein’s theory
of general relativity even gravitational waves may be created by distorting the fabric of the space-time
continuum, although these have never been experimentally detected due to measurement difficulties.
In this chapter we will discuss mechanical waves. These waves may be created by suddenly distorting an elastic
medium. Its surroundings will exert a restoring force on the displaced portion of the medium, pulling it back
towards the equilibrium position. As we have seen in the previous chapter this may lead to vibrations. But in
this case vibrations are not limited to the displaced section. As its surroundings exert a force upon it, it will also
exert an opposite force on its surroundings. Due to this force, the surrounding medium may also start to vibrate,
and the disturbance may propagate trough the medium.
An interesting feature of this type of motion is that although the disturbance may travel large distances, the
movement of a given piece of the medium is limited: it is oscillating around its own equilibrium position. The
thunder of lightning can be heard from a distance of several kilometres, but this does not mean that individual
air molecules travel this far. They collide with other particles, passing over their energy and momentum. It is
through these collisions that the disturbance reaches our ears, while the air as a whole does not move.
The direction of the oscillation can be either parallel or perpendicular to the direction into which the wave is
traveling. Longitudinal waves occur when the direction of the distortion is parallel to the propagation, like in
case of sound waves in air. In other cases the medium is oscillating at right angles to the direction into which the
wave is traveling. These are called transverse waves. It must be noted however that transverse mechanical
waves are not possible in all media since the restoring force must be parallel to the displacement. In case of
transverse waves this means that the restoring force has to be perpendicular to the direction of propagation. In
other words a shearing stress must exist in the medium, and this is not possible in all materials. For example no
shearing stress may exist in gases, thus sound waves in air are always longitudinal waves.
As we are living in a three dimensional world there are two directions that are perpendicular to the direction of
propagation. This means that in case of transverse waves the oscillations at each point may be the superposition
of two perpendicular oscillations. The frequency of these perpendicular oscillations must be the same, but their
initial phases might be different. The situation is very similar to what we have discussed in section 6.7.3. The
Lissajous figures created by the superposition of perpendicular oscillations of the same frequency may form a
straight line (when the phase difference is 0 or ), a circle (when the phase difference is or
) or an
ellipse (in case of any other arbitrary phase difference). Accordingly transverse waves might have linear
polarisation, circular polarisation or elliptical polarisation.
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8.1. 7.1 Sine wave
We have started the discussion of oscillatory motion with the simple harmonic oscillator, as it was the easiest to
describe in mathematical terms. For the same reason the first type of wave we will discuss in more detail is the
one dimensional sine wave, which can be described by:
where
is the displacement,
is the amplitude,
is the angular frequency, and
is called the wave
number. If is fixed, the displacement is a sinusoidal function of time. In other words, each point of the
medium is doing harmonic oscillations, with an initial phase depending on the position. If we fix the time (like
taking a snapshot of the wave) one can see, that the displacement is also periodic in space. The spatial period of
the wave (the distance between two consecutive corresponding points – like tow crests or zero crossings) is
called the wavelength and it is usually marked by . While the period of the oscillation is determined by the
angular frequency ( ), the wavelength depends on the wave number ( ):
The velocity of propagation can be determined by following the movement of a certain point of the wave, like a
crest or a trough. In other words we shall determine how x should change as a function of time, to keep the
argument of the sin() function (the phase) constant:
As it is the rate at which the phase of the wave propagates through space, it is called the phase velocity.
In case of simple harmonic oscillations, the acceleration of the particle was proportional to the displacement. In
case of a sine wave every point of the medium is doing a simple harmonic oscillation, thus:
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As y(x,t) depends on two parameters, we may also calculate its differentials with respect to the space coordinate
( ):
Comparing (6) and (8) gives:
Since we already know, that
is the phase velocity of the wave:
For sine waves the second differentials of the displacement with respect to time and to the space coordinate are
proportional to each other, and their ratio is the square of the velocity of the wave. Systems that can be
described by similar equations have the potential to exhibit wave motion, thus (10) is called a wave equation
8.2. 7.2 Transverse wave on a string
A string under tension is one of the simplest mechanical systems that can exhibit wave motion. Imagine a thin
wire with a linear mass density , fixed at one and, and pulled with a constant
force at the other end. When
a section of the string is displaced in a perpendicular direction, the tension in the string pulls it back towards the
original, straight position. In other words it represents a restoring force, and we may expect the displaced section
to start oscillating. But the sections of the string are not separated from each other. If one section starts
oscillating, it will exert a periodic force on the neighbouring sections, forcing them to start oscillating, too.
To give a mathematical description of the phenomenon, consider an infinitesimally small section of the string.
Both ends are pulled by the same
force, but in slightly different directions. (If a transverse wave is traveling
in the string, it is no longer straight, but slightly curved.) Applying Newton’s law of motion in the horizontal
direction:
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where
is the mass of the section,
is its length, and
function changes very slowly. If
,
direction is zero. In the vertical direction:
If the distortion is small (
is its acceleration. At small angles the cosine
. Thus the acceleration in the horizontal
) the sine function may be approximated by the tangent of the same angle. As
the tangent is basically the steepness of the curve, it may be replaced by
:
Using this approximation (13) becomes:
If the section is infinitesimally small (
):
(17) is a wave equation, and shows, that if the distortion is small, transverse sine waves are going to travel in the
string with
velocity. This may be verified by substituting (1) into (17):
As the equation must hold for all values of
and
:
8.3. 7.3 Energy transport by mechanical waves
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When a sine wave is traveling in a medium, each point is doing a simple harmonic oscillation. As we have seen
in the previous chapter the energy of a harmonic oscillator is:
where
is the amplitude of the oscillation, m is the mass of the oscillating body, and
is the angular
frequency. Using this, the energy of the wave in an infinitesimally small section of the medium is:
where
is the mass of the section,
wave travelling by
velocity needs
is its length, and
is the linear mass density of the string. A
time to travel dl distance. Therefore:
The power transmitted by the wave is:
The power transmitted by a wave traveling in a three dimensional medium may be calculated in a similar
manner. Consider a small
section of the medium:
where
is the density of the medium. The power transmitted by the wave trough a unit area of the medium is:
Human senses - such as hearing and vision - measure intensities on a logarithmic scale. This is the reason why
the intensity of certain types of waves is usually measured not in
, but in other, special units, on a
logarithmic scale. For example the intensity level of sound is usually given in decibels (dB). By definition:
where
. A
increase in the noise level actually means that the power of sound waves
reaching our ears is increased by one order of magnitude. Our ears have a very impressive dynamic range:
(
) is usually the smallest sound level that a human may detect. A whisper is around
), while normal talking is approximately
(
(
). Our eras can tolerate noise levels up to
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) for an extended period of time, without any lasting damage, and only noises louder than
(
) are immediately painful. Measuring a quantity on such a wide dynamic range on a linear scale would be
very difficult even with modern instruments. That is why most biological systems measure such quantities on a
logarithmic scale.
