Mechanics 1 CHAPTER 1 MECHANICS OBJECTIVES After reading this chapter, you should be able to: • Understand the measurements of mass, volume, density • Explain the determining the coefficient of friction • Focus on vector forces • Define particles and rigid body • Identify work and energy • Understand the collision, projectile motion, ballistic pendulum and centripetal force INTRODUCTION Mechanics, science concerned with the motion of bodies under the action of forces, including the special case in which a body remains at rest. Of first concern in the problem of motion are the forces that bodies exert on one another. This leads to the study of such topics as gravity, electricity, and magnetism, according to the nature of the forces involved. Given the forces, one can seek the manner in which bodies move under the action of forces; this is the subject matter of mechanics proper. 2 Experimental Physics I Historically, mechanics was among the first of the exact sciences to be developed. Its internal beauty as a mathematical discipline and its early remarkable success in accounting in quantitative detail for the motions of the Moon, Earth, and other planetary bodies had enormous influence on philosophical thought and provided impetus for the systematic development of science. Mechanics may be divided into three branches: statics, which deals with forces acting on and in a body at rest; kinematics, which describes the possible motions of a body or system of bodies; and kinetics, which attempts to explain or predict the motion that will occur in a given situation. Alternatively, mechanics may be divided according to the kind of system studied. The simplest mechanical system is the particle, defined as a body so small that its shape and internal structure are of no consequence in the given problem. More complicated is the motion of a system of two or more particles that exert forces on one another and possibly undergo forces exerted by bodies outside of the system. 1.1 MEASUREMENTS OF MASS, VOLUME, DENSITY 1.1.1 Mass Mass can be best understood as the amount of matter present in any object or body. Everything we see around us has mass. For example, a table, a chair, your bed, a football, a glass, and even air has mass. That being said, all objects are light or heavy because of their mass. We will learn what is mass, how to calculate it, and its examples while discovering interesting facts about it. Mass is the most basic property of matter and it is one of the fundamental quantities. Mass is defined as the amount of matter present in a body. The SI unit of mass is the kilogram (kg). The formula of mass can be written as: Mass = Density × Volume Note The mass of a body is constant; it doesn’t change at any time. Only in certain extreme cases when a huge amount of energy is given or taken from a body, the mass may be impacted. For example, in a nuclear reaction, a tiny amount of matter is converted into a huge amount of energy, this reduces the mass of the substance. Mechanics 3 1.1.2 Volumes Volumes of liquids are measured with the aid of graduated cylinders, pipets, and burets. For the time being you will only be using the graduated cylinder for direct determination of liquid volumes. The measurements of volumes of solid objects is not as straightforward as it is with liquids. In some cases, when dealing with regular solids such as cubes, cylinders, or prisms, it is quite possible to calculate the volumes by measuring the pertinent dimensions of the object. Some formulas for the calculation of volumes of regular solids are given below: where: Ρ = length in centimeters w = width in centimeters h = height in centimeters r = radius in centimeters π = 3.14 ... Volumes of liquids are expressed in units of milliliters (mL) while volumes calculated from the geometry of the object will be given in cubic centimeters (cm3 or cc). For our purposes 1 mL is equal to 1 cm3. In the case of irregularly shaped objects a method known as volume by liquid displacement can be used. If we fill a 100 mL graduated cylinder to the 50 mL mark and if we then introduce the solid object into the cylinder, and assuming that the object does not float, the liquid level will then rise to a new mark. The difference between the two liquid levels represents the desired volume. In another version of the same method, a container is filled completely with liquid, usually water, and when the object is submerged a certain amount of liquid overflows and can be captured. The volume of liquid can either be measured directly or can be weighed. Knowing the mass of liquid and the density we can get the volume. This method of determining volume is easy to carry out 4 Experimental Physics I and is more accurate due to the high accuracy possible in weighing. In the case of objects that float, it is of course necessary to make sure that all of the substance is completely submerged in the liquid by pushing them down. 1.1.3 Density Knowing the mass and volume of an object allows the calculation of its density. Density is defined as the mass divided by the volume of the object. Mass should be expressed in units of grams and volume in units of mL or cm3. Note, that the density of water is 1 g≅ mL-1. The general physical law relating to floating objects is known as Archimedes’ Principle. It states that a body wholly or partly immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced. buoyant force=(den.of fluid)(acc.due to gravity)(vol.of immersed object) In the special case of floating objects one can also express the principle as: “A floating body displaces a volume of liquid whose weight equals that of the floating body.” Archimedes’ Principle can be used to determine specific gravities of solids. 1.2 DETERMINING THE COEFFICIENT OF FRICTION Friction is a resistive force that prevents two objects from sliding freely against each other. The coefficient of friction (fr) is a number that is the ratio of the resistive force of friction (Fr) divided by the normal or perpendicular force (N) pushing the objects together. It is represented by the equation: fr = Fr/N. Mechanics 5 There are different types and values for the coefficient of friction, depending on the type of resistive force. You can determine the coefficient of friction through experiments, such as measuring the force required to overcome friction or measuring the angle at which an object will start to slide off an incline. There are also charts of common coefficients of friction available. 1.2.1 Different types of Coefficient The different types of friction are static, kinetic, deformation, molecular and rolling. Each has its own coefficient of friction. Static Coefficient Static friction is the force that holds back a stationary object up to the point that it just starts moving. Thus, the static coefficient of friction concerns the force restricting the movement of an object that is stationary on a relatively smooth, hard surface. Kinetic Coefficient Once you overcome static friction, kinetic friction is the force holding back regular motion. This, kinetic fiction coefficient of friction concerns the force restricting the movement of an object that is sliding on a relatively smooth, hard surface. Deformation Coefficient The deformation coefficient of friction concerns the force restricting the movement of an object that is sliding or rolling and one or both surfaces are relatively soft and deformed by the forces. Molecular Coefficient Molecular coefficient of friction concerns the force restricting the movement of an object that is sliding on an extremely smooth surface or where a fluid is involved. Rolling coefficient The rolling coefficient of friction combines static, deformation and molecular coefficients of friction. This coefficient of friction can be made quite low. 6 Experimental Physics I 1.2.2 Experiments to Determine Coefficient There are a number of experiments you can do to determine the coefficient of friction between two materials. You can directly measure the forces involved or you can use some indirect methods, measuring such things as incline of a ramp or time to stop. Direct Measurements An experiment to determine the coefficient of friction would be to use some force to push to materials together and then measure that force. You could apply this force by squeezing a pair of pliers, by applying the brakes in your car, or by using the force of gravity to apply a weight on an object. Then you can try to move one object and measure the necessary force. This could be trying to pull a strip of wood from the grip of the pliers, trying to move a car wheel when the brakes are applied, or pulling a weighted object along the floor. A scale or similar device can be used to measure the forces. Measuring the Squeezing Force If you can measure the force you apply to push the materials together, you can determine their static coefficient of friction. Thus, if you took a pair of pliers and squeezed it so that it applied 18 pounds of force on a piece of wood, and it took 9 pounds of force to pull the wood from the squeezed pliers, then the static coefficient of friction of the wood and the pliers would be 9 pounds / 18 pounds = 0.5. Thus, no matter how hard you squeezed the pliers, it would always take 0.5 times that force to pull the wood out. Important A pair pliers is a simple machine. It has a mechanical advantage of about 3 times. Thus you would only have to squeeze the handles with 6 pounds of force to create a force of 18 pounds at the pliers head. Using the Force of Gravity Since it is difficult to measure the force that you squeeze, a more common way to measure the force between objects is to use the weight of one object. An object’s weight is the force is exerts on another object, caused by gravity. If the weight is W in pounds or newtons, the friction equation for an object sliding across a material on the ground can be rewritten as: Mechanics 7 Once you know the weight of the object you are sliding, you can use a scale to measure the force it takes to move the object. For example, measure the weight of a book. Then use the scale to measure the force required to start the book sliding along a table. From these two measurements, you can determine the static coefficient of friction between the book and the table. You can verify that the friction equation is true by adding a second book and repeating the measurement. The force required to pull two books should be twice as much as for one book. To measure the static coefficient of friction, you take the value of the force just as the object starts to move. Doing the same experiment with sliding or kinetic friction, you want to take your reading when the object is sliding at an even velocity. Otherwise, you will be adding in acceleration force effects. Indirect Measurements There are several indirect methods to determine the coefficient of friction. A method to determine the static coefficient of friction is to measure the angle at which an object starts to slide on an incline or ramp. A method to determine the kinetic coefficient of friction is to measure the time is takes to stop an object. Using an Incline You can use an object on an incline to determine the static coefficient of friction by finding the angle at which the force of gravity overcomes the static friction. Perpendicular Force Reduced: When an object is placed on an incline, the force perpendicular between the surfaces is reduced, according to the angle of the incline. 8 Experimental Physics I The force required to overcome friction (Fr) equals the coefficient of friction (u) times the cosine of the incline angle (cos(a)) times the weight of the object (W). There are mathematical tables that give the values of cosines for various angles. Gravity contributes to sliding: Note that when an object is on an incline, the force of gravity contributes to causing the object to slide down the ramp or incline. Let’s call that force (FG), and it is equal to the weight of the object (W) times the sine of the angle (sin(a)) Tangent of angle determines coefficient: If you put the ramp at a steep enough angle, Fg will become greater than Fr and the object will slide down the incline. The angle at which it just starts to slide is determined from the equation: Dividing both sides of the equation by W and cos(a), we get the equation for the static coefficient of friction fr where tan(a) is the tangent of angle (a) and equals sin(a)/cos(a). There are mathematical tables for determining the tangent, sine and cosine of various angles. Using Time You can also use a stopwatch to determine the kinetic or rolling coefficient of friction. But it is not easy to do. If you have an object moving at some velocity v and you let it roll or slide along a surface until it stopped. You could then measure the time t it takes to stop to determine its coefficient of friction. Mechanics 9 From the Force Equation, F = m*a, where a is the acceleration. Since the object is starting at some velocity v and decelerating until v = 0, then the force of friction can be written as: Fr = m*v/t If the object weighs W pounds, and W = m*g, where g is the gravity constant 32 ft/sec/sec (9.8 m/s2, then the Friction Equation is: Combining the two equations for Fr, we get: Thus, if a car is moving at 64 feet per second and takes 4 seconds to come to a stop, its coefficient of friction is: fr = 64 / (32 x 4) = 0.5 1.3 VECTOR FORCES A force is defined as any cause that tends to alter the state of rest of a body or its state of uniform motion in a straight line. A force can be defined quantitatively as the product of the mass of the body that the force is acting on and the acceleration of the force. P = ma where P = applied force m = mass of the body (kg) a = acceleration caused by the force (m/s2) The SI units for force are therefore kg m/s2, which is designated a Newton (N). The following multiples are often used: 1 kN = 1 000 N, 1 MN = 1 000 000 N All objects on earth tend to accelerate toward the center of the earth 10 Experimental Physics I due to gravitational attraction; hence the force of gravitation acting on a body with the mass (m) is the product of the mass and the acceleration due to gravity (g), which has a magnitude of 9.81 m/s2: F = mg = v rg where: F = force (N) m = mass (kg) g = acceleration due to gravity (9.81m/s2) v = volume (m³) r = density (kg/m³) All forces (interactions) between objects can be placed into two broad categories: • • contact forces, and forces resulting from action-at-a-distance Contact forces are those types of forces that result when the two interacting objects are perceived to be physically contacting each other. Examples of contact forces include frictional forces, tensional forces, normal forces, air resistance forces, and applied forces. Action-at-a-distance forces are those types of forces that result even when the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite their physical separation. Examples of action-at-a-distance forces include gravitational forces. For example, the sun and planets exert a gravitational pull on each other despite their large spatial separation. Even when your feet leave the earth and you are no longer in physical contact with the earth, there is a gravitational pull between you and the Earth. Electric forces are action-at-a-distance forces. For example, the protons in the nucleus of an atom and the electrons outside the nucleus experience an electrical pull towards each other despite their small spatial separation. And magnetic forces are action-at-a-distance forces. For example, two magnets can exert a magnetic pull on each other even when separated by a distance of a few centimeters. Mechanics 11 Vector Most forces have magnitude and direction and can be shown as a vector. The point of application must also be specified. A vector is illustrated by a line, the length of which is proportional to the magnitude on a given scale, and an arrow that shows the direction of the force. Vector Addition The sum of two or more vectors is called the resultant. The resultant of two concurrent vectors is obtained by constructing a vector diagram of the two vectors. The vectors to be added are arranged in tip-totail fashion. Where three or more vectors are to be added, they can be arranged in the same manner, and this is called a polygon. A line drawn to close the triangle or polygon (from start to finishing point) forms the resultant vector. The subtraction of a vector is defined as the addition of the corresponding negative vector. 12 Experimental Physics I 1.3.1 Types of Force In simple terms, a force is a push, a pull, or a drag on an object. There are three main types of force: • • • An applied force is an interaction of one object on another that causes the second object to change its velocity. A resistive force passively resists motion and works in a direction opposite to that motion. An inertial force resists a change in velocity. It is equal to and in an opposite direction of the other two forces. There is no such thing as a unidirectional force or a force that acts on only one object. There must always be two objects involved, acting on each other. One object acts on the other, while the second resists the action of the first. Applied Force An applied force is an interaction that causes the second object to change its velocity. Force Equation The force required to overcome the inertia of an object is according to the equation: F = ma where: • • • F is the force m is the mass of the object a is the acceleration caused by the force Types of Applied Force There are several types of applied force: The most common form of force is a push through physical contact. For example, you can push on a door to open it. An object can also collide with another object, exerting a force and causing the second object to accelerate. This is another type of push and can be called an impulse force, since the time interval is very short. You can pull on an object to change its velocity. Gravitation, magnetism, and static electricity are some of the pulling forces that act at a distance with no physical Mechanics 13 contact required to move objects. Finally, if two objects or materials are in contact, one can drag the other along by friction or other means. Resistive Force A resistive force passively inhibits or resists the motion of an object. It is a form of push-back. It is considered passive, since it only responds to actions on the object. Friction and fluid resistance are the major resistive forces. Friction When an object is being pushed along the surface of another object or material, the resistive force of friction pushes back on the first object to resist its motion. Fluid Resistance Fluid resistance pushes back on the moving object, which is basically trying to plow through the fluid. It also included friction on the surface of the object. Air resistance and water resistance are common forms of fluid resistance. Inertial Force An inertial force works against a change in velocity, caused by an applied force, as well as a resistive force. Against Applied Force According to Newton’s Third Law of Motion or the Action-Reaction Law: Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body. This is often stated as: “For every action there is an equal and opposite reaction.” When you push on an object, an equal inertial force pushes back. This is the resistance to acceleration. Hints The more mass something has the more inertia it has. You can think of inertia as a property that makes it hard to push something around. 