8.4. 7.4 Group velocity
As we have seen in the previous chapter superposition of two oscillations of similar frequencies creates a beat.
A similar phenomenon can be observed with waves. Imagine two waves travelling in the same medium with
slightly different angular frequencies and wave numbers. (This is actually a very realistic model. In practice it is
impossible to create a perfectly monochromatic wave: usually we have to deal with the superposition of several
waves with slightly different parameters.) The wave that we can observe is going to be the superposition of the
two waves:
using the trigonometric identity
:
where is the average of
and , and
is the average of
and
. The superposition of the two waves
gives a wave whose wave number and angular frequency is the average of the wave numbers and angular
frequencies of the two original waves (this is called the carrier wave), but whose amplitude is not constant, but
modulated by a sinusoidal function. Since this modulation function also depends on both and , its nodes
(the positions where the amplitude is minimal) and antinodes (the positions where the amplitude is maximal) are
also going to move in the medium.
Two waves traveling in the same medium with similar angular frequencies are going to have similar wave
numbers and phase velocities, too. Since the angular frequency and wave number of the carrier wave is the
average of those of the two waves, its velocity is also going to be similar.
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The groups formed by the modulation are going to travel with a different velocity:
If the frequencies and wave numbers of the waves are infinitesimally close to each other
This is called the group velocity and it is a very important parameter not only in physics, but also in the field of
telecommunication. Information is usually transmitted by the modulation of some sort of wave. The type of
carrier waves may vary from sound waves to electromagnetic waves, and different types of modulation
techniques are in use, but in most cases the velocity with which information can be transmitted from one point
to another is the same as the velocity of the groups formed by the modulation. Even though electromagnetic
waves travel at the speed of light, this does not mean that information can be transmitted at the same velocity.
For most media the group velocity is lower than the phase velocity, therefore the speed of information
transmission is also lower.
It must be noted however that there are exceptions from this rule: the group velocity is not necessarily the same
as the “signal velocity”. The
function is called the dispersion relationship. For most materials the slope of
this function decreases for higher frequencies, thus the group velocity is lower than the phase velocity. But there
are some exotic materials with a so called anomal dispersion relationship. In these materials the group velocity
may be higher than the phase velocity (or the speed of light). This does not mean however that these materials
enable faster than light information transmission. As a thumb rule the signal velocity cannot be higher than the
lesser of the phase and group velocities.
8.5. 7.5 Wave packets
Another interesting parameter that can be understood by considering the superposition of waves is the amount of
information that can be transmitted in a unit of time. Although actual communication protocols are very
complicated the rate of information transmission depends on the duration of pulses that can be formed by
modulating the carrier wave. (The shorter the pulses the more data can be transmitted in a unit of time.)
In section 6.7.4 we have used Fourier analysis to describe anharmonic oscillations as the superposition of a large
number of sinusoidal components. In a similar manner we may attempt to construct short pulses from the
superposition of sine waves. (This kind of construct is usually referred to as a wave packet.) The problem is that
Fourier analysis is applicable only to periodic functions, and a pulse is not periodic. This problem may be
remedied by treating the non-periodic function describing the pulse as a periodic function with an infinitely long
period. (If the period is infinitely long, the function never repeats itself...) As the frequency difference between
Fourier components is the inverse of the period, the Fourier components of an aperiodic function are infinitely
close to each other. In other words the components do not belong to discrete frequencies: they form a continuous
distribution. This distribution is called the Fourier transform of the function.
where
is the time,
is the frequency, and
is the function whose Fourier transform is
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Let us consider the Fourier transform of a Gaussian-shaped pulse (In nature quantities tend to have a so called
normal distribution which is described by the Gaussian function. Therefore it is reasonable to discuss Gaussianshaped pulses. Of course, other pulse shapes may also be described in a similar fashion.):
The duration of the pulse is determined by the constant
, which is the full with at half maximum of the
Gaussian function. The Fourier transform of this function is:
This function gives the amplitudes of the harmonic components whose superposition is the
shaped pulse. This is another Gaussian function whose full width at half maximum is
:
wide Gaussian-
It must be noted, that information is transmitted by the modulation of a carrier wave. This means, that the
components, that make up the pulse should centre around the frequency of the carrier wave ( ). In other words
we shall shift the
function by
. The pulse formed by these components is:
Of course this is not a pure Gaussian-shaped pulse, since it is modulated be the
factor, which is
basically a sinusoidal modulation with the frequency of the carrier wave. Nonetheless the envelope of the pulse
is still a Gaussian function whose full width at half maximum satisfies:
This means that in order to construct short pulses (and reach high data transmission rates) we have to include
Fourier components from a wide frequency range. The wider the available frequency window, the shorter the
pulses may become.