14 Experimental Physics I Likewise, when swinging an object on a rope around you in a circle, you pull on the rope to change the direction of motion. In turn, you can feel a pull on the rope. Against Resistive Force When a resistive force like friction, slows down the motion of an object, the inertial force will push in the opposite direction and tend to keep the object moving. The main type of force is an applied force, which is an interaction of one object on another that causes the second object to change its velocity. Other types of forces include a resistive force that passively resists motion and an inertial force that resists a change in velocity. There must always be two objects involved in a force, acting on each other. 1.3.2 Principle for Number of Forces Acting On a Body or Point If a number of forces acting simultaneously on a body then the effect produced by all the forces will be same as produced by a single resultant of all the forces. FR = F1 + F2 FR = 30 + (-20) F2 is –ve, is opposite or in –ve direction FR = 10 KN Two forces F1 and F2 are acting simultaneously opposite to each other on a body (a). The resultant of both, FR will produce the same effect on the body as produced by F1 and F2 (b). Mechanics 15 1.3.3 Causes of Force Forces may arise from a number of different effects, including i. ii. iii. iv. v. vi. Gravity; Electromagnetism or electrostatics; Pressure exerted by fluid or gas on part of a structure Wind or fluid induced drag or lift forces; Contact forces, which act wherever a structure or component touches anything; Friction forces, which also act at contacts. Some of these forces can be described by universal laws. For example, gravity forces can be calculated using Newton’s law of gravitation; electrostatic forces acting between charged particles are governed by Coulomb’s law; electromagnetic forces acting between current carrying wires are governed by Ampere’s law; buoyancy forces are governed by laws describing hydrostatic forces in fluids. Some forces have to be measured. For example, to determine friction forces acting in a machine, you may need to measure the coefficient of friction for the contacting surfaces. Similarly, to determine aerodynamic lift or drag forces acting on a structure, you would probably need to measure its lift and drag coefficient experimentally. Contact forces are pressures that act on the small area of contact between two objects. Contact forces can either be measured, or they can be calculated by analyzing forces and deformation in the system of interest. 1.3.4 Forces in Translational Equilibrium 16 Experimental Physics I There are many objects we do not want to see in motion. In the Figure above, the mountain climbers want their ropes to keep them from moving downward. We construct buildings and bridges to be as motionless as possible. We want the acceleration (and velocity) of these objects to be zero. For an object to be in static equilibrium (that is, motionless) the right-hand side of Newton’s Second Law, ∑F=ma, must be zero. Thus, ∑F=0. This equation is simple enough when an object is held with a single support. In an earlier example, we depicted Joe Loose hanging by a single rope (Figure below). Joe’s goal was to remain hanging in equilibrium (just like the climbers in the photograph). The force of gravity pulling Joe down was exactly balanced by the tension in the rope that supported him. But Joe won’t be hanging for very long, will he? You can see that the rope is slowly fraying against the mountainside (recall the original problem). Soon it will snap. But Joe’s in luck, because a rescue team has come to his aid. They arrive just in time to secure two more ropes to the mountain side and toss Joe the slack to tie around his waist before his rope snaps! Joe is saved. But does Joe thank the rescue team like any sane person would? No. Instead, still in midair, he pulls out a pad and pencil from his back pocket in order to analyze the forces acting on him (Figure below). Mechanics 17 In order for Joe to remain in equilibrium, he must not move in the x− or y−directions. This means that the sum of all forces in the x− direction must add to zero. And the sum of all forces in the y−direction must add to zero. The procedure for solving problems with forces in equilibrium is as follows: 1. 2. 3. Place Figure above in a coordinate plane with the object at the origin. Resolve the tension vectors T1 and T2 into their x− and y− components. Use Newton’s Second Law: ∑Fx=0 and ∑Fy=0. In order to solve this problem, we’ll need more information, including the angles that the ropes make with the vertical. The information is provided below, along with Figure below. θ1=45° θ2=30° Mg=800 N Find T1 and T2 The solution requires solving a set of simultaneous equations. First, we find the components of vectors T1 and T2. T1x=T1 sin 45∘ and T1y=T1 cos 45∘ T2= T2 sin 30∘ and T2y=T2 cos 30∘ 18 Experimental Physics I Next we apply Newton’s Second Law. ∑Fx:T2 sin 30° −T1 sin45° = 0 ∑Fy:T2 cos 30°+T1 cos45° = 800 The first equation can be quickly simplified to give T2=2–√T1. T2 is then substituted in the second equation and T1 is found. Once T1 is found, T2 can easily be computed using T2=2–√T1 T1 = 414.11 = 414 = 410 N T2 = 585.56 = 586 = 590 N 1.4 PARTICLES AND RIGID BODY Particles are objects that have mass, position, and velocity, and respond to forces, but that have no spatial extent. Because they are simple, particles are by far the easiest objects to simulate. Despite their simplicity, particles can be made to exhibit a wide range of interesting behavior. For example, a wide variety of non-rigid structures can be built by connecting particles with simple damped springs. A rigid body is an idealized model of an object that has a definite and unchanging shape and size. In reality, real-world bodies are constantly interacting with the environment, undergoing forces that can twist, stretch, or squeeze them in ways that would make precise calculations involving them quite impractical. The concept of a rigid body is particularly crucial when considering rotational motion. Some key points related to a rigid body and rotational motion are: • Every part of a rigid body has the same angular velocity. Mechanics • • 19 In a rotating rigid body, the velocity and acceleration of any point can be calculated using the distance from the axis of rotation and the body’s angular velocity and angular acceleration. You can calculate the kinetic energy of a rotating rigid body using the angular velocity of the body and its moment of inertia. 1.4.1 Systems of Particles Systems of particles means such things as a swarm of bees, a star cluster, a cloud of gas, an atom, a brick. A brick is indeed composed of a system of particles – atoms − which are constrained so that there is very little motion (apart from small amplitude vibrations) of the particles relative to each other. In a system of particles there may be very little or no interaction between the particles (as in a loose association of stars separated from each other by large distances) or there may be (as in the brick) strong forces between the particles. Momentum Hints Rigid body analysis is more complex and also has to take into account moments and rotational motions. In actuality, no bodies are truly particles, but some bodies can be approximated as particles to simplify analysis. The momentum of a particle is defined as the product of its mass times its velocity. It is a vector quantity. The momentum of a system is the vector sum of the momenta of the objects which make up the system. If the system is an isolated system, then the momentum of the system is a constant of the motion and subject to the principle of conservation of momentum. The basic definition of momentum applies even at relativistic velocities but then the mass is taken to be the relativistic mass. The most common symbol for momentum is p. The SI unit for momentum is kg m/s. 20 Experimental Physics I The mathematical definition of momentum is consistent with our intuitive, everyday notion of “momentum.” If two cars have equal masses but one has twice the velocity of the other, it has twice the momentum. And if a truck has three times the mass of a car and the same velocity, it has three times the momentum. Newton’s First Law states that, in the absence of external forces, the velocity of a particle remains constant. Expressed in terms of momentum, the First Law therefore states that the momentum remains constant: P = [constant] (no external forces) Thus, we can say that the momentum of the particle is conserved. Of course, we could equally well say that the velocity of this particle is conserved; but the deeper significance of momentum will emerge when we study the motion of a system of several particles exerting forces on one another. We will find that the total momentum of such a system is conserved—any momentum lost by one particle is compensated by a momentum gain of some other particle or particles. To express the Second Law in terms of momentum, we note that since the mass is constant, the time derivative of above equation is or But, according to Newton’s Second Law, ma equals the force; hence, the rate of change of the momentum with respect to time equals the force: dp =F dt This equation gives the Second Law a concise and elegant form. We can also express Newton’s Third Law in terms of momentum. Since the action force is exactly opposite to the reaction force, the rate of change of momentum generated by the action force on one body is exactly opposite to the rate of change of momentum generated by the reaction force on the other body. Hence, we can state the Third Law as follows: Mechanics 21 Whenever two bodies exert forces on each other, the resulting changes of momentum are of equal magnitudes and of opposite directions. This balance in the changes of momentum leads us to a general law of conservation of the total momentum for a system of particles. The total momentum of a system of n particles is simply the (vector) sum of all the individual momenta of all the particles. Thus, if p1 = m1v1, p2 = m2v2, . . . , and pn = mnvn are the individual momenta of the particles, then the total momentum is P = p1 + p2 + ………. + pn The simplest of all many-particle systems consists of just two particles exerting some mutual forces on one another (Figure 1). Let us assume that the two particles are isolated from the rest of the Universe so that, apart from their mutual forces, they experience no extra forces of any kind. According to the above formulation of the Third Law, the rates of change of p1 and p2 are then exactly opposite: Figure 1. Two particles exerting mutual forces on each other. The net change of momentum of the isolated particle pair is zero. The rate of change of the sum p1+ p2 is therefore zero, since the rate of change of the first term in this sum is canceled by the rate of change of the second term: 22 Experimental Physics I This means that the sum p1+ p2 is a constant of the motion: p1+ p2 = [constant] This is the Law of Conservation of Momentum. Note that Newton’s Third Law is an essential ingredient for establishing the conservation of momentum: the total momentum is constant because the equality of action and reaction keeps the momentum changes of the two particles exactly equal in magnitude but opposite in direction—the particles merely exchange some momentum by means of their mutual forces. Thus, for our particles, the total momentum P at some instant equals the total momentum P’ at some other instant, so P = P’ Conservation of momentum is a powerful tool which permits us to calculate some general features of the motion even when we are ignorant of the detailed properties of the interparticle forces. Center of Mass The motion of a rotating ax thrown between two jugglers looks rather complicated, and very different from the standard projectile motion. Experiments have shown that one point of the ax follows a trajectory described by the standard equations of motion of a projectile. This special point is called the center of mass of the ax. The position of the center of mass of a system of two particles with mass m1 and m2, located at position x1 and x2, respectively, is defined as = xcm m1 x1 + m2 x2 1 = ∑m x m1 + m2 M i i i Since we are free to define our coordinate system in whatever way is convenient, we can define the origin of our coordinate system to coincide with the left most object (Figure 2). The position of the center of mass is now Mechanics xcm = 23 m2 d m1 + m2 This equation shows that the center of mass lies between the two masses, closest to the heavier mass. Figure 2. Position of the center of mass in one dimension. In general, for a system with more than two particles, the position of the center of mass will satisfy the following relation xmin ≤ xcm ≤ xmax The definition of the center of mass in one dimension can be easily generalized to three dimensions xcm = 1 ∑m x M i i i ycm = 1 ∑m y M i i i zcm = 1 ∑m z M i i i or in vector notation rcm = 1 ∑ m r M i ii 24 Experimental Physics I For a rigid body, the summation will be replaced by an integral 1 M rcm = ∫ rdm v Suppose we are dealing with a number of objects. Figure 3 shows a system consisting of 4 masses, m1, m2, m3 and m4, located at x1, x2, x3 and x4, respectively. The position of the center of mass of m1 and m2 is given by xcm1,2 = m1 x1 + m2 x2 m1 + m2 The position of the center of mass of m3 and m4 is given by xcm 3,4 = m3 x3 + m4 x4 m3 + m4 The position of the center of mass of the whole system is given by xcm = m1 x1 + m2 x2 + m3 x3 + m4 x4 m1 + m2 + m3 + m4 This can be rewritten as = xcm 1 m1 + m2 + m3 + m4 m x + m4 x4 m1 x1 + m2 x2 + ( m3 + m4 ) 3 3 ( m1 + m2 ) m + m m3 + m4 1 2 Using the center of mass of m1 and m2 and of m3 and m4 we can express the center of mass of the whole system as follows xcm = (m 1 + m2 ) xcm1,2 + ( m3 + m4 ) xcm 3,4 (m 1 + m2 ) + ( m3 + m4 ) Figure 3. Location of 4 masses. Mechanics 25 This shows that the center of mass of a system can be calculated from the position of the center of mass of all objects that make up the system. The Motion of the Center of Mass The definition of the center of mass of a system of particles can be rewritten as Mrcm = ∑ mi ri i where M is the total mass of the system. Differentiating this equation with respect to time shows Mvcm = ∑ mi vi i where vcm is the velocity of the center of mass and vi is the velocity of mass mi. The acceleration of the center of mass can be obtained by once again differentiating this expression with respect to time Macm = ∑ mi ai i where acm is the acceleration of the center of mass and ai is the acceleration of mass mi. Using Newton’s second law we can identify mi ai with the force acting on mass mi. This shows that Ma = cm F ∑F ∑= i i ext This equation shows that the motion of the center of mass is only determined by the external forces. Forces exerted by one part of the system on other parts of the system are called internal forces. According to Newton’s third law, the sum of all internal forces cancel out (for each interaction there are two forces acting on two parts: they are equal in magnitude but pointing in an opposite direction and cancel if we take the vector sum of all internal forces). 