This explains why telecommunication corporations are willing to pay billions to obtain the right to use certain
frequency windows. Although we usually say that - for example - cell phones transmit data at 900 MHz
frequency, in reality they use not just a single frequency but a range of frequencies around 900 MHz. Each
provider has its own frequency window, for which it pays a concession fee to the government. The wider this
window is, the faster your mobile internet connection may become.
Based on this, we may also understand why the number of radio and television stations is limited. Although in
layman terms we usually say that a station transmits at a given frequency, this refers to the frequency of the
carrier wave. In reality the station uses a range of frequencies around this frequency, to modulate the carrier
wave. This is why the frequencies of radio stations cannot be arbitrarily close to each other lest they interfere.
It must also be noted that (45) describes a much more basic principle, which has a wide range of implications
beyond telecommunication. Equations like (45) are usually referred to as uncertainty relations. Although we
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may be able to form wave packets by combining a large number of sinusoidal components, the length of these
wave packets cannot be infinitely short, and they always include a large number of different frequencies. The
dimensions of the wave packet in the time domain and in the frequency domain are related to each other. The
shorter the pulse is the wider range of frequencies it must include. When we use only a narrow frequency range,
the pulse is going to get longer.
It is easy to see, how a similar relation can be deuced between the length of a pulse (its
range of wavenumber ( ) components that must be combined to form the wave packet:
dimension), and the
If we try to measure the position of the wave packet, the precision of our measurements will be limited by
.
In a similar fashion
limits the precision of wavenumber measurements. Equation (46) shows that the position
and wave number of the wave packet cannot be measured at arbitrarily high precision at the same time. The
large number of sinusoidal components that are required to form a short pulse makes wavenumber
measurements uncertain. On the other hand, if we use only a narrow range of wavenumber components
increases and the pulse becomes blurred, which decreases the precision of position measurements.
In general, where waves are concerned there are certain parameters that are connected to each other. These
parameters cannot be measured at the same time at arbitrarily high precision. As we will see during the
discussion of quantum mechanics, the quantum states of physical systems are represented by wave functions,
and quantum uncertainties are closely related to the principles that we have discussed above.
8.6. 7.6 Standing waves
Imagine two waves of the same frequency traveling in the same medium in opposite directions. (Such situations
may arise when the wave is reflected back from the end of the medium.)
The superposition of the two waves is:
Using the same trigonometric identity as in the previous section:
The superposition of the two waves is a so called standing wave. Each point of the medium is oscillating with
the same frequency, but unlike other waves, in case of standing waves, the initial phase is the same in every
point of the medium. This means that the wave is not moving. This is because the position and time
dependencies of the disturbance are separated into two terms. All points are oscillating in unison, with an
amplitude determined by the position. (The situation is slightly similar to that what we have seen in section 7.4,
but in this case the group velocity is zero: the groups are not moving.) The positions where the amplitude of the
oscillation is minimal are called nodes, the ones, where the amplitude is maximal, are called anti-nodes.
Most musical instruments are designed to efficiently form certain types of standing waves and reject others. For
example in case of string instruments - such as guitars and violins - the ends of the strings are held tight. This
means that the standing wave must have nodes at both ends, otherwise its energy dissipates away very quickly.
When the player uses the instrument many waves with different frequencies are created, but only those may
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form stable standing waves, that have nodes at both ends. The distance between two neighbouring nodes is ,
therefore only those waves are stable, whose half wavelength fits integer times into the length of the string. The
wavelength ( ) is determined by the wave number ( ), which in turn depends on the velocity of the wave ( )
and its frequency ( ):
The so called principal mode is the standing wave with the lowest frequency that is stable in the string. It has
nodes at both ends, but there are no further nodes in between. Therefore the wavelength is twice the length of
the string:
This is called the fundamental frequency of the instrument. The first harmonic has one extra node between the
ends:
The second harmonic has two nodes between the ends
And so on... In general:
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Only waves with these frequencies form stable oscillations in the string, all other frequencies are rejected. The
phenomenon is a type of resonance, and such instruments are commonly referred to as resonators. The ratio of
the intensities of the stable harmonics (or overtones) depends on the design of the instrument, and it is
characteristic to each resonator. It is these intensity ratios that give each musical instrument its own unique tone.
Other types of musical instruments have different resonators. For example a pan flute consists of small tubes
closed at one end, and open at the other end, so that the player can blow air into it. The standing waves forming
in such a resonator needs to have a node at the closed end an anti-node at the open end. The distance between a
node and an anti-node is a quarter wavelength, thus:
The first overtone has an additional node between the two ends:
The second overtone has two nodes between the ends:
In general:
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Resonators open at both ends, have anti-nodes at both ends. The principal mode has a single node in between:
The first harmonic has two nodes between the anti-nodes at the ends:
The second harmonic:
In general:
8.7. 7.7 The Doppler Effect
When a racing car drives through the finish line the spectators on the stands will hear a shift in the frequency in
its sound: it seems to be higher when the car races towards the spectators and drops quickly as it drives past. The
phenomenon is called Doppler Effect, and it is detected when the source or the observer is moving with respect
to the medium in which the wave is traveling.
To better understand the phenomenon imagine a stationary observer and a sound source moving towards the
observer at a given
velocity. If the sound source would be stationary, the wavelength ( ) of the sound waves
could be calculated from the ratio of the speed of sound ( ) and the frequency ( ) emitted by the source (
). But the movement of the source compresses the waves and changes their wavelengths. This can be
understood by calculating the distance between two crests of the wave emitted by the source. During one period
(
) the crest of the wave emitted by the source travels
distance towards the observer. But in the same
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time the source is also moving, and it gets closer to the observer by
distance. Because of this, when
the next crest of the wave leaves the source, its distance from the previous one is not , but:
Due to this shift in the wavelength of the emitted sound, the observer is detecting a shift in the frequency of the
sound:
The Doppler Effect is also observed when the source is stationary, and the observer is moving towards it. To
determine the frequency detected by the observer we have to calculate how many cycles of the wave reaches
him or her in a unit of time. (You may imagine this by following the crests of the sound waves. The detected
frequency is the number of crests reaching the observer in a unit of time.) Assume that the velocity of the wave
is
and its wavelength is
. The wave requires
detected by a stationary observer is
is
time to travel
distance. Therefore the frequency
. (If a crest reaches the observer every
.) But when the observer is moving towards the source with a
time, the frequency
velocity it will encounter a further
oscillations in the same unit of time. Therefore the frequency detected by the moving observer is
(85) and (86) can be combined into a single formula:
where
sound,
is the frequency emitted by the source,
is the frequency detected by the observer,
is the velocity of the observer and
is the velocity of the source, respectively.
is the speed of
The phenomenon has widespread practical applications. Speed traps are important tools in the hands of the
police to enforce speed limits. Doppler radars may measure wind speed. Modern ultrasonic imaging devices
may utilise this principle to measure blood flow in our veins. Even the expansion of the universe was detected
by the red shift in the spectrum of distant stars due to the Doppler Effect. (It must be noted however, that the
formula for electromagnetic waves is different form (87) due to relativistic effects.)
9. 8 First law of thermodynamics and related subjects
- György Hárs
The subject of study is the thermodynamic system, which is separated from the environment by the boundary.
The boundary does not allow material transport between the environment and the system so the mass contained
is constant. In general the boundary allows transfer of heat and mechanical work. The heat transfer may be
inhibited by insulation and so the system is called "thermally isolated" or in other words "adiabatic" system. The
mechanical interaction may also be excluded by using rigid boundary, which system is called "mechanically
isolated". If both thermal and mechanical isolation is active then the system is considered "isolated" system.
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The thermodynamic system contains gas in this chapter with definite primary state parameters such as pressure (
, temperature (
and volume (
. These parameters are considered primary concepts with no further
definition in the phenomenological discussion of the subject. The pressure and the temperature are intensive,
driving force-like parameters. The volume is extensive, quantity-like parameter. Microphysical substantiation of
the state parameters is provided by the kinetic theory of gases.
9.1. 8.1 Ideal gas equation
The ideal gas equation is an empirical law with units to be substituted as follows:
One mol of material consists of 6 10 pieces (Avogadro number
of particles. This is called the amount of
substance. The molar mass (
is the mass of one mol substance, which is equal with the same number of
grams as the atomic mass number of the particle. In System International the molar mass should be substituted
in kilograms per mol. The first and second version of the ideal gas law contain the Boltzmann constant ( , the
last two use the universal gas constant ( . The temperature is measured in (
Kelvin degrees which is the
absolute temperature scale with the starting value at -273 degrees Celsius. The following relations can be found
between the constants:
and
9.2. 8.2 The internal energy of the gas U
The particles of the gas make chaotic motion in the container. This motion represents kinetic energy which is
considered the internal energy of the gas. The calculation is carried out in the framework of the kinetic theory of
gases with the following basic assumptions.
• The particles interact by elastic collisions with the chamber walls and with each other.
• No external potential is applied, so the total energy is kinetic.
• Uniform spatial distribution is in the chamber.
• There is no dedicated direction (isotropic structure).
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Particles bounce back from the wall of the chamber. Let us calculate the amount of linear momentum transferred
to the wall in a short dt time. The x direction is normal to the wall,
is the component of the velocity. Half
of those particles reach the wall in dt time which are in .dt proximity of the wall, since the other half moves
opposite direction from the wall. Multiplying this thickness with the half of the density and the area, the total
colliding particles result. The mass of the particle is denoted . The variation of momentum is 2
due to
bouncing back with negative . So ultimately the total momentum (dI) transferred to the wall is as follows:
The time derivative of the momentum is the force:
The pressure is the ratio of the force over the area:
This is true in all three directions.
Let us summarize. The root means square velocity is the Pythagorean sum of the component velocities:
The pressure can be expressed:
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The ideal gas equation can also be taken into consideration:
The right hand sides of the equations are equal:
After some ordering the kinetic energy of the particle is revealed:
In the argument above the kinetic energy of the particle was associated with three translational coordinates,
which represent three thermodynamic freedom degrees. The conclusion demonstrates the principle of
equipartition. According to this principle, all kinds of energy storage capabilities contain equal amount of
energy which is one half kT. In reality the particles may also rotate in three normal directions, so maximum six
thermodynamic freedom degrees are at disposal. If in general case the thermodynamic freedom degree is
denoted
the total kinetic energy stored in one particle is as follows:
If one considers
pieces of particles the value of the internal energy (
results.
Let us use the relation expressed earlier:
With this the expression of the internal energy can be transformed:
Let us introduce the molar heat capacitance of constant volume with the following definition:
So ultimately the internal energy of the gas is revealed:
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The internal energy is a secondary state parameter of the gas, which means that it is unambiguously generated
from a primary state parameter (concretely from the temperature). The final parameter is
the initial is .
The variation of the internal energy at finite and infinitesimal temperature variations is as follows:
9.3. 8.3 The p-V diagram
The p-V diagram is a phase-plane on which each point represents a state of thermodynamic equilibrium. The
horizontal axis is the volume the vertical is the pressure. Due to the ideal gas law points of identical temperature
are located on hyperbolas which are called isotherms. The higher the temperature the further is the hyperbola
from the origin. The thermodynamic process which progresses reasonably slowly to reach equilibrium state at
all times is called reversible process, provided no heat dissipation happened due to friction. Therefore the
process can be displayed by a solid line on the p-V diagram. Rapidly occurring irreversible processes
(irreversible jumps) with finite variations do not reach equilibrium during the process they reach equilibrium in
the final state only. State parameters such as and
are defined for equilibrium, so these parameters are nonexisting during irreversible jump. A doted line from the starting point to the final point shows the jump on the pV plane.