26 Experimental Physics I Figure 4. Internal and External Forces acting on a System of Particles. The previous equations show that the center of mass of a system of particles acts like a particle of mass M, and reacts like a particle when the system is exposed to external forces. They also show that when the net external force acting on the system is zero, the velocity of the center of mass will be constant. Energy of a System of Particles The total kinetic energy of a system of particles is simply the sum of the individual kinetic energies of all the particles, 1 1 1 K = m1 v12 + m2 v22 + mn vn2 2 2 2 Since for the momentum of a system of particles resembles the expression for the momentum of a single particle, we might be tempted to guess that the equation for the kinetic energy for a system of particles also can be expressed in the form of the translational kinetic energy of 1 2 MvCM 2 the center of mass , resembling the kinetic energy of a single particle. The total kinetic energy of a system of particles is usually larger 1 2 MvCM 2 than . We can see this in the following simple example: Consider two automobiles of equal masses moving toward each other at equal speeds .The velocity of the center of mass is then zero, and consequently 1 2 MvCM 2 = 0. However, since each automobile has a positive kinetic energy, the total kinetic energy is not zero. Mechanics 27 If the internal and external forces acting on a system of particles are conservative, then the system will have a potential energy. We saw above that for the specific example of the gravitational potential energy near the Earth’s surface, the potential energy of the system took the same form as for a single particle, U = MgyCM. But this form is a result of the particular force (uniform and proportional to mass); in general, the potential energy for a system does not have the same form as for a single particle. Unless we specify all of the forces, we cannot write down an explicit formula for the potential energy; but in any case, this potential energy will be some function of the positions of all the particles. The total mechanical energy is the sum of the total kinetic energy and the total potential energy. This total energy will be conserved during the motion of the system of particles. Note that in reckoning the total potential energy of the system, we must include the potential energy of both the external forces and the internal forces. We know that the internal forces do not contribute to the changes of total momentum of the system, but these internal forces, and their potential energies, contribute to the total energy. For instance, if two particles are falling toward each other under the influence of their mutual gravitational attraction, the momentum gained by one particle is balanced by momentum lost by the other, but the kinetic energy gained by one particle is not balanced by kinetic energy lost by the other—both particles gain kinetic energy. In this example the gravitational attraction plays the role of an internal force in the system, and the gain of kinetic energy is due to a loss of mutual gravitational potential energy 1.4.2 Rotation of a Rigid Body A body is rigid if the particles in the body do not move relative to one another. Thus, the body has a fixed shape, and all its parts have a fixed position relative to one another. A hammer is a rigid body, and so is a baseball bat. A baseball is not rigid—when struck a blow by the bat, the ball suffers a substantial deformation; that is, different parts of the ball move relative to one another. However, the baseball can be regarded as a rigid body while it flies through the air—the air resistance is not sufficiently large to produce an appreciable deformation of the ball. This example indicates that whether a body can be regarded as rigid depends on the circumstances. Nobody is absolutely rigid; when subjected to a sufficiently large force, anybody will suffer some deformation or perhaps even break into several pieces. 28 Experimental Physics I Motion of a Rigid Body A rigid body can simultaneously have two kinds of motion: it can change its position in space, and it can change its orientation in space. Change of position is translational motion, this motion can be conveniently described as motion of the center of mass. Change in orientation is rotational motion; that is, it is rotation about some axis. Hints When a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position As an example, consider the motion of a hammer thrown upward (Figure 5). The orientation of the hammer changes relative to fixed coordinates attached to the ground. Instantaneously, the hammer rotates about a horizontal axis, say, a horizontal axis that passes through the center of mass. In figure 5, this horizontal axis sticks out of the plane of the page and moves upward with the center of mass. The complete motion can then be described as a rotation of the hammer about this axis and a simultaneous translation of the axis along a parabolic path. Figure 5. A hammer in free fall under the influence of gravity. The center of mass of the hammer moves with constant vertical acceleration g, just like a particle in free fall. Mechanics 29 In this example of the thrown hammer, the axis of rotation always remains horizontal, out of the plane of the page. In the general case of motion of a rigid body, the axis of rotation can have any direction and can also change its direction. To describe such complicated motion, it is convenient to separate the rotation into three components along three perpendicular axes. The three components of rotation are illustrated by the motion of an aircraft (Figure 6): the aircraft can turn left or right (yaw), it can tilt to the left or the right (roll), and it can tilt its nose up or down (pitch). Figure 6. Pitch, roll, and yaw motions of an aircraft. Rotation about a Fixed Axis Figure 7 shows a rigid body rotating about a fixed axis, which coincides with the z axis. During this rotational motion, each point of the body remains at a given distance from this axis and moves along a circle centered on the axis. To describe the orientation of the body at any instant, we select one particle in the body and use it as a reference point; any particle can serve as reference point, provided that it is not on the axis of rotation. The circular motion of this reference particle (labeled P in figure 7) is then representative of the rotational motion of the entire body, and the angular position of this particle is representative of the angular orientation of the entire body. 30 Experimental Physics I Figure 7. The four blades of this fan are a rigid body rotating about a fixed axis, which coincides with the z axis. The reference particle P in this rigid body moves along a circle around this axis. Figure 8. Motion of a reference particle P in the rigid body rotating about a fixed axis. The axis is indicated by the circled dot O. The radius of the circle traced out by the motion of the reference particle is R. Figure 8 shows the rotating rigid body as seen from along the axis of rotation. The coordinates in figure 8 have been chosen so the z axis coincides with the axis of rotation, whereas the x and y axes are in the plane of the circle traced out by the motion of the reference particle. The angular position of the reference particle—and hence the angular orientation of the entire rigid body—can be described by the position Mechanics 31 angle f between the radial line OP and the x axis. Conventionally, the angle is taken as positive when reckoned in a counterclockwise direction (as in figure 8). We will usually measure this position angle in radians, rather than degrees. By definition, the angle in radians is the length s of the circular arc divided by the radius R, or φ= s R In Figure 8, the length s is the distance traveled by the reference particle from the x axis to the point P. Note that if the length s is the circumference of a full circle, then s = 2pR, and f = s/R = 2pR/R =2p. Thus, there are 2p radians in a full circle; that is, there are 2p radians in 360° 2p radians = 360° Accordingly, 1 radian equals 360/2p, or 1 radian = 57.3° When a rigid body rotates, the position angle f changes in time. The body then has an angular velocity w.The definition of the angular velocity for rotational motion is mathematically analogous to the definition of velocity for translational motion. The average angular velocity ω is defined as ω= ∆φ ∆t where Df is the change in the angular position and Dt the corresponding change in time. The instantaneous angular velocity is defined as ω= dφ dt According to these definitions, the angular velocity is the rate of change of the angle with time. The unit of angular velocity is the radian per second (1 radian/s).The radian is the ratio of two lengths, and hence it is a pure number; thus, 1 radian/s is the same thing as 1/s. However, to prevent confusion, it is often useful to retain the vacuous label radian as a reminder that angular motion is involved. 32 Experimental Physics I If the body rotates with constant angular velocity, then we can also measure the rate of rotation in terms of the ordinary frequency f, or the number of revolutions per second. Since each complete revolution involves a change f of by 2p radians, the frequency of revolution is smaller than the angular velocity by a factor of 2p: f = ω 2π This expresses the frequency in terms of the angular velocity. The unit of rotational frequency is the revolution per second (1 rev/s). Like the radian, the revolution is a pure number, and hence 1 rev/s is the same thing as 1/s. But we will keep the label rev to prevent confusion between rev/s and radian/s. As in the case of planetary motion, the time per revolution is called the period of the motion. If the number of revolutions per second is f, then the time per revolution is 1/f, that is, T= 1 f If the angular velocity of a rigid body is changing, the body has an angular acceleration a. The rotational motion of a ceiling fan that is gradually building up speed immediately after being turned on is an example of accelerated rotational motion. The mathematical definition of the average angular acceleration is, again, analogous to the definition of acceleration for translational motion. If the angular velocity changes by Dw in a time Dt, then the average angular acceleration is α= ∆ω ∆t and the instantaneous angular acceleration is α= dω dt Thus, the angular acceleration is the rate of change of the angular velocity. The unit of angular acceleration is the radian per second per second, or radian per second squared (1 radian/s2). Mechanics 33 Since the angular velocity w is the rate of change of the angular position f, the angular acceleration given by above equation can also be written d 2φ α= 2 dt The angular velocity and acceleration of the rigid body; that is, they give the angular velocity and acceleration of every particle in the body. It is interesting to focus on one of the particles and evaluate its translational speed and acceleration as it moves along its circular path around the axis of rotation of the rigid body. If the particle is at a distance R from the axis of rotation (figure 9), then the length along the circular path of the particle is, according to the definition of angle, s = fR Figure 9. The instantaneous translational velocity of a particle in a rotating rigid body is tangent to the circular path. Since R is a constant, the rate of change of s is entirely due to the rate of change of f, so ds dφ = R dt dt Here ds/dt is the translational speed v with which the particle moves along its circular path, and df/dt is the angular velocity w; hence above equation is equivalent to 34 Experimental Physics I v = wR This shows that the translational speed of the particle along its circular path around the axis is directly proportional to the radius: the farther a particle in the rigid body is from the axis, the faster it moves. We can understand this by comparing the motions of two particles, one on a circle of large radius R1, and the other on a circle of smaller radius R2 (figure 10). For each revolution of the rigid body, both of these particles complete one trip around their circles. But the particle on the larger circle has to travel a larger distance, and hence must move with a larger speed. Figure 10. Several particles in a rigid body rotating about a fixed axis and their velocities. For a particle at a given R, the translational speed is constant if the angular velocity is constant. This speed is the distance around the circular path (the circumference) divided by the time for one revolution (the period), or Since 2p/T = 2pf = w. If v is changing, that the rate of change of v is proportional to the rate of change of w: Mechanics 35 A rate of change of the speed along the circle implies that the particle has an acceleration along the circle, called a tangential acceleration. According to the last equation, this tangential acceleration is atangential = aR Note that, besides this tangential acceleration directed along the circle, the particle also has a centripetal acceleration directed toward the center of the circle. We know that the centripetal acceleration for uniform circular motion is With v = wR, this becomes acentripital = w2R The net translational acceleration of the particle is the vector sum of the tangential and the centripetal accelerations, which are perpendicular (figure 11); thus, the magnitude of the net acceleration is Figure 11. A particle in a rotating rigid body with an angular acceleration has both a centripetal acceleration acentripetal and a tangential acceleration atangential. The net instantaneous translational acceleration anet is then the vector sum of acentripetal and atangential. 36 Experimental Physics I Although we have here introduced the concept of tangential acceleration in the context of the rotational motion of a rigid body, this concept is also applicable to the translational motion of a particle along a circular path or any curved path. For instance, consider an automobile (regarded as a particle) traveling around a curve. If the driver steps on the accelerator (or on the brake), the automobile will suffer a change of speed as it travels around the curve. It will then have both a tangential and a centripetal acceleration. Review of Vector Operations Addition (a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 ) Subtraction (a1 , b1) − (a2 , b2 ) = (a1 − a2 , b1 − b2 ) Scalarmultiplication k ⋅ (a, b) =(k ⋅ a, k ⋅ b) In vector addition, we add the corresponding components. In vector subtraction, we subtract the corresponding components. 1.4 ROTATIONAL EQUILIBRIUM/TORQUE Equilibrium is the state of being balanced physically, and it is an important factor for being in motion. Explore the relationship between motion and equilibrium, focusing on the two types: translational and rotational. 1.4.1 Torque Up to this point in physics, we haven’t allowed forces to act in such a way as to cause rotation. The forces acting on a grocery cart traveling along an aisle could only be exerted in such a way that no rotation of the cart was produced. As you know from experience, this can be accomplished by pushing straight ahead on the center of the handle of the cart or by pushing equally on the two ends of the handle. (This assumes no wobbly, squeaky wheels—generally a bad assumption.) But as we know, bad things can happen at the end of the aisle when paths cross and carts can end up spinning off into the produce section. When this occurs, the cart moves along some new path and often spins. Forces that tend to cause rotation are said to apply a torque to the objects they act on. Consider the three forces, F, in Figures 12a–c. All three act in the plane of the page. The rod is hinged so that it can rotate around its axis of rotation, which is normal (perpendicular) to the page. Mechanics 37 Figure 12a. Torque on a rod. Figure 12b. Force perpendicular to r. Maximum torque. Figure 12c. Force collinear to r. Zero torque. The axis of rotation (×) is the point (or line) around which we want to calculate the torque. We can choose any point we like, but there will usually be a best choice, which will make our calculations simplest. Or there may be a point specified in an assignment (e.g., «Find the torque around the door hinge.»).The line of action of the force is an extended line collinear with the force. This is usually shown as a dotted line in figures. The lever arm, r⊥, is the shortest distance between the axis of rotation and the line of action of the force. It is a line drawn from the axis of rotation so as to hit the line of action at a right angle. (Sometimes the character ℓ is used to represent the lever arm.)We define torque, τ (tau), as the product of the lever arm, r⊥, and the force, F. 38 Experimental Physics I τ = r ⊥F Another quantity, which sadly has no name, is r. r is the distance between the axis and the point of application of the force. Notice in Figures 12a–c that r and r⊥ are sometimes, but not always, equal. In Figure 12b, where the force is perpendicular to r, r and r⊥ are equal. Thus, the torque is simply rF. This is the maximum torque possible for a given r and F. The other extreme occurs when the line of action passes through the axis of rotation. In this case, r and r⊥ = 0. Thus, the torque is also zero. Often, 0 < r⊥ < r ; thus, 0 < r⊥F < rF, as in Figure 12a. 1.4.2 Rotational Equilibrium We know that a body with no net force acting on it will be in translational equilibrium and have zero translational acceleration. As shown in Figure 13a, two equal but opposite forces are able to cancel, resulting in no translational acceleration. Figure 13a. But if you give a spin to a ball by giving it a twist with your fingers, you can give it a rotational acceleration even though the net force acting is zero. In Figure 13b, the two equal forces, F1 and F2, acting on the ball will not cause a translational acceleration, but they will cause a rotational acceleration. Each force produces a counterclockwise torque, thus producing a nonzero torque. Figure 13b. Mechanics 39 For a body to be in rotational equilibrium, the net torque on it must equal zero. This is the case in both Figures 13c and 13d. Figure 13c. In Figure 13c, the two equal forces F1 and F2 act with equal lever arms r⊥. Force F2 applies a negative, clockwise (cw), torque of –r⊥F2. Force F1 applies a positive, counterclockwise (ccw), torque of r⊥F1. ∑ τ= ( −r⊥ F2 ) + r⊥ F1= 0 Figure 13d. In Figure 13d, the two nonequal forces F1 and F2 act with nonequal lever arms r⊥ and 2r⊥. F2 = 2F1. Force F2 applies a negative, clockwise, torque of –r⊥F2. Force F1 applies a positive, counterclockwise, torque of 2r⊥F1. ∑ τ = ( −r F ) ⊥ 2 ∑τ=0 + 2r⊥ F1 = ( −r ⊥ 2F1 ) + 2r⊥ F1 40 Experimental Physics I We will be working with systems that are in rotational equilibrium; that is, where the vector sum of the torques about any axis equals zero. ∑τ any axis = 0 This is a vector equation where positive torques are defined as torques that would tend to produce a counterclockwise rotation and negative torques would tend to produce a clockwise rotation. The First Condition Equilibrium. ∑F any direction = 0 The two equations can also be used together in situations where forces act to put a system in both types of equilibrium. For example, when you lean against a wall, there are forces exerted on you by the wall, the ground, and Earth’s gravity. 