9.4. 8.4 Expansion work of the gas
If the volume of a thermodynamic system increases then the system carries out work on the environment. This is
called the work of the gas which is a positive number upon expansion. The infinitesimal amount of work can be
expressed as follows:
If a finite expansion process is concerned the work of the gas is the integral
of the infinitesimal contributions.
The work of gas (
can be viewed as the area under the curve in the p-V diagram. The advantage is that
the integral is positive or negative simultaneously with the work of the gas.
9.5. 8.5 First law of thermodynamics
The variation of internal energy can be caused by two effects. Either heat transfer (
or external mechanical
work (
can change the internal energy of the gas. The transferred heat is obviously positive if the system
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accepted heat. The mechanical work can either be external work, which is positive upon compression and work
of the gas, which is positive upon expansion. One has to be careful not to make confusion. From practical point
of view it is advised to insist on using the work of the gas.
The first law of thermodynamics is essentially the conservation of energy. Most practical forms for infinitesimal
and finite cases are as follows:
The above described form of the first law can be interpreted as follows: The transferred heat (
is utilized
for two purposes. Partly the internal energy (
of the system is increased and partly the gas carried out
mechanical work (
on the environment.
sectionThermodynamic processes
In this section reversible processes are discussed. Let us transform the ideal gas equation in the following
manner:
If the amount of the gas does not change during a process the right hand side of the above equation is constant.
Accordingly can be written:
This is called the united law of gases. Historically it is worth to mention the more specific laws which date back
to those scientists who discovered the actual laws.
9.5.1. 8.5.1 Isochoric process
Here the volume is constant during the process. The volume cancels out from the united law of gases:
This is the Gay-Lussac II. law.
The work of the gas is zero since there is no change in the volume (
increased by the total amount of the heat transferred.
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0). The internal energy of the gas is
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This equation explains the why the notation
heat capacitance of the gas.
is used. In isochoric process this quantity shows up as molar
In any processes however the variation of the internal energy can be expressed by means of
formula.
with the above
9.5.2. 8.5.2 Isobaric process
Here the pressure is constant during the process. The pressure cancels out from the united law of gases:
This is the Gay-Lussac I. law. (The name with hyphen corresponds to one person.)
The work of expansion is the mere product of the pressure and the change in the volume.
Let us use the ideal gas equation.
The difference of the above equations is as follows:
Now the first law can be used:
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Accordingly the molar heat capacitance in case of isobaric process is denoted
.
9.5.3. 8.5.3 Isothermal process
Here the temperature is constant during the process. The temperature cancels out from the united law of gases:
This is the Boyle-Mariotte law. (The name with hyphen corresponds to two persons who independently
discovered this law.)
In here the internal energy is constant (
0). The pressure is expressed from the ideal gas equation.
The work of the gas can be calculated by integrating the pressure between the border volumes.
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The first law looks like as it follows:
According to this result the total heat transferred has been turned to mechanical work.
9.5.4. 8.5.4 Adiabatic process
It needs to be emphasized that reversible adiabatic process what we are dealing with. The word adiabatic on it
own means a process without heat exchange with the environment however the mechanical work can be
transferred to the system. In contrast to this neither heat nor mechanical work can be transferred to the system
which is said to be isolated.
Since there is no heat exchange with the environment (
written as follows:
0), the infinitesimal form of the first law can be
Earlier the constant volume molar heat capacitance has already been introduced:
After substitution one can find:
On the other hand consider the temperature derivative of the ideal gas equation.
The right hand side of the last equation can be recognized in the equation above in parenthesis. Let us substitute
it.
Here dT cancels out.
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The variation of the product can be separated.
Here we introduce the adiabatic exponent denoted by the Greek letter (kappa
This is a separable differential equation:
Let us integrate from the initial value to the final value:
Since the natural logarithm function is unambiguous so the arguments are equal.
In general can be written the exponential equation of the adiabatic process:
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The equation of the adiabatic process can also be expressed by means of other primary state parameters as well.
Let us combine it with the universal gas law.
After dividing the equations
Express the
cancels out:
from the second equation and substitute back to the first one:
9.6. 8.6 Summary of the molar heat capacitances
The molar heat capacitances have been used extensively at the discussion of the thermodynamic processes. The
following parameters have been discussed: (
8.3 J/molK)
The thermodynamic freedom degree is the number of the energy storing capabilities each of which can contain
kT kinetic energy, according to the equipartition principle. The noble gases with mono-atomic molecule have
three freedom degrees, which are the
translational directions. The diatomic molecules such as the air and
lot of others have the three
translations and two additional rotational axes except for one direction which
connects the centers of the atoms. The exception is explained by the fact that looking at the molecule in the
central line it looks like a point without extension, thus it lacks of moment of inertia. Higher number of atoms in
the molecule generates six thermodynamic freedom degrees since three translational and three rotational
motions are all available.
9.7. 8.7 The Carnot cycle
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The Carnot cycle is a reversible cycle which consists of two isothermal and two adiabatic processes. This is the
classical example of the heat engine which is capable of generating mechanical work from a certain part of the
heat.
The corner points of the cycle are denoted
and
. The points
and
are located on the
isotherm while the
and
points are located on
isotherm. The
temperature is higher than
therefore
isotherm is closer to the origin of the p-V plane. The process starts in point
• AB section: (work output is positive in this section)
This is an isothermal expansion on
temperature. The heat intake from the heat bath is
. The variation of
the internal energy is zero (
0) therefore the transferred heat is fully converted to mechanical work
• BC section: (work output is positive in this section)
Adiabatic expansion (
0) as long as the temperature cools down to
positive mechanical work on the expense of its own internal energy.
temperature. The gas carries out
• CD section: (work output is negative in this section)
Here an isothermal compression takes place on
temperature. The variation of the internal energy is zero (
0) therefore mechanical work
is fully converted to heat
which in turn has been transferred
to the sink.