1.5 WORK AND ENERGY The concepts of work and energy are closely tied to the concept of force because an applied force can do work on an object and cause a change in energy. Energy is defined as the ability to do work. 1.5.1 Work The concept of work in physics is much more narrowly defined than the common use of the word. Work is done on an object when an applied force moves it through a distance. In our everyday language, work is related to expenditure of muscular effort, but this is not the case in the language of physics. A person that holds a heavy object does no physical work because the force is not moving the object through a distance. Work, according to the physics definition, is being accomplished while the heavy object is being lifted but not while the object is stationary. Another example of the absence of work is a mass on the end of a string rotating in a horizontal circle on a frictionless surface. The centripetal force is directed toward the center of the circle and, therefore, is not moving the object through a distance; that is, the force is not in the direction of motion of the object. (However, work was done to set the mass in motion.) Mathematically, work is W = F · x, where F is the applied force and x is the distance moved, that is, displacement. Work is a scalar. The SI unit for work is the joule (J), which is newton‐meter or kg m/s 2. Mechanics 41 If work is done by a varying force, the above equation cannot be used. Figure shows the force‐versus‐displacement graph for an object that has three different successive forces acting on it. The force is increasing in segment I, is constant in segment II, and is decreasing in segment III. The work performed on the object by each force is the area between the curve and the x axis. The total work done is the total area between the curve and the x axis. For example, in this case, the work done by the three successive forces is shown in below Figure. In this example, the total work accomplished is (1/2)(15)(3) + (15) (2) + (1/2)(15)(2) = 22.5 + 30 + 15; work = 67.5 J. For a gradually changing force, the work is expressed in integral form, W = ∫ F · dx. 1.5.2 Dot or Scalar Product The dot product, also called the scalar product, of two vector s is a number ( scalar quantity) obtained by performing a specific operation on the vector components. The dot product has meaning only for pairs of vectors having the same number of dimensions. The symbol for dot product is a heavy dot ( ). In the two-dimensional Cartesian plane, vectors are expressed in terms of the x -coordinates and y -coordinates of their end points, assuming they begin at the origin ( x , y ) = (0,0). The dot product of two vectors is determined by multiplying their x -coordinates, then multiplying their y -coordinates, and finally adding the two products. Thus, in the above example: 42 Experimental Physics I A B = (2 x -4) + (5 x -3) = -8 - 15 = -23 B C = (-4 x 5) + (-3 x -5) = -20 + 15 = -5 C A = (5 x 2) + (-5 x 5) = 10 - 25 = -15 In polar coordinates, vectors are expressed in terms of length (magnitude) and direction. When expressed in this format, the dot product of two vectors is equal to the product of their lengths, multiplied by the cosine of the angle between them. For any two vectors A and B , A B = B A . That is, the dot product operation is commutative; it does not matter in which order the operation is performed. Work done by a force Work is done whenever a force moves something over a distance. You can calculate the energy transferred, or work done, by multiplying the force by the distance moved in the direction of the force. Energy transferred = work done = force x distance moved in the direction of the force When energy is transferred from chemical energy stored in muscles to ‘uphill energy’ in a raised load, or to ‘elastic energy’ in stretched springs, the energy transferred is a measure of how much work has been done. Energy transferred = mgh This second equation is illustrated by raising kilograms onto different height shelves. You can show that the equation is a good summary of what happens. It takes account of the mass, the height Mechanics raised and whether the kilogram is raised on the Earth or the Moon. The useful thing which you get from fuels by burning them is the transfer of energy released, to some other energy store such as a raised load, or a moving body. However, not all the energy available does a useful job. If you lift a lot of bricks, you can get too hot. As well as transferring energy to the raised bricks, some of the energy generated in your muscles warms you up. The transfer of energy is not 100% efficient and not all the energy transferred is represented by mgh. Nor do you know how much total energy is stored by things being ‘uphill’. You can only calculate energy that is transferred. 43 Important The term work was introduced in 1826 by the French mathematician GaspardGustave Coriolis as “weight lifted through a height”, which is based on the use of early steam engines to lift buckets of water out of flooded ore mines. The SI unit of work is the joule (J). 1.5.3 Potential Energy An object can store energy as the result of its position. For example, the heavy ball of a demolition machine is storing energy when it is held at an elevated position. This stored energy of position is referred to as potential energy. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not drawn), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. This stored energy of position is referred to as potential energy. Potential energy is the stored energy of position possessed by an object. 44 Experimental Physics I Gravitational Potential Energy Gravitational potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation: PEgrav = mass • g • height PEgrav = m *• g • h In the above equation, m represents the mass of the object, h represents the height of the object and g represents the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity. Since many of our labs are done on tabletops, it is often customary to assign the tabletop to be the zero height position. Again this is merely arbitrary. If the tabletop is the zero position, then the potential energy of an object is based upon its height relative to the tabletop. For example, a pendulum bob swinging to and from above the tabletop has a potential energy that can be measured based on its height above the tabletop. By measuring the mass of the bob and the height of the bob above the tabletop, the potential energy of the bob can be determined. Mechanics 45 Since the gravitational potential energy of an object is directly proportional to its height above the zero position, a doubling of the height will result in a doubling of the gravitational potential energy. A tripling of the height will result in a tripling of the gravitational potential energy. Elastic Potential Energy The second form of potential energy that we will discuss is elastic potential energy. Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy. Springs are a special instance of a device that can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k). Fspring = k • x Such springs are said to follow Hooke’s Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs that relates the amount of elastic potential 46 Experimental Physics I energy to the amount of stretch (or compression) and the spring constant. The equation is: PEspring = 0.5 • k • x2 Where k = spring constant x = amount of compression (relative to equilibrium position) To summarize, potential energy is the energy that is stored in an object due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position. 1.5.4 Kinetic Energy Kinetic energy, form of energy that an object or a particle has by reason of its motion. If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy. Kinetic energy is a property of a moving object or particle and depends not only on its motion but also on its mass. The kind of motion may be translation (or motion along a path from one place to another), rotation about an axis, vibration, or any combination of motions. Translational kinetic energy of a body is equal to one-half the product of its mass, m, and the square of its velocity, v, or 1/2mv2. This formula is valid only for low to relatively high speeds; for extremely high-speed particles it yields values that are too small. When the speed of an object approaches that of light (3 × 108 metres per second, or 186,000 miles per second), its mass increases, and the Mechanics 47 laws of relativity must be used. Relativistic kinetic energy is equal to the increase in the mass of a particle over that which it has at rest multiplied by the square of the speed of light. The unit of energy in the metre-kilogram-second system is the joule. A two-kilogram mass (something weighing 4.4 pounds on Earth) moving at a speed of one metre per second (slightly more than two miles per hour) has a kinetic energy of one joule. In the centimetre-gram-second system the unit of energy is the erg, 10−7 joule, equivalent to the kinetic energy of a mosquito in flight. Other units of energy also are used, in specific contexts, such as the still smaller unit, the electron volt, on the atomic and subatomic scale. For a rotating body, the moment of inertia, I, corresponds to mass, and the angular velocity (omega), ω, corresponds to linear, or translational, velocity. Accordingly, rotational kinetic energy is equal to one-half the product of the moment of inertia and the square of the angular velocity, or 1/2Iω2. 1.5.5 Mechanical Energy Mechanical energy is often confused with Kinetic and Potential Energy. We will try to make it very easy to understand and know the difference. Before that, we need to understand the word ‘Work’. Work’ is done when a force acts on an object to cause it to move, change shape, displace, or do something physical. For, example, if I push a door open for my pet dog to walk in, work is done on the door (by causing it to open). But what kind of force caused the door to open? Here is where Mechanical Energy comes in. Mechanical energy is the sum of kinetic and potential energy in an object that is used to do work. In other words, it is energy in an object due to its motion or position, or both. In the ‘open door’ example above, I possess potential chemical energy (energy stored in me), and by lifting my hands to push the door, my action also had kinetic energy (energy in the motion of my hands). By pushing the door, my potential and kinetic energy was transferred into mechanical energy, which caused work to be done (door opened). Here, the door gained mechanical energy, which caused the door to be displaced temporarily. Note that for work to be done, an object has to supply a force for another object to be displaced. Here is another example of a boy with an iron hammer and nail. In the illustration below… 48 Experimental Physics I (1) (2) (3) The iron hammer on its own has no kinetic energy, but it has some potential energy (because of its weight). To drive a nail into the piece of wood (which is work), he has to lift the iron hammer up, (this increases its potential energy because if it is high position). And force it to move at great speed downwards (now has kinetic energy) to hit the nail. The sum of the potential and kinetic energy that the hammer acquired to drive in the nail is called the Mechanical energy, which resulted in the work done. 1.5.6 Power The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation. Power = Work / time or P=W/t The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts. Mechanics 49 Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. Thus, the power of a machine is the work/time ratio for that particular machine. A car engine is an example of a machine that is given a power rating. The power rating relates to how rapidly the car can accelerate the car. Suppose that a 40-horsepower engine could accelerate the car from 0 mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four times the horsepower could do the same amount of work in one-fourth the time. That is, a 160-horsepower engine could accelerate the same car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same amount of work, power and time are inversely proportional. The power equation suggests that a more powerful engine can do the same amount of work in less time. A person is also a machine that has a power rating. Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time. A common physics lab involves quickly climbing a flight of stairs and using mass, height and time information to determine a student’s personal power. Despite the diagonal motion along the staircase, it is often assumed that the horizontal motion is constant and all the force from the steps is used to elevate the student upward at a constant speed. Thus, the weight of the student is equal to the force that does the work on the student and the height of the staircase is the upward displacement. Suppose that Ben Pumpiniron elevates his 80kg body up the 2.0-meter stairwell in 1.8 seconds. If this were the case, then we could calculate Ben’ spower rating. It can be assumed that Ben must apply an 800-Newton downward force upon the stairs to elevate his body. By so doing, the stairs would push upward on Ben’s body with just enough force to lift his body up the stairs. It can also be assumed that the angle between the force of the stairs on Ben and Ben’s displacement is 0 degrees. With these two approximations, Ben’s power rating could be determined as shown below. Work 784 N2.0 m = Time 1.8 seconds Power = 871 Watts Power = Ben’s power rating is 871 Watts. He is quite a horse. 50 Experimental Physics I Another Formula for Power The expression for power is work/time. And since the expression for work is force*displacement, the expression for power can be rewritten as (force*displacement)/time. Since the expression for velocity is displacement/time, the expression for power can be rewritten once more as force*velocity. This is shown below. Work Force Displcement = Time Time Displcement Power = Force Time Power = Force Velocity Power = This new equation for power reveals that a powerful machine is both strong (big force) and fast (big velocity). A powerful car engine is strong and fast. A powerful piece of farm equipment is strong and fast. A powerful weightlifter is strong and fast. A powerful lineman on a football team is strong and fast. A machine that is strong enough to apply a big force to cause a displacement in a small amount of time (i.e., a big velocity) is a powerful machine. 1.5.7 Conservative and Nonconservative Forces It is important to know the difference between conservative and nonconservative forces. The work a conservative force does on an object is path-independent; the actual path taken by the object makes no difference. Fifty meters up in the air has the same gravitational potential energy whether you get there by taking the steps or by hopping on a Ferris wheel. That is different from the force of friction, which dissipates kinetic energy as heat. When friction is involved, the path you take matters — a longer path will dissipate more kinetic energy than a short one. For that reason, friction is a nonconservative force. For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises h meters into the air. You can take two ways up the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, “Guess what — our potential energy compared to before is mgh greater.” “Mine, too,” you say, looking out over the view. You pull out this equation: Mechanics 51 ∆PE = mg(h f − h i ) This equation basically states that the actual path you take when going vertically from hi to hf does not matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity does not matter, gravity is a conservative force. Here is another way of looking at conservative and nonconservative forces. Say you are vacationing in the Alps and your hotel is at the top of Mt. Newton. You spend the whole day driving around — down to a lake one minute, to the top of a higher peak the next. At the end of the day, you end up back at the same location: your hotel on top of Mt. Newton. What is the change in your gravitational potential energy? In other words, how much network did gravity perform on you during the day? Gravity is a conservative force, so the change in your gravitational potential energy is 0. Because you have experienced no net change in your gravitational potential energy, gravity did no network on you during the day. The road exerted a normal force on your car as you drove around, but that force was always perpendicular to the road (meaning no force parallel to your motion), so it did not do any work, either. Conservative forces are easier to work with in physics because they do not “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy. If you have to deal with nonconservative forces such as friction, including air friction, the situation is different. If you are dragging something over a field carpeted with sandpaper, for example, the force of friction does different amounts of work on you depending on your path. A path that is twice as long will involve twice as much work to overcome friction. What is really not being conserved around a track with friction is the total potential and kinetic energy, which taken together is mechanical energy. When friction is involved, the loss in mechanical energy goes into heat energy. You can say that the total amount of energy does not change if you include that heat energy. However, the heat energy dissipates into the environment quickly, so it is not recoverable or convertible. For that and other reasons, physicists often work in terms of mechanical energy. 