• DA section: (work output is negative in this section)
Adiabatic compression (
0) as long as the temperature warms up to
increased the internal energy of the gas.
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The total amount of work carried out by the cycle
above: Apparently
and
cancel out.
is the sum of the mechanical works of the processes
The above formula can be simplified further by considering the fact that the BC and DA section are adiabatic
processes. According to the laws of adiabatic process the following equations can be written: (See at the end of
the section "adiabatic process".)
After dividing the two equations the temperatures cancel out on both sides:
This can be substituted to the expression of the work of the cycle.
The work of the cycle can be viewed as the unusual rectangular area on the
plane. If the direction of the
circumference is clockwise then the cycle carries out a positive work on the environment. In other words this is
a heat engine.
The thermal efficiency of the heat engine (
is defined as the ratio of the work of the cycle (
over the
heat intake from the heat bath (
. In other words this is that percentage of the heat intake which has been
converted to mechanical work during the cycle. The rest of the heat intake necessarily goes to the sink.
Many terms cancel out.
The work of the cycle can be expressed in an alternative way as well:
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The efficiency is as follows:
The two expressions of the efficiency are equal:
After some ordering an important formula is the result.
The two terms of the above equation are called "reduced heat", which are the ratio of the heat and the actual
temperature at which the heat transfer took place. This equation declares the fact that the sum of the reduced
heats for the Carnot cycle is zero. This has far reaching consequences in connection with the concept of entropy.
10. 9 The entropy and the second law of
thermodynamics - György Hárs
At the end of the previous chapter an important relation has been revealed.
The two terms of the above equation are called "reduced heat", which are the ratio of the heat and the actual
temperature at which the heat transfer took place. The "reduced heat condition" declares the fact that the sum of
the reduced heats for the Carnot cycle is zero. This has far reaching consequences in connection with the
concept of entropy.
10.1. 9.1 The entropy
Consider the p-V phase plane. Let us draw in solid line numerous isotherms in equidistant small temperature
steps. Similarly draw in doted line numerous adiabatic curves with small equidistant increments. Now an
arbitrary reversible cycle is plotted on the top of the isotherm and adiabatic grid.
The cycle on the figure can be approximated with little motions partly on the isotherm and partly on the
adiabatic curves. This way the arbitrary cycle is composed of several tiny Carnot cycles. The reduced heat
condition is valid for each of them. The reduced heat intake at the high temperature side for a tiny Carnot cycle
is equal to the reduced heat egress at the low temperature side. Therefore the reduced heat contributions cancel
out by pairs, thus the total sum of the reduced heats is equal to zero for the whole arbitrary cycle. If one makes
the grid infinitely fine the sum of reduced heats is converted to the closed loop integral. Accordingly this can be
written:
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If the arbitrary closed loop integral is zero this is equivalent with the statement that the integral between two
points does not depend on the path of the integration.
Let us break the closed loop with two points
is the final point.
Integrating from
to
and
. Point
will be considered the starting point while
is the negative of the opposite direction integration.
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The integral depends on the initial and final points only.
Since the integral is independent of the path therefore the concept of entropy (
can be introduced. Variation
of the entropy can be calculated on the most convenient path, since the actual path is unimportant. First an
isochoric process takes us to the intermediate point (
then an isothermal process reaches the final point ( .
In isochoric process
therefore the following integral is evaluated
After this an isothermal process follows:
.
Here we use the ideal gas equation in the following form:
So the total entropy variation is the sum of the two sub-processes:
The entropy variation can be expressed by any two of the three (
made the (
version. The (
and the (
versions follow.
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primary state parameters. We have
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The ideal gas equation for the final and the initial states are as follows:
Let us divide them.
Take the natural logarithm of both sides:
The above equation can substituted into the formula of the entropy variation.
Accordingly:
Similarly
Now the above equation is substituted to the entropy variation:
Accordingly:
Direct physical meaning can be associated to the variation of entropy between two points of the
plane.
By choosing a universal initial point as reference point, the entropy function can be converted to a secondary
state parameter. This time any point on the
plane is characterized by a single entropy value. Practical
choice can be the following reference point. (
273K,
10 Pa,
2.266 10 m /mol).
10.2. 9.2 The isentropic process
Let us find out where those points are located on the
words we are looking for the isentropic curves. Here
expressed as follows:
plane, which have identical entropy value. In other
is the condition. The entropy variation is
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Here we recall the adiabatic exponent (
molar heat capacitances.
which has already been introduced in chapter 8 as the ratio of the
This is a known formula, which was derived first at the discussion of the reversible adiabatic process in the
previous chapter. Now it has been proven that the isentropic process is identical with the reversible adiabatic
process.
10.3. 9.3 The microphysical meaning of entropy
Let us make some elementary combinatorial calculations to reveal the fundamental roots of entropy. We have
two compartments (1 and 2) and four particles (
4) with the names a, b, c and d. From the point of view of
micro states the particles are distinguishable, which means that a new micro state is generated if two particles
replace each other. However these micro states represent the same macro state since in macro state only the
number of the particles count. Accordingly, from point of view of macro states the particles are
indistinguishable. The concept of thermodynamic weight is associated with the macro state and it is defined as
the number of those micro states which generate the given macro state. In the table below all the possible micro
states are listed as well as the corresponding thermodynamic weights.
In present case the thermodynamic weight (
can be expressed as follows:
In general case there are several elementary cells. The number of cells is denoted with . A series of numbers
provide the characteristics of the macro state
where the numbers represent the
numbers of particles in the elementary cells from 1 to . The corresponding thermodynamic weight is as
follows:
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In terms of combinatorics the above formula is a permutation with repetition. In the numerator the total number
of permutation shows up provided all the particles are distinguishable. In the denominator there is the divider
that contains those cases which differ in the order of particles within the cell. These cases do not constitute new
macro state since only the amount of particles in the cell counts.
10.4. 9.4 Gay-Lussac experiment
The gas subjected to an isolated irreversible jump in this classical experiment. The phenomena will be studied
by both phenomenological and statistical thermodynamics.
10.4.1. 9.4.1 Phenomenological approach
Consider a gas container which consists of two internal compartments with volumes
and . Between the
volumes there is a valve that can be operated remotely. The whole container is thermally isolated from the
environment. One of the volumes (
contains gas with known parameters. Vacuum is in the other
compartment (
. Suddenly the valve is opened the gas flows partly to the empty volume. During the gas flow
a whistle noise can be heard. Thermometers attached to both compartments show that
cooled down and
warmed up by some extent. When the noise died out and some minutes passed both thermometers show the
initial temperature, in accordance with the first law of thermodynamics. No work has been done by the gas no
heat has been transferred to the system so the total internal energy must have been conserved. This means that
the temperature is necessarily unchanged.
Though the temperature is identical with that in the initial state, this is far of being an isothermal process. This is
an isolated irreversible jump without work done. The initial and the final states are in thermal equilibrium
therefore drawing these two points on the
plane is justified. However due to the irreversible jump it is
not is not allowed to draw any solid line much a rather a doted line from the initial to the final point.
Since the entropy is a state parameter the variation of entropy can be expressed without any respect to the events
which took the system from the initial to the final state (here I did not use the word "process" intentionally). The
known formula from the previous page can readily be used with the condition that
and
.
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The first term cancels out due to the identical temperatures values.
The final volume is obviously bigger than the initial therefore the entropy variation is positive. The irreversible
jump takes place necessarily to the right hand side direction to a higher entropy value adiabatic curve. In
addition the initial and final states are on the same isotherm.
10.4.2. 9.4.2 Statistical approach
In the figure below the gas container of the experiment is shown. Let us subdivide the total
volume to very
small identical cells with the volume denoted with . The number of cells
and
comes out as follows:
The actual emerging macro states carry the highest thermodynamic weight. In these macro states all the cells
contain the same amount of particles. Here
is the total number of particles. The initial and final macro states
are the following:
The corresponding thermodynamic weights can be readily written:
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It is worth to be are aware of the fact that 0!
1. Let us calculate the ratio of the thermodynamic weights.
An approximate formula for the natural logarithm of the factorial will be used here:
The relative error of this formula is tending to zero when goes to the infinity. In present case when
extremely high the approximation is especially accurate. Sketch of proof follows at the end of this section.
is
Multiply the equation with the Boltzmann constant.
Compare it with the variation of entropy stated earlier:
The right hand sides match perfectly so the left hand sides are equal:
The above formula is called the Boltzmann equation. Now this has been proven for the special case of the GayLussac experiment QED.
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10.5. 9.5 The Boltzmann equation
The Boltzmann equation is a fundamentally significant equation which establishes the bridge between
phenomenological and the statistical approach of the thermodynamics. Its importance is far greater than that it
has just been proven for.
Here
is the Boltzmann constant
,
and
are the thermodynamic weights of the
and
macro states. The message of Boltzmann
equation in connection with the Gay-Lussac experiment is clear: The entropy increases according to the
phenomenological formula. Therefore the variation of entropy is positive so the logarithm value in the
Boltzmann equation should also be positive. Accordingly the ratio of thermodynamic weights is bigger than unit
which means that far greater thermodynamic weight belongs to the final macro state than to the initial macro
state. The fact can be generally stated that in the isolated system at irreversible jump the final equilibrium state
is located on a higher value isentropic curve on the p-V plane. The thermodynamic weight of the final state is
bigger than that of the initial. At irreversible jumps the system is tending to the state of highest possible
thermodynamic weight which is the final equilibrium macro state.
10.6. 9.6 Approximate formula
a sketch of proof:
The sum of logarithms of integers can be approximated with the following integral:
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10.7. 9.7 Equalization process
So far the subject of study was a single system consisting of gaseous particles. In equalization processes more
thermodynamic systems take part, typically two systems will be considered here. Initially these systems are in
thermal equilibrium on their own. Suddenly they are united and the equilibrium state of both systems vanishes.
Violent irreversible actions can proceed thus no equilibrium state parameters can be defined. Strictly speaking
temperature, pressure, internal energy and entropy are not existing parameters in course of the events. After
some time these actions are gradually relaxing and the united system occupies its newly born equilibrium state
parameters.
10.7.1. 9.7.1 Equalization between gaseous components
Let us consider two separated gas systems. They contain the same type of gas with the following equilibrium
state parameters
and
The gas systems are united by removing the separating wall between
them so the volumes are added. We wonder the final state parameters and the variation of entropy.
The ideal gas equation will be used in the following form:
Here
denotes the number of moles.
The internal energy, the volume, the number of moles and the entropy are added together upon uniting the
systems. The parameters without subscripts characterize the united system.
The final temperature can be written accordingly:
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This is a weighted arithmetic mean in terms of mathematics where the weights are the corresponding mole
numbers.
Now the pressure of the united system needs to be found. This is expressed from the ideal gas equation:
Two terms cancel out. The pressure finally can be expressed.
This is a weighted harmonic mean in terms of mathematics where the weights are the corresponding nT
products. For later use it is worth to express the
over ratio in the final state. This is a weighted arithmetic
mean of the
ratios in the initial states where the weights are the corresponding moles.
Now the entropy variations need to be dealt with. The term variation means the difference when the initial state
parameters are subtracted from the final state parameters.
The total entropy variation of the equalization is the sum of the two variations above:
The universal gas constant is factored out.
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For convenience reasons a new parameter is introduced here:
With this parameter the formula can be transformed:
Let us organize the fraction under a single logarithm.
The total variation of entropy due to the unification is finally revealed.