52 Experimental Physics I Gravity Gravity is the most common conservative force, and to demonstrate that it is conservative is relatively simple. Consider first a ball thrown up into the air. On the ball’s trip upward, gravity works against the motion of the ball, producing a total work of - mgh. This negative work causes the ball to slow down until it stops, reverses direction and begins to fall. During its fall, the force of gravity is in the same direction as the motion of the ball, and the gravitational force does positive work of magnitude mgh , accelerating ball until it reaches the ground with the same speed with which it left. What is the net work done by gravity on the ball over this closed loop? Zero, as we expect by our first principle of conservative forces. What about our second principle? Let us construct two alternative paths for a ball being thrown up into the air: Friction Friction is the most common nonconservative force, and we will demonstrate why it is not conservative. Consider a crate on a rough floor, of weight W. The crate is pushed from one end of the floor to the other, a distance of h meters, and then back to its original spot. What is the net work done on the crate? At all times the friction opposes the motion of the crate, exerting a force of μk W at all times. Thus the total work done over the trip is simply (- 2) (μk W) (h) = - 2hwμk , clearly not equal to zero. The network by friction over a closed path is not zero, and it is nonconservative. Is friction path independent? We expect not, because we know it is nonconservative. To prove the suspicion, simply consider two possible ways to move a crate between two points on a rough floor. One is a straight line, one is a somewhat longer route. No matter the path, the force is the same at all times that the crate is moving. The difference, however, is that friction acts over a longer distance in the case of the second path, causing a greater network to be done. Thus friction is not path independent, and we confirm that it is nonconservative. Mechanics 53 1.6 COLLISION Collision means two objects coming into contact with each other for a very short period. In other words, collision is a reciprocative interaction between two masses for a very short interval wherein the momentum and energy of the colliding masses changes. While playing carroms, you might have noticed the effect of a striker on coins when they both collide. Collision involves two masses m1 and m2. The v1i is the speed of particle m1, where the subscript ‘i’ implies initial. The particle with mass m2 is at rest. In this case, the object with mass m1 collides with the stationary object of mass m2. As a result of this collision the masses m1 and m2 move in different directions. 1.6.1 Elastic and Inelastic Collisions In an elastic collision, the objects separate after impact and don’t lose any of their kinetic energy. Kinetic energy is the energy of motion and is covered in detail elsewhere. The law of conservation of momentum is very useful here, and it can be used whenever the net external force on a system is zero. Figure 14 shows an elastic collision where momentum is conserved. 54 Experimental Physics I Figure 14. The diagram shows a one-dimensional elastic collision between two objects. Perfectly elastic collisions can happen only with subatomic particles. Everyday observable examples of perfectly elastic collisions don’t exist—some kinetic energy is always lost, as it is converted into heat transfer due to friction. However, collisions between everyday objects are almost perfectly elastic when they occur with objects and surfaces that are nearly frictionless, such as with two steel blocks on ice. Now, to solve problems involving one-dimensional elastic collisions between two objects, we can use the equation for conservation of momentum. First, the equation for conservation of momentum for two objects in a one-dimensional collision is p1 + p2 = p′1 + p′ 2 (Fnet = 0). Substituting the definition of momentum p = mv for each initial and final momentum, we get m1 v1 + m2 v 2 = m1 v′1 + m2 v′ 2 , where the primes (‘) indicate values after the collision; In some texts, you may see i for initial (before collision) and f for final (after collision). The equation assumes that the mass of each object does not change during the collision. Mechanics 55 Now, let us turn to the second type of collision. An inelastic collision is one in which objects stick together after impact, and kinetic energy is not conserved. This lack of conservation means that the forces between colliding objects may convert kinetic energy to other forms of energy, such as potential energy or thermal energy. The concepts of energy are discussed more thoroughly elsewhere. For inelastic collisions, kinetic energy may be lost in the form of heat. Figure 15 shows an example of an inelastic collision. Two objects that have equal masses head toward each other at equal speeds and then stick together. The two objects come to rest after sticking together, conserving momentum but not kinetic energy after they collide. Some of the energy of motion gets converted to thermal energy, or heat. Figure 15. A one-dimensional inelastic collision between two objects. Momentum is conserved, but kinetic energy is not conserved. (a) Two objects of equal mass initially head directly toward each other at the same speed. (b) The objects stick together, creating a perfectly inelastic collision. In the case shown in this figure, the combined objects stop; This is not true for all inelastic collisions. 56 Experimental Physics I Since the two objects stick together after colliding, they move together at the same speed. This lets us simplify the conservation of momentum equation from m1 v1 + m2 v 2 = m1 v′1 + m2 v′ 2 to m1 v1 + m2 v 2 =( m1 + m2 ) v′ for inelastic collisions, where v′ is the final velocity for both objects as they are stuck together, either in motion or at rest. Solving Collision Problems In one-dimensional collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are twodimensional collisions, and just as we did with two-dimensional forces, we will solve these problems by first choosing a coordinate system and separating the motion into its x and y components. One complication with two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass each other, they will spin in circles. We will not consider such rotation until later, and so for now, we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin. We start by assuming that Fnet = 0, so that momentum p is conserved. The simplest collision is one in which one of the particles is initially at rest. The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in Figure 16. Because momentum is conserved, the components of momentum along the xand y-axes, displayed as px and py, will also be conserved. With the chosen coordinate system, py is initially zero and px is the momentum of the incoming particle. Mechanics 57 Figure 16. A two-dimensional collision with the coordinate system chosen so that m2 is initially at rest and v1 is parallel to the x-axis. Now, we will take the conservation of momentum equation, p1 + p2 = p′1 + p′2 and break it into its x and y components. Along the x-axis, the equation for conservation of momentum is p1x + p2x = p′1x + p′ 2x . In terms of masses and velocities, this equation is m1 v 1 x + m2 v 2 x = m1 v ′1 x + m2 v ′ 2 x . But because particle 2 is initially at rest, this equation becomes m = v m1 v ′1 x + m2 v ′ 2 x . 1 1x The components of the velocities along the x-axis have the form v cos θ. Because particle 1 initially moves along the x-axis, we find v1x = v1. Conservation of momentum along the x-axis gives the equation where θ1 and θ2 are as shown in Figure 16. 58 Experimental Physics I Along the y-axis, the equation for conservation of momentum is p1 y + p2 y = p′1 y + p′ 2 y , or m1 v 1 y + m2 v 2 y = m1 v ′1 y + m2 v ′ 2 y . But v1y is zero, because particle 1 initially moves along the x-axis. Because particle 2 is initially at rest, v2y is also zero. The equation for conservation of momentum along the y-axis becomes = 0 m1 v ′1 y + m2 v ′ 2 y. The components of the velocities along the y-axis have the form v sin θ. Therefore, conservation of momentum along the y-axis gives the following equation: 1.6.2 Conservation of Momentum It is important we realize that momentum is conserved during collisions, explosions, and other events involving objects in motion. To say that a quantity is conserved means that it is constant throughout the event. In the case of conservation of momentum, the total momentum in the system remains the same before and after the collision. where forces acting on the objects produced large changes in momentum. Why is this? The systems of interest considered in those problems were not inclusive enough. If the systems were expanded to include more objects, then momentum would in fact be conserved in those sample problems. It is always possible to find a larger system where momentum is conserved, even though momentum changes for individual objects within the system. For example, if a football player runs into the goalpost in the end zone, a force will cause him to bounce backward. His momentum is obviously greatly changed, and considering only the football player, we would find that momentum is not conserved. However, the system can be expanded to contain the entire Earth. Surprisingly, Earth also recoils— conserving momentum—because of the force applied to it through the Mechanics 59 goalpost. The effect on Earth is not noticeable because it is so much more massive than the player, but the effect is real. Next, consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—in the example shown in Figure 17 of one car bumping into another. Both cars are coasting in the same direction when the lead car, labeled m2, is bumped by the trailing car, labeled m1. The only unbalanced force on each car is the force of the collision, assuming that the effects due to friction are negligible. Car m1 slows down as a result of the collision, losing some momentum, while car m2 speeds up and gains some momentum. If we choose the system to include both cars and assume that friction is negligible, then the momentum of the two-car system should remain constant. Now we will prove that the total momentum of the two-car system does in fact remain constant, and is therefore conserved. Figure 17. Car of mass m1 moving with a velocity of v1 bumps into another car of mass m2 and velocity v2. As a result, the first car slows down to a velocity of v′1 and the second speeds up to a velocity of v′2. The momentum of each car is changed, but the total momentum ptot of the two cars is the same before and after the collision if you assume friction is negligible. Using the impulse-momentum theorem, the change in momentum of car 1 is given by Äp1 = F1Ät , where F1 is the force on car 1 due to car 2, and Δt is the time the force acts, or the duration of the collision. 60 Experimental Physics I Similarly, the change in momentum of car 2 is Δp2=F2Δt where F2 is the force on car 2 due to car 1, and we assume the duration of the collision Δt is the same for both cars. We know from Newton’s third law of motion that F2 = –F1, and so Δp2=−F1Δt=−Δp1. Therefore, the changes in momentum are equal and opposite, and Δp1+Δp2=0. Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is, p1+p2=constant p1+p2=p′1+p′2, where p′1 and p′2 are the momenta of cars 1 and 2 after the collision. This result that momentum is conserved is true not only for this example involving the two cars, but for any system where the net external force is zero, which is known as an isolated system. The law of conservation of momentum states that for an isolated system with any number of objects in it, the total momentum is conserved. In equation form, the law of conservation of momentum for an isolated system is written as ptot=constant or ptot = p′tot , where ptot is the total momentum, or the sum of the momenta of the individual objects in the system at a given time, and p′tot is the total momentum some time later. The conservation of momentum principle can be applied to systems as diverse as a comet striking the Earth or a gas containing huge numbers of atoms and molecules. Conservation of momentum appears to be violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not. Mechanics 61 1.7 PROJECTILE MOTION Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. The motion of falling objects, as covered in Problem-Solving Basics for One-Dimensional Kinematics, is a simple one-dimensional type of projectile motion in which there is no horizontal movement. We consider two-dimensional projectile motion, such as that of a football or other object for which air resistance is negligible. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. Where vertical and horizontal motions were seen to be independent. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. (This choice of axes is the most sensible, because Hints acceleration due to gravity is vertical— thus, there will be no acceleration along We used the horizontal axis when air resistance is the notation A to negligible.) As is customary, we call the represent a vechorizontal axis the x-axis and the vertical tor with compoaxis the y-axis. Figure 18 illustrates the nents Ax and Ay. notation for displacement, where s is defined If we continued this format, we to be the total displacement and x and y are would call displaceits components along the horizontal and ment s with comvertical axes, respectively. The magnitudes ponents sx and sy. of these vectors are s, x, and y. However, to simpli- fy the notation, we To describe motion we must deal will simply reprewith velocity and acceleration, as well as sent the component with displacement. We must find their vectors as x and y. components along the x– and y-axes, too. We will assume all forces except gravity (such as air resistance and friction, for example) are negligible. The components of acceleration are then very simple: ay = –g = –9.80 m/s2. (Note that this definition assumes that the upwards direction is defined as the positive direction. If you arrange the coordinate system instead such that the downwards direction is positive, then acceleration due to gravity takes a positive value.) Because gravity is vertical, ax=0. Both accelerations are constant, so the kinematic equations can be used. 62 Experimental Physics I Figure 18. The total displacement s of a soccer ball at a point along its path. The vector s has components x and y along the horizontal and vertical axes. Its magnitude is s, and it makes an angle θ with the horizontal. Given these assumptions, the following steps are then used to analyze projectile motion: Step 1. Resolve or break the motion into horizontal and vertical components along the x- and y-axes. These axes are perpendicular, so Ax = A cos θ and Ay = A sin θ are used. The magnitude of the components of displacement s along these axes are x and y. The magnitudes of the components of the velocity v are Vx = V cos θ and Vy = v sin θ where v is the magnitude of the velocity and θ is its direction, as shown in 2. Initial values are denoted with a subscript 0, as usual. Step 2.Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms: Step 3. Solve for the unknowns in the two separate motions— one horizontal and one vertical. Note that the only common variable between the motions is time t. The problem solving procedures here are the same as for one-dimensional kinematics and are illustrated in the solved examples below. Step 4. Recombine the two motions to find the total displacement s and velocity v. Because the x – and y -motions are perpendicular, we determine these vectors by using the techniques outlined in the Vector Addition and Subtraction: Analytical Methods and employing = A Ax 2 + Ay 2 Mechanics 63 and θ = tan−1 (Ay/Ax) in the following form, where θ is the direction of the displacement s and θv is the direction of the velocity v: Figure 19. (a) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because ax=0 and vx is thus constant. (c) The velocity in the vertical direction begins to decrease as the object rises; at its highest point, the vertical velocity is zero. As the object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d) The x – and y -motions are recombined to give the total velocity at any given point on the trajectory. In solving part (a) of the preceding example, the expression we found for y is valid for any projectile motion where air resistance is negligible. Call the maximum height y=h; then, h= v0 y 2 2g 64 Experimental Physics I This equation defines the maximum height of a projectile and depends only on the vertical component of the initial velocity. One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic. He used it to predict the range of a projectile. On level ground, we define range to be the horizontal distance R traveled by a projectile. Galileo and many others were interested in the range of projectiles primarily for military purposes—such as aiming cannons. However, investigating the range of projectiles can shed light on other interesting phenomena, such as the orbits of satellites around the Earth. Let us consider projectile range further. Figure 20. Trajectories of projectiles on level ground. (a) The greater the initial speed v0, the greater the range for a given initial angle. (b) The effect of initial angle θ0 on the range of a projectile with a given initial speed. Note that the range is the same for 15º and 75º, although the maximum heights of those paths are different. How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed v0, the greater the range, as shown in Figure 20(a). The initial angle θ0 also has a dramatic effect on the range, as illustrated in Figure 20(b). For a fixed initial speed, such as might be produced by a cannon, the maximum range is obtained with θ0 = 45º. This is true only for conditions neglecting air resistance. Mechanics 65 If air resistance is considered, the maximum angle is approximately 38º. Interestingly, for every initial angle except 45º, there are two angles that give the same range—the sum of those angles is 90º. The range also depends on the value of the acceleration of gravity g. The lunar astronaut Alan Shepherd was able to drive a golf ball a great distance on the Moon because gravity is weaker there. The range R of a projectile on level ground for which air resistance is negligible is given by R= v 0 2 sin2θ0 g where v0 is the initial speed and θ0 is the initial angle relative to the horizontal. When we speak of the range of a projectile on level ground, we assume that R is very small compared with the circumference of the Earth. If, however, the range is large, the Earth curves away below the projectile and acceleration of gravity changes direction along the path. The range is larger than predicted by the range equation given above because the projectile has farther to fall than it would on level ground. (See Figure 21.) If the initial speed is great enough, the projectile goes into orbit. This is called escape velocity. This possibility was recognized centuries before it could be accomplished. When an object is in orbit, the Earth curves away from underneath the object at the same rate as it falls. The object thus falls continuously but never hits the surface. In Addition of Velocities, we will examine the addition of velocities, which is another important aspect of two-dimensional kinematics and will also yield insights beyond the immediate topic. Figure 21. Projectile to satellite. In each case shown here, a projectile is launched from a very high tower to avoid air resistance. With increasing initial speed, the range increases and becomes longer than it would be on level ground because the Earth curves away underneath its path. With a large enough initial speed, orbit is achieved. 66 Experimental Physics I 1.8 BALLISTIC PENDULUM A ballistic pendulum is a device for measuring a bullet’s momentum, from which it is possible to calculate the velocity and kinetic energy. Ballistic pendulums have been largely rendered obsolete by modern chronographs, which allow direct measurement of the projectile velocity. It can be used to measure any transfer of momentum. For example, a ballistic pendulum was used by physicist C. V. Boys to measure the elasticity of golf balls and by physicist Peter Guthrie Tait to measure the effect that spin had on the distance a golf ball traveled. In a perfectly inelastic collision, a bullet is fired into the stationary pendulum, which captures the bullet and absorbs its energy. The stationary pendulum now moves with a new velocity just after the collision. While not all of the energy from the bullet is transformed into kinetic energy for the pendulum (some is used as heat and deformation energy) , the momentum of the system is conserved. By measuring the height of the pendulum’s swing, the potential energy of the pendulum when it stops can be measured. In the case of a pendulum total mechanical energy is conserved. So kinetic energy of the pendulum (after firing) is fully converted to potential energy. Thus the pendulum’s initial velocity can be calculated. Using the law of conservation of momentum, the velocity of the bullet can be computed. From the law of conservation of mechanical energy of the pendulum; 1 (m + M)V 2 = (m + M)gh 2 where, m- Mass of bullet M- Mass of Pendulum h- Maximum height reached by the pendulum v- Velocity of the bullet V- Velocity of pendulum g-Gravity of earth According to the law of conservation of momentum; Mechanics 67 mv = (M + m)V mv = (M + m) (2gh) Velocity of the bullet is given by; 1 (M + m) (2gh) m M = v ( + m) (2gh) m = v 1.9 CENTRIPETAL FORCE A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as «a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre”. In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits. Formula The magnitude of the centripetal force on an object of mass m moving at tangential speed v along a path with radius of curvature r is: = Fc ma = c a c = lim ∆t → 0 mv 2 r | ∆v | ∆t 68 Experimental Physics I Where ac is the centripetal acceleration and ∆v is the difference between the velocity vectors. Since the velocity vectors in the above diagram have constant magnitude and since each one is perpendicular to its respective position vector, simple vector subtraction implies two similar isosceles triangles with congruent angles – one comprising a base of ∆v and a leg length of v, and the other a base of ∆r (position vector difference) and a leg length of r. | ∆v | | ∆r | = v r ∆v |= v | ∆r | r v | ∆r |. r | ∆v | v | ∆r | | ∆r | v2 a c = lim = lim = ω lim = vω = ∆t → 0 ∆t ∆t → 0 ∆t r ∆t →0 ∆t r Therefore, || ∆v | can be substituted with The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle (the circle that best fits the local path of the object, if the path is not circular). The speed in the formula is squared, so twice the speed needs four times the force. The inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. This force is also sometimes written in terms of the angular velocity ω of the object about the center of the circle, related to the tangential velocity by the formula v = ωr so that F= mrω2 . c Expressed using the orbital period T for one revolution of the circle, ω= 2π T the equation becomes Mechanics 69 2 2π Fc = mr . T In particle accelerators, velocity can be very high (close to the speed of light in vacuum) so the same rest mass now exerts greater inertia (relativistic mass) thereby requiring greater force for the same centripetal acceleration, so the equation becomes: Fc = γmv 2 r where γ= 1 1− v2 c2 is the Lorentz factor. Thus the centripetal force is given by: Fc = γmvω which is the rate of change of relativistic momentum γmv. EXERCISE Answer the following questions: 1. Certain force acting on a 20 kg mass changes its velocity from 5 m s–1 to 2 m s–1. Calculate the work done by the force. 2. The potential energy of a freely falling object decreases progressively. Does this violate the law of conservation of energy? Why? 3. What is the volume of 35.7 g of water? 4. What is a Collision? Explain the types of collision. 5. What is projectile motion? 6. Discuss about the centripetal force 7. An object of mass, m is moving with a constant velocity, v. How much work should be done on the object in order to bring the object to rest? 70 Experimental Physics I 8. 9. 10. Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force? Calculate the centripetal force on the end of a 100 m (radius) wind turbine blade that is rotating at 0.5 rev/s. Assume the mass is 4 kg. Give an estimate of the accuracy with which you can measure volumes with your 50 mL graduated cylinder. MULTIPLE CHOICE QUESTIONS Tick the correct answer. 1. What is termed as the quantity of matter contained in a body? a. Density b. Volume c. Mass d. Specific gravity 2. Which is called mass per unit volume of a substances? a. Mass b. Weight c. Volume d. Density 3. An iron sphere of mass 10 kg has the same diameter as an aluminium sphere of mass is 3.5 kg. Both spheres are dropped simultaneously from a tower. When they are lo m above the ground, they have the same. a. acceleration b. momenta c. potential energy d. kinetic energy 4. The work done on an object does not depend upon the a. displacement b. force applied c. angle between force and displacement d. initial velocity of the object Mechanics 5. 71 If speed of a car becomes 2 times, its kinetic energy becomes a. 4 times b. 8 times c. 16 times d. 12 times 6. Work done by friction a. increases kinetic energy of body b. decreases kinetic energy of body c. increases potential energy of body d. decreases potential energy of body. 7. Which of the following is a type of motion? a. Circular b. Rectilinear c. Periodic d. All the above 8. A body of mass m, projected at an angle of θ from the ground with an initial velocity of v, acceleration due to gravity is g, what is the maximum horizontal range covered? a. R = v2 (sin 2θ)/g b. R = v2 (sin θ)/2g c. R = v2 (sin 2θ)/2g d. R = v2 (sin θ)/g 9. On calculating which of the following quantities, the mass of the body has an effect in simple projectile motion? a. Velocity b. Force c. Time of flight d. Range 10. When do we get maximum range in a simple projectile motion? a. When θ = 45° b. When θ = 60° c. When θ = 90° d. When θ = 0° 72 Experimental Physics I ANSWER 1. (c) 2. (d) 3. (a) 4. (d) 5. (a) 6. (b) 7. (a) 8. (a) 9. (b) 10. (a) REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. Carroll, Sean M., “From Eternity to Here”, Penguin Group, 2010 Eastlake, Charles N., “An Aerodynamicist’s View of Lift, Bernoulli, and Newton”, The Physics Teacher 40, 166 (March 2002). Gonzalez, Guillermo and Richards, Jay W., The Privileged Planet, Regnery Publishing, 2004. Klarreich, Erica, “Navigating Celestial Currents”, Science News 167, 250, April 16, 2005. Sears, Zemansky,Young and Freedman, University Physics, 10th Ed., Addison-Wesley, 2000 von Arx, William S., An Introduction to Physical Oceanography, Addison-Wesley, 1962. Watts, Robert G. and Ferrer, Ricardo, The lateral force on a spinning sphere: Aerodynamics of a curveball, American Journal of Physics 55, 40, Jan 1987. Weinberg, S. (May 1, 2005). The Quantum Theory of Fields, Volume 1: Foundations (1st ed.). Cambridge University Press. p. xxi. Young, H.D. and Freedman, R. A., University Physics, 11th Ed., Pearson, 2004. Heat 39 CHAPTER 2 HEAT OBJECTIVES After reading this chapter, you should be able to: • Define thermal linear expansion • Explain calorimetry and the specific heat of a metal • Discuss about heat of fusion of ice • Elaborate heat of vaporization of water Heat, energy that is transferred from one body to another as the result of a difference in temperature. If two bodies at different temperatures are brought together, energy is transferred—i.e., heat flows—from the hotter body to the colder. The effect of this transfer of energy usually, but not always, is an increase in the temperature of the colder body and a decrease in the temperature of the hotter body. A substance may absorb heat without an increase in temperature by changing from one physical state (or phase) to another, as from a solid to a liquid (melting), from a solid to a vapour (sublimation), from a liquid to a vapour (boiling), or 40 Experimental Physics I from one solid form to another (usually called a crystalline transition). The important distinction between heat and temperature (heat being a form of energy and temperature a measure of the amount of that energy present in a body) was clarified during the 18th and 19th centuries. Heat is one of the essential energy forms on Earth for the survival of different lives. Heat transfer takes place from one body to the other because of the difference in temperature according to thermodynamics. We use heat energy in our day to day activities such as cooking, transportation, ironing, recreation and much more. Heat energy also plays a crucial role when it comes to nature. The occurrence of rain, wind, change in the seasons, etc. is all dependent on the gradient that is created because of the uneven heating of various regions. In this article, we will discuss what is meant by heat, what is Latent heat and what is used to measure heat. 2.1 THERMAL LINEAR EXPANSION Thermal linear expansion is the process by which solid objects expand in length as a result of a transfer of energy into that object due to heat. When heat flows into a solid object, the individual molecules and atoms composing that solid begin to vibrate with greater energies thereby increasing the distance between neighboring atoms. This expansion between atoms and molecules collectively increases the length of any long solid object, such as a metal rod. The length that the solid object elongates upon heating or compresses upon cooling is dependent upon three factors. It depends on the original length, which means that a solid object with a longer length will elongate a greater distance. It depends on the change in temperature, which basically means that objects will expand more for greater increases in temperatures. Finally, the expansion also depends on the type of object we are using, which is given by the coefficient of linear expansion. 2.1.1 Linear Expansion Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. (An example of this is the buckling of railroad track, as seen in. ) Atoms and molecules in a solid, for instance, constantly oscillate around its equilibrium point. This kind of excitation is called thermal motion. When a substance is heated, its constituent particles begin moving more, thus maintaining a greater average separation with their neighboring particles. The degree of Heat 41 expansion divided by the change in temperature is called the material’s coefficient of thermal expansion; it generally varies with temperature. Figure 1: Thermal expansion of long continuous sections of rail tracks is the driving force for rail buckling. Expansion, Not Contraction Why does matter usually expand when heated? The answer can be found in the shape of the typical particle-particle potential in matter. Particles in solids and liquids constantly feel the presence of other neighboring particles. This interaction can be represented mathematically as a potential curve. Fig 2 illustrates how this interparticle potential usually takes an asymmetric form rather than a symmetric form, as a function of particle-particle distance. Note that the potential curve is steeper for shorter distance. In the diagram, (b) shows that as the substance is heated, the equilibrium (or average) particleparticle distance increases. Materials which contract or maintain their shape with increasing temperature are rare. This effect is limited in size, and only occurs within limited temperature ranges. Linear Expansion To a first approximation, the change in length measurements of an object (linear dimension as opposed to, for example, volumetric dimension) due to thermal expansion is related to temperature change by a linear expansion coefficient. It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write: 42 Experimental Physics I Figure 2: Typical inter-particle potential in condensed matter (such as solid or liquid). where L is a particular length measurement and dL/dT is the rate of change of that linear dimension per unit change in temperature. From the definition of the expansion coefficient, the change in the linear dimension ΔL over a temperature range ΔT can be estimated to be: This equation works well as long as the linear-expansion coefficient does not change much over the change in temperature. If it does, the equation must be integrated. 2.1.2 Area Expansion Objects expand in all dimensions. That is, their areas and volumes, as well as their lengths, increase with temperature. We learned about the linear expansion (in one dimension) in the previous Atom. Objects expand in all dimensions, and we can extend the thermal expansion for 1D to two (or three) dimensions. That is, their areas and volumes, as well as their lengths, increase with temperature. Heat 43 Area thermal expansion coefficient The area thermal expansion coefficient relates the change in a material’s area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, we may write: , where is some area of interest on the object, and dA/dT is the rate of change of that area per unit change in temperature. The change in the linear dimension can be estimated as: . This equation works well as long as the linear expansion coefficient does not change much over the change in temperature ΔT. If it does, the equation must be integrated. Figure 3: In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. Relationship to linear thermal expansion coefficient For isotropic materials, and for small expansions, the linear thermal expansion coefficient is one half of the area coefficient. To derive the relationship, let’s take a square of steel that has sides of length L. The original area will be A = L2,and the new area, after a temperature increase, 44 Experimental Physics I will be The approximation holds for a sufficiently small ΔL campared to L. Since from the equation above (and from the definitions of the thermal coefficients), we get . 2.1.3 Volume Expansion Substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. illustrates that, in general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Such substances that expand in all directions are called isotropic. For isotropic materials, the area and linear coefficients may be calculated from the volumetric coefficient (discussed below). Figure 5: Volumetric Expansion: In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. (c) Volume also increases, because all three dimensions increase. Heat 45 Mathematical definitions of these coefficients are defined below for solids, liquids, and gasses: The subscript p indicates that the pressure is held constant during the expansion. In the case of a gas, the fact that the pressure is held constant is important, as the volume of a gas will vary appreciably with pressure as well as with temperature. For a solid, we can ignore the effects of pressure on the material, thus the volumetric thermal expansion coefficient can be written: where V is the volume of the material, and is dV/dT the rate of change of that volume with temperature. This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 °C. This is an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 °C, or 0.004% per degree C. Relationship to Linear Thermal Expansion Coefficient For isotropic material, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient. To derive the relationship, let’s take a cube of steel that has sides of length L. The original volume will be V = L3,and the new volume, after a temperature increase, will be: The approximation holds for a sufficiently small ΔLΔL compared to L. Since: 46 Experimental Physics I Important (and from the definitions of the thermal coefficients), we arrive at: Special Properties of Water Objects will expand with increasing temperature, but water is the most important exception to the general rule. In the case of a gas, expansion depends on how the pressure changed in the process because the volume of a gas will vary appreciably with pressure as well as temperature. Special Properties of Water In general, objects will expand with increasing temperature. However, a number of materials contract on heating within certain temperature ranges; this is usually called negative thermal expansion, rather than “thermal contraction. ” Water is the most important exception to the general rule. Water has this unique characteristic because of the particular nature of the hydrogen bond in H2O. Density of Water as Temperature Changes At temperatures greater than 4ºC (40ºF) water expands with increasing temperature (its density decreases). However, it expands with decreasing temperature when it is between +4ºC and 0ºC (40ºF to 32ºF). Water is densest at +4ºC. Figure 6: Water Density vs. Temperature: The density of water as a function of temperature. Note that the thermal expansion is actually very small. The maximum density at +4ºC is only 0.0075% greater than the density at 2ºC, and 0.012% greater than that at 0ºC. Heat 47 Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to 4ºC it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of 4ºC. If the temperature in the surface layer drops below 4ºC, the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over, while the bottom may remain at 4ºC. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in 4ºC water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water. Figure 7: Temperature in a Lake: Temperature distribution in a lake on warm and cold days in winter. Ice Versus Water The solid form of most substances is denser than the liquid phase; thus, a block of most solids will sink in the liquid. However, a block of ice floats in liquid water because ice is less dense. Upon freezing, the density of water decreases by about 9%. 2.1.4 Thermal Expansion of Solids and Liquids The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the 48 Experimental Physics I hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes. What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it. An increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension. 2.1.5 Linear Thermal Expansion—Thermal Expansion in One Dimension The change in length ΔL is proportional to length L. The dependence of thermal expansion on temperature, substance, and length is summarized in the equation ΔL = αLΔT,where ΔL is the change in length L, ΔT is the change in temperature, and α is the coefficient of linear expansion, which varies slightly with temperature. Table 1 lists representative values of the coefficient of linear expansion, which may have units of 1/ºC or 1/K. Because the size of a kelvin and a degree Celsius are the same, both α and ΔT can be expressed in units of kelvins or degrees Celsius. The equation ΔL = αLΔT is accurate for small changes in temperature and can be used for large changes in temperature if an average value of α is used. Heat 49 Table 1: Thermal Expansion Coefficients at 20ºC Material Coefficient of linear Coefficient of volume expansion β(1/ expansion α(1/ºC) ºC) Solids Aluminum 25 × 10– 6 75 × 10– 6 Brass 19 × 10– 6 56 × 10– 6 Copper 17 × 10– 6 51 × 10– 6 Gold 14 × 10– 6 42 × 10– 6 Iron or Steel 12 × 10– 6 35 × 10– 6 Invar (Nickel-iron alloy) 0.9 × 10– 6 2.7 × 10– 6 Lead 29 × 10– 6 87 × 10– 6 Silver 18 × 10– 6 54 × 10– 6 Glass (ordinary) 9 × 10– 6 27 × 10– 6 Glass (Pyrex®) 3 × 10– 9 × 10– 6 Quartz 0.4 × 10– 6 1 × 10– 6 Concrete, Brick ~12 × 10– 6 ~36 × 10– 6 Marble (average) 2.5 × 10– 6 7.5 × 10– 6 6 Liquids Ether 1650 × 10– 6 Ethyl alcohol 1100 × 10– 6 Petrol 950 × 10– 6 Glycerin 500 × 10– 6 Mercury 180 × 10– 6 Water 210 × 10– 6 Gases Air and most other gases at atmospheric pressure 3400 × 10– 6 Example. Calculating linear thermal expansion: the golden gate bridge The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge is exposed to temperatures ranging from –15ºC to 40ºC. What is its change in length between these temperatures? Assume that the bridge is made entirely of steel. 50 Experimental Physics I Strategy Use the equation for linear thermal expansion ΔL = αLΔT to calculate the change in length , ΔL. Use the coefficient of linear expansion, α, for steel from Table 1, and note that the change in temperature, ΔT, is 55ºC. Solution Plug all of the known values into the equation to solve for ΔL. Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small. 2.1.6 Thermal Expansion in Two and Three Dimensions Objects expand in all dimensions, That is, their areas and volumes, as well as their lengths, increase with temperature. Holes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighboring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbors must get slightly larger, so the hole gets slightly larger). In general, objects will expand with increasing temperature. Water is the most important exception to this rule. Water expands with increasing temperature (its density decreases) when it is at temperatures greater than 4ºC (40ºF). However, it expands with decreasing temperature when it is between +4ºC and 0ºC (40ºF to 32ºF). Water is densest at +4ºC. Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to 4ºC it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of 4ºC. If the temperature in the surface layer drops below 4ºC, the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in 4ºC water Heat 51 beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water. 2.1.7 Thermal Stress Thermal stress is created by thermal expansion or contraction. Thermal stress can be destructive, such as when expanding gasoline ruptures a tank. It can also be useful, for example, when two parts are joined together by heating one in manufacturing, then slipping it over the other and allowing the combination to cool. Thermal stress can explain many phenomena, such as the weathering of rocks and pavement by the expansion of ice when it freezes. Forces and pressures created by thermal stress are typically as great as that in the example above. Railroad tracks and roadways can buckle on hot days if they lack sufficient expansion joints. Power lines sag more in the summer than in the winter, and will snap in cold weather if there is insufficient slack. Cracks open and close in plaster walls as a house warms and cools. Glass cooking pans will crack if cooled rapidly or unevenly, because of differential contraction and the stresses it creates. (Pyrex® is less susceptible because of its small coefficient of thermal expansion.) Nuclear reactor pressure vessels are threatened by overly rapid cooling, and although none have failed, several have been cooled faster than considered desirable. Biological cells are ruptured when foods are frozen, detracting from their taste. Repeated thawing and freezing accentuate the damage. Even the oceans can be affected. A significant portion of the rise in sea level that is resulting from global warming is due to the thermal expansion of sea water. Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coatingmetal interface. Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. 52 Experimental Physics I Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion, and are closer to those of teeth. 2.2 CALORIMETRY AND THE SPECIFIC HEAT OF A METAL A calorimeter is a device that is in use for measuring the warmth of chemical reactions or physical changes also as heat capacity. The most common types of calorimeters are differential scanning calorimeters, titration calorimeters, isothermal micro calorimeters, and accelerated rate calorimeters. A normal calorimeter usually consists of a thermometer. This thermometer is again attached to a metal container filled with water suspended above a combustion chamber. It is one of the measurement devices useful in the study of thermodynamics, chemistry, and biochemistry. 2.2.1 Procedure of Calorimeter Now let us find the enthalpy change per mole of a substance A in a reaction between two substances A and B. Here, both the substances A and B, are separately added to a calorimeter and the initial and final temperatures (before the reaction has started and after it’s finished) are noted. Multiplying the natural process by the mass and heat capacities of the substances gives worth for the energy given off or absorbed during the reaction. Dividing the energy change by what percentage moles of A were present gives its enthalpy change of the reaction. q=Cv(Tf−Ti)q=Cv(Tf−Ti) Heat 53 Where q is that the amount of warmth consistent with the change in temperature measured in joules and Cv is that the heat capacity of the calorimeter which is measured in units of energy per temperature (Joules/Kelvin). 2.2.2 History In 1761 Black introduced the thought of heat of transformation which causes the creation of the primary ice-calorimeters. In the year 1780, a French nobleman and chemist Lavoisier performed an experiment in which he used the warmth from the guinea pig’s respiration to melt snow surrounding his apparatus, showing that respiratory gas exchange is the combustion, almost like a candle burning. 2.2.3 Types of Calorimeter Adiabatic Calorimeters An adiabatic calorimeter is a calorimeter which helps to examine a runaway reaction. As a result of an adiabatic environment, any heat generated by the fabric sample under test causes the sample to extend in temperature, thus fueling the reaction. The adiabatic calorimeter is actually a wrong term because it’s not fully adiabatic. Some amount of heat is usually lost by the sample to the sample holder. A mathematical correction factor, referred to as the phi-factor, are often wont to adjust the calorimetric result to account for these heat losses. Reaction Calorimeters A reaction calorimeter may be a calorimeter during which a reaction is initiated within a closed insulated container. Reaction heats are measured and therefore the heat content is obtained by integrating heat flow versus time. This is the quality utilized in industry to live heats since industrial processes are engineered to run at constant temperatures. There are four, primary types of methods for measuring the warmth in reaction calorimeter: 1) Heat Flow Calorimeter – The cooling/heating jacket plays an important role in controlling either the temperature of the method or the temperature of the jacket. Heat is measured by monitoring the temperature difference between heat transfer fluid and therefore the process fluid. 54 Experimental Physics I 2) 3) 4) Heat Balance Calorimeter – The cooling/heating jacket controls the temperature of the method. Heat is measured by monitoring the warmth gained or lost by the warmth transfer fluid. Power Compensation – Power compensation uses a heater placed within the vessel to take care of a continuing temperature. The energy supplied to the present heater is often varied as reactions require and therefore the calorimetry signal is only derived from this electric power. Constant Flux Calorimeter – Constant flux calorimetry (or COFLUX) springs from heat balance calorimetry and uses specialized control mechanisms to take care of a continuing heat flow (or flux) across the vessel wall. Bomb Calorimeters A bomb may be a sort of constant-volume calorimeter utilized in measuring the warmth of combustion of a specific reaction. Electrical energy is employed to ignite the fuel; because the fuel is burning, it’ll heat up the encompassing air, which expands and escapes through a tube that leads the air out of the calorimeter. When the air is escaping through the copper tube it’ll also heat up the water outside the tube. The temperature difference of the water allows for calculating the calorie content of the fuel. In brief, a bomb consists of a little cup to contain the sample, oxygen, a chrome steel bomb, water, a stirrer, a thermometer, the dewar or insulating container (to prevent heat be due to the calorimeter to the surroundings) and ignition circuit connected to the bomb. By using chrome steel for the bomb, the reaction will occur with no volume change observed. Calvet-type Calorimeters The detection is predicated on a three-dimensional fluxmeter sensor. The fluxmeter element consists of a hoop of several thermocouples serial. The alternative thermopile having high thermal conductivity surrounds the experimental space within the calorimetric block. The thermopiles arranged radially guarantees an almost complete integration of the warmth. The calibration of the calorimetric detectors may be a key parameter and has got to be performed very carefully. Heat 55 For this Calvet-type calorimeters, a selected calibration, named as Joule effect or electrical calibration, has been developed and used to beat all sorts of issues. The main advantages of this sort of calibration are as follows: 1. 2. 3. 4. It is an absolute calibration. The use of standard materials for calibration is not necessary. The calibration is often performed at a continuing temperature, within the heating mode and within the cooling mode. It can be applied to any experimental vessel volume. It is a very accurate calibration. Adiabatic Calorimeters Adiabatic calorimeters measure the change in enthalpy of a reaction occurring in solution. During the reaction, the no heat exchange with the surroundings is allowed and the atmospheric pressure remains constant. Differential Scanning Calorimeter In this differential scanning calorimeter or DSC, heat flows into a sample which is usually contained within a small aluminium capsule or ‘pan’. This heat flow is measured differentially, i.e., by comparing it to the flow into an empty reference pan. Isothermal Titration Calorimeter In an isothermal titration calorimeter, the warmth of reaction is employed to follow a titration experiment. This technique is gaining high importance mainly within the field of biochemistry because it facilitates the determination of substrate binding to enzymes. The technique is usually utilized in the pharmaceutical industry to characterize potential drug candidates. 2.2.4 Specific Heat and Heat Capacity Heat capacity is a measure of the amount of heat energy required to change the temperature of a pure substance by a given amount. Heat Capacity Heat capacity is an intrinsic physical property of a substance that measures the amount of heat required to change that substance’s 56 Experimental Physics I temperature by a given amount. In the International System of Units (SI), heat capacity is expressed in units of joules per kelvin (J⋅K−1). Heat capacity is an extensive property, meaning that it is dependent upon the size/mass of the sample. For instance, a sample containing twice the amount of substance as another sample would require twice the amount of heat energy (Q) to achieve the same change in temperature (ΔT) as that required to change the temperature of the first sample. Molar and Specific Heat Capacities There are two derived quantities that specify heat capacity as an intensive property (i.e., independent of the size of a sample) of a substance. They are: • • the molar heat capacity, which is the heat capacity per mole of a pure substance. Molar heat capacity is often designated CP, to denote heat capacity under constant pressure conditions, as well as CV, to denote heat capacity under constant volume conditions. Units Important Specific heat capacity is a measure of the amount of heat necessary to raise the temperature of one gram of a pure substance by one degree K. of molar heat capacity are . the specific heat capacity, often simply called specific heat, which is the heat capacity per unit mass of a pure substance. This is designated cP and cV and its units are given in . Heat, Enthalpy, and Temperature Given the molar heat capacity or the specific heat for a pure substance, it is possible to calculate the amount of heat required to raise/ lower that substance’s temperature by a given amount. The following two formulas apply: q=mcpΔT q=nCPΔT In these equations, m is the substance’s mass in grams (used when calculating with specific heat), and n is the number of moles of substance (used when calculating with molar heat capacity). Heat 57 Constant-Volume Calorimetry Constant-volume calorimeters, such as bomb calorimeters, are used to measure the heat of combustion of a reaction. The Bomb Calorimeter Bomb calorimetry is used to measure the heat that a reaction absorbs or releases, and is practically used to measure the calorie content of food. A bomb calorimeter is a type of constant-volume calorimeter used to measure a particular reaction’s heat of combustion. For instance, if we were interested in determining the heat content of a sushi roll, for example, we would be looking to find out the number of calories it contains. In order to do this, we would place the sushi roll in a container referred to as the “bomb”, seal it, and then immerse it in the water inside the calorimeter. Then, we would evacuate all the air out of the bomb before pumping in pure oxygen gas (O2). After the oxygen is added, a fuse would ignite the sample causing it to combust, thereby yielding carbon dioxide, gaseous water, and heat. As such, bomb calorimeters are built to withstand the large pressures produced from the gaseous products in these combustion reactions. A schematic representation of a bomb calorimeter used for the measurement of heats of combustion. The weighed sample is placed in a crucible, which in turn is placed in the bomb. The sample is burned completely in oxygen under pressure. The sample is ignited by an iron wire ignition coil that glows when heated. The calorimeter is filled with fluid, usually water, and insulated by means of a jacket. The temperature of the water is measured with the thermometer. From the change in temperature, the heat of reaction can be calculated. Once the sample is completely combusted, the heat released in the reaction transfers to the water and the calorimeter. The temperature change of the water is measured with a thermometer. The total heat given off in the reaction will be equal to the heat gained by the water and the calorimeter: 