The newborn equilibrium parameters show up in the numerator while the initial parameters are in the
denominator. The purpose is to prove the second law of thermodynamics in mathematical precision. The second
law states that the variation of entropy is always positive at irreversible jumps of an isolated system.
Let us consider the numerator which is associated with the final state parameters. The temperature over pressure
has already been expressed above from the original formulae.
According to the well-known mathematical theorem weighted geometric mean is always not bigger than the
weighted arithmetic mean.
So replace the numerator with a smaller or equal quantity. This way the calculated formula is inevitably
diminished or unchanged.
Organize the fraction under a single root sign.
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Here the pressures cancel out.
Here
denotes the thermodynamic freedom degree which can be 3, 5 and 6.
Let us substitute the value of the final temperature to the numerator.
In the numerator the weighted arithmetic mean of the initial temperatures is present. In the denominator the
weighted geometric mean of the same temperatures shows up with the identical weights. Here again one has to
refer to the mathematical theorem of the inequality between the arithmetic geometric means. The numerator is
surely not smaller than the denominator so the fraction is not smaller than unit, which means that the logarithm
value is a non-negative number. So ultimately the second law of thermodynamics has been proven for the
concrete case of the isolated irreversible jump at gas equalization. Q.E.D.
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Fig. 9.7
Some special cases of the general entropy variation formula will be discussed here.
First the mole numbers of the initial components are equal. (
The initial temperatures can be equal too: (
The final pressure can be written for the actual conditions: This is the harmonic mean of the initial pressures.
This is substituted to the entropy variation above.
Here we should refer to the mathematical theorem that the harmonic mean is always smaller or equal than the
geometric mean. So the entropy variation is non-negative.
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Finally the initial pressures are considered equal.
The final temperature can be written for the actual conditions:
Here we should refer to the mathematical theorem that the arithmetic mean is always greater or equal than the
geometric mean. So the entropy variation is non-negative again.
10.7.2. 9.7.2 Equalization of non-gaseous materials without phase transition
Heat equalization between liquid-solid and solid-solid components is studied mostly in this point. In general the
liquid-liquid equalization process is not part of this discussion except for the case when both components are
water for instance, when the mixing is not associated with growth of entropy.
The product of mass and specific heat is the heat capacitance denoted capital
.
The internal energies are added together.
The temperature after equalization is denoted
without subscript.
Now the entropy variation is studied. The infinitesimal heat transfer can be used to calculate the entropy
variation. The actual temperature of the heat transfer is denoted
.
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The total entropy variation of the equalization is the sum of the two components.
After mathematical transformations this can be written:
The final result is as follows:
Let us substitute the final temperature calculated earlier:
In the numerator of the fraction the weighted arithmetic mean of the initial temperatures can be found, in the
denominator the weighted geometric mean of the same temperatures show up with the identical weights. Now
the mathematical theorem of the inequality between the arithmetic and geometric mean needs to be referred to.
According this theorem the arithmetic mean is never smaller than the geometric mean. Ultimately the fraction is
never smaller than unit which means that the entropy variation of the equalization process is never negative. The
second law of thermodynamics has been proven for this case. Q.E.D.
The non-negative result of the entropy variation can also be understood by verifying an infinitesimal entropy
transfer from the warmer component to the colder one. The warmer component carried out a dQ heat egress at
temperature while the colder component took up the same dQ heat at a lower
temperature. The
infinitesimal entropy egress dS and intake dS can be written as follows:
The total infinitesimal entropy production is the sum of the formulae above:
The difference in the numerator is obviously non-negative thus the original statement is proven.
10.7.3. 9.7.3 Ice cubes in the water
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The next problem contains phase transition which is associated with latent heat of melting. This is a known
parameter of any material. The actual latent heat
is calculated as a product:
. The
entropy variation is the ratio of the latent heat over the melting point temperature in Kelvin degrees.
Frequent practice is to cool down the drinks by means of ice cubes.
We have a glass of water (
0.03 kg, T
-13
0.3 kg) at room temperature
27
300K). Three ice cubes
260K) are dropped into the water. After the cubes melted we want to find the final temperature and to determine
the variation of the entropy of the system.
The chosen reference state is water at zero Celsius
relative to the reference state.
270K). The internal energies
The total internal energies of the initial components are equal with that of the final state.
After the system got into equilibrium the new temperature
is formed.
The numerical constants are as follows:
After substitution the final temperature is revealed:
The final temperature is 289.8 K which is 16.8 Celsius. So the drink is cooler by 10.2 C.
The entropy variation of the water in the glass (
is negative.
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The entropy variation of the ice cubes (
is positive.
The entropy production of the equalization is the sum of the variations:
After some transformations can be written as follows:
The numerical constants can be substituted:
The entropy variation of the equalization is positive as expected according to the second law.
10.8. 9.8 The second law of thermodynamics
In previous points the increase of entropy in isolated systems during irreversible events has been demonstrated
for several concrete cases. The second law of thermodynamics is an empirically validated postulate. Increase of
entropy in the isolated system is statistically substantiated by the Boltzmann equation by declaring that highest
thermodynamic weight belongs to the thermal equilibrium state.
The most accepted statement of the second law is as follows:
In isolated system (no heat and no work exchange with the environment) at naturally occurring irreversible
events the total entropy production is positive. In limit case the entropy production can be zero in reversible
process.
To be isolated is a fundamentally important condition in the statement above. If the system is not isolated, the
entropy variation can be anything. This can be highly negative too. By cooling and compressing the gas the
entropy is dropping fast all the way when it gets liquefied.
There are some related statements of the second law with the common feature of denying the possibility of the
followings: Perpetual motion machine of the second kind, Heat engine operating cyclically with a single heat
reservoir, Heat engine operating cyclically to convert heat to work in full extent, Heat that moves to the warmer
object from the colder one on its own. These statements are merely the consequences of the "most accepted"
version of the second law above.
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