58 Experimental Physics I qrxn=−qcal Keep in mind that the heat gained by the calorimeter is the sum of the heat gained by the water, as well as the calorimeter itself. This can be expressed as follows: qcal=mwaterCwaterΔT+CcalΔT where Cwater denotes the specific heat capacity of the water , and Ccal is the heat capacity of the calorimeter (typically in ). Therefore, when running bomb calorimetry experiments, it is necessary to calibrate the calorimeter in order to determine Ccal. Since the volume is constant for a bomb calorimeter, there is no pressure-volume work. As a result: ΔU=qV Important Change in enthalpy can be calculated based on the change in temperature of the solution, its specific heat capacity, and mass. where ΔU is the change in internal energy, and qV denotes the heat absorbed or released by the reaction measured under conditions of constant volume. (This expression was previously derived in the “Internal Energy and Enthalpy ” section.) Thus, the total heat given off by the reaction is related to the change in internal energy (ΔU), not the change in enthalpy (ΔH) which is measured under conditions of constant pressure. The value produced by such experiments does not completely reflect how our body burns food. For example, we cannot digest fiber, so obtained values have to be corrected to account for such differences between experimental (total) and actual (what the human body can absorb) values. Constant-Pressure Calorimetry A constant-pressure calorimeter measures the change in enthalpy of a reaction at constant pressure. Heat 59 A constant-pressure calorimeter measures the change in enthalpy of a reaction occurring in a liquid solution. In that case, the gaseous pressure above the solution remains constant, and we say that the reaction is occurring under conditions of constant pressure. The heat transferred to/from the solution in order for the reaction to occur is equal to the change in enthalpy (ΔH=qP), and a constant-pressure calorimeter thus measures this heat of reaction. In contrast, a bomb calorimeter ‘s volume is constant, so there is no pressure-volume work and the heat measured relates to the change in internal energy (ΔU=qV). A simple example of a constant-pressure calorimeter is a coffee-cup calorimeter, which is constructed from two nested Styrofoam cups and a lid with two holes, which allows for the insertion of a thermometer and a stirring rod. The inner cup holds a known amount of a liquid, usually water, that absorbs the heat from the reaction. The outer cup is assumed to be perfectly adiabatic, meaning that it does not absorb any heat whatsoever. As such, the outer cup is assumed to be a perfect insulator. A styrofoam cup with an inserted thermometer can be used as a calorimeter, in order to measure the change in enthalpy/heat of reaction at constant pressure. Calculating Specific Heat Data collected during a constant-pressure calorimetry experiment can be used to calculate the heat capacity of an unknown substance. We already know our equation relating heat (q), specific heat capacity (C), and the change in observed temperature (ΔTΔT): q=mCΔTq=mCΔT 60 Experimental Physics I 2.3 HEAT OF FUSION OF ICE Latent heat, energy absorbed or released by a substance during a change in its physical state (phase) that occurs without changing its temperature. The latent heat associated with melting a solid or freezing a liquid is called the heat of fusion; that associated with vaporizing a liquid or a solid or condensing a vapour is called the heat of vaporization. The latent heat is normally expressed as the amount of heat (in units of joules or calories) per mole or unit mass of the substance undergoing a change of state. For example, when a pot of water is kept boiling, the temperature remains at 100 °C (212 °F) until the last drop evaporates, because all the heat being added to the liquid is absorbed as latent heat of vaporization and carried away by the escaping vapour molecules. Similarly, while ice melts, it remains at 0 °C (32 °F), and the liquid water that is formed with the latent heat of fusion is also at 0 °C. The heat of fusion for water at 0 °C is approximately 334 joules (79.7 calories) per gram, and the heat of vaporization at 100 °C is about 2,230 joules (533 calories) per gram. Because the heat of vaporization is so large, steam carries a great deal of thermal energy that is released when it condenses, making water an excellent working fluid for heat engines. Latent heat arises from the work required to overcome the forces that hold together atoms or molecules in a material. The regular structure of a crystalline solid is maintained by forces of attraction among its individual atoms, which oscillate slightly about their average positions in the crystal lattice. As the temperature increases, these motions become increasingly violent until, at the melting point, the attractive forces are no longer sufficient to maintain the stability of the crystal lattice. However, additional heat (the latent heat of fusion) must be added (at constant temperature) in order to accomplish the transition to the even more-disordered liquid state, in which the individual particles are no longer held in fixed lattice positions but are free to move about through the liquid. A liquid differs from a gas in that the forces of attraction between the particles are still sufficient to maintain a longrange order that endows the liquid with a degree of cohesion. As the temperature further increases, a second transition point (the boiling point) is reached where the long-range order becomes unstable relative to the largely independent motions of the particles in the much larger volume occupied by a vapour or gas. Once again, additional heat (the latent heat of vaporization) must be added to break the long-range order Heat 61 of the liquid and accomplish the transition to the largely disordered gaseous state. Latent heat is associated with processes other than changes among the solid, liquid, and vapour phases of a single substance. Many solids exist in different crystalline modifications, and the transitions between these generally involve absorption or evolution of latent heat. The process of dissolving one substance in another often involves heat; if the solution process is a strictly physical change, the heat is a latent heat. Sometimes, however, the process is accompanied by a chemical change, and part of the heat is that associated with the chemical reaction. 2.3.1 Heat of Fusion Formula The heat of fusion of any substance is the important calculation of the heat. It is the change in the value of the enthalpy by providing energy i.e. heat, for a specific quantity of the substance. It will change its state from a solid to a liquid keeping the pressure constant. The heat of fusion of any sample will measure the amount of heat that needs to be introduced to convert its crystalline fraction into the disordered state. ‘Heat of fusion’ measures the amount of energy required to melt a given amount of a solid at its melting point temperature. In other words, it also represents the amount of energy given up when a given mass of liquid solidifies. For example, water has a heat of fusion of 80 calories per gram. It means that it takes 80 calories of energy to melt 1 gram of ice at the temperature of zero degrees C into the water at zero degrees C. Heat of fusion values will differ for the different materials. For example, we may see that heat gained by ice is equal to the heat lost by the water. We denote the Heat of fusion by the symbol ΔHf. 62 Experimental Physics I When a solid material turns into the liquid, then it is what we know as melting. This melting process will need an increase in energy to allow the solid-state particles to break free from each other. This energy input is the heat of fusion. The heat of fusion is not the same for all substances, but it is a constant value for each individual kind of substance. The Formula for the Heat of Fusion: We compute it as: ΔHf heat of fusion q Heat m mass Examples for Heat of Fusion Formula 1: Calculate the heat in Joules which is required to melt 26 grams of the ice. It is given here that heat of fusion of water is 334 J/g i.e. equals to 80 cal per gram. Solution: Given parameters are, Mass, m = 26 g We know that, Rearranging the formula, q=m×ΔHf, =26×334 = 8684 Joules. Thus heat required will be 8684 Joules. 2: What will be the heat of fusion for the water, if it takes 668 Joules of the heat energy to melt 2 grams? Solution: Known values are, Q = 668 joules Heat 63 M = 2 grams Formula is: = 334 J per gram. Thus heat of fusion will be 334 J per gram. 3: What mass of water will be melted at zero degrees C, if 1500 J of heat energy is applied? Use Heat of Fusion Formula. Solution: The heat of fusion for water is applicable here and the equation has to be rearranged to solve it for the mass. Here, Hf=1500J Q = 334 C per gram Then, = 4.49 gram Mass of water will be 4.49 gram. 2.3.2 How to Measure Heat of Fusion of Ice “Heat” represents the thermal energy of molecules in a substance. Water freezes at 0 degrees Celsius. But the temperature of an ice cube 64 Experimental Physics I can fall well below that. When an ice cube is removed from a freezer, the cube’s temperature increases as it absorbs heat from its surroundings. But once the ice cube reaches 0 C, it begins to melt and its temperature stays at 0 throughout the melting process, even though the ice cube continues to absorb heat. This occurs because the thermal energy absorbed by the ice cube is consumed by water molecules separating from each other during melting. The amount of heat absorbed by a solid during its melting phase is known as the latent heat of fusion and is measured via calorimetry. Data Collection Place an empty Styrofoam cup on a balance and record the mass of the empty cup in grams. Then fill the cup with about 100 milliliters, or about 3.5 ounces, of distilled water. Return the filled cup to the balance and record the weight of the cup and water together. Place a thermometer into the water in the cup, wait about 5 minutes for the thermometer to come to thermal equilibrium with the water, then record the temperature of the water as the initial temperature. Place two or three ice cubes on a paper towel to remove any liquid water on the surfaces of the cubes, then quickly transfer the cubes to the Styrofoam cup. Use the thermometer to gently stir the mixture. Observe the temperature reading on the thermometer. It should begin to drop almost immediately. Continue stirring and record the lowest temperature indicated on the thermometer before the temperature begins to rise. Record this value as the “final temperature.” Heat 65 Remove the thermometer and return the Styrofoam cup once again to the balance and record the mass of the cup, water and melted ice together. Calculations Determine the mass of water in the cup by subtracting the mass of the empty cup from the weight of the cup and water together, as collected in step 1. For example, if the empty cup weighed 3.1 grams and the cup and water together weighed 106.5 grams, then the mass of the water was 106.5 - 3.1 = 103.4 g. Calculate the temperature change of the water by subtracting the initial water temperature from the final water temperature. Thus, if the initial temperature was 24.5 C and the final temperature was 19.2 C, then deltaT = 19.2 - 24.5 = -5.3 C. Calculate the heat, q, removed from the water according to the equation q = mc(deltaT), where m and deltaT represent the mass and temperature change of the water, respectively, and c represents water’s specific heat capacity, or 4.184 joules per gram per degree Celsius, or 4.187 J/g-C. Continuing the example from steps 1 and 2, q = ms(deltaT) = 103.4 g * 4.184 J/g-C * -5.3 C = -2293 J. This represents the heat removed from the water, hence its negative sign. By the laws of thermodynamics, this means that the ice cubes in the water absorbed +2293 J of heat. Determine the mass of the ice cubes by subtracting the mass of the cup and water from the mass of the cup, water and ice cubes together. If the cup, water and ice together weighed 110.4 g, then the mass of the ice cubes was 110.4 g - 103.4 g = 7.0 g. Find the latent heat of fusion, Lf, according to Lf = q ÷ m by dividing the heat, q, absorbed by the ice, as determined in step 3, by the mass of ice, m, determined in step 4. In this case, Lf = q / m = 2293 J ÷ 7.0 g = 328 J/g. Compare your experimental result to the accepted value of 333.5 J/g. 2.3.4 Heats of Fusion and Solidification Suppose you hold an ice cube in your hand. It feels cold because heat energy leaves your hand and enters the ice cube. What happens to the ice cube? It melts. However, the temperature during a phase change remains constant. So the heat that is being lost by your hand does not raise the temperature of the ice above its melting temperature of 0°C. Rather, all the heat goes into the change of state. Energy is absorbed dur- 66 Experimental Physics I ing the process of changing ice into water. The water that is produced also remains at 0°C until all of the ice is melted. 2.3.5 Heats of Fusion and Solidification All solids absorb heat as they melt to become liquids. The gain of heat in this endothermic process goes into changing the state rather than changing the temperature. The molar heat of fusion of a substance is the heat absorbed by one mole of that substance as it is converted from a solid to a liquid. Since the melting of any substance absorbs heat, it follows that the freezing of any substance releases heat. The molar heat of solidification of a substance is the heat released by one mole of that substance as it is converted from a liquid to a solid. Since fusion and solidification of a given substance are the exact opposite processes, the numerical value of the molar heat of fusion is the same as the numerical value of the molar heat of solidification, but opposite in sign. In other words, . The Figure below shows all of the possible changes of state along with the direction of heat flow during each process. Heat 67 From left to right, heat is absorbed from the surroundings during melting, evaporation, and sublimation. Form right to left, heat is released to the surroundings during freezing, condensation, and deposition. Every substance has a unique value for its molar heat of fusion, depending on the amount of energy required to disrupt the intermolecular forces present in the solid. When 1 mol of ice at 0°C is converted to 1 mol of liquid water at 0°C, 6.01 kJ of heat are absorbed from the surroundings. When 1 mol of water at 0°C freezes to ice at 0°C, 6.01 kJ of heat are released into the surroundings. The molar heats of fusion and solidification of a given substance can be used to calculate the heat absorbed or released when various amounts are melted or frozen. Sample Problem Heat of Fusion Calculate the heat absorbed when 31.6 g of ice at 0°C is completely melted. Step 1: List the known quantities and plan the problem . Known • • • mass = 31.6 g ice molar mass H 2 O( s ) = 18.02 g/mol molar heat of fusion = 6.01 kJ/mol Unknown • The mass of ice is first converted to moles. This is then multiplied by the conversion factor of in order to find the kJ of heat absorbed. Step 2: Solve . Step 3: Think about your result . The given quantity is a bit less than 2 moles of ice, and so just less than 12 kJ of heat is absorbed by the melting process. 68 Experimental Physics I 2.3.6 Determining the Heat of Fusion of Ice When a sample of ice of mass m completely undergoes a phase change from solid into liquid, i.e. melts to water, the total energy Q which the ice absorbs from its environment is proportional to the heat of fusion Lf , the heat transfer for a unit mass, and the mass m: In the other direction, when the phase change is from liquid to solid, the sample must release the same amount of energy. For water at its normal freezing or melting temperature Important This lab employs a double-wall calorimeter as shown in Figure to measure the heat of fusion for ice. The calorimeter consists of an aluminum container, a reservoir, a plastic lid and an insulator ring. The reservoir holds a maximum of 150 ml water. The clear plastic lid has 3 access holes. It includes a cork with a hole for holding a thermometer, and a hole for a stirrer. Water is poured into the reservoir, and the initial equilibrium temperature T1 is reached after heat transfer among all the devices is completed. Ice of mass m is then put into the water, and then absorbs heat from water. In the meanwhile, the water releases heat, and as a result, the temperature of the whole system decreases until the system It is important to prevent any unnecessary heat transfer between the environment and the system. When you measure the mass for ice or put the ice into the water, do it as soon as possible because you do not want the ice to get heat from the air. Also, when you stir the water with the ice don’t do it too fast because you don’t want to give your energy to the system Heat 69 starts to absorb heat from air surrounding the system. At this turning point, the temperature is T2. The temperature then increases again. The total heat transfer can be broken into two parts: heat given off and heat absorbed (if we assume the system is closed). The heat absorbed goes into • • the ice, to turn it into ice water the resulting ice water, to raise it to T2. The heat given off comes from • • • the reservoir the water in the reservoir the thermometer all of which start out at T1. Assuming the system is closed, all of the heat given off must be absorbed. The heat equation for the system can then be expressed as: where • • • • • mw is the mass of water mA is the mass of the reservoir and stirrer c is the specific heat of water cA is the specific heat of the (aluminum) reservoir q is the heat released by the thermometer which is equal to: where V is the volume of the thermometer which can be determined by Archimedes’ Principle. Method 1. 2. 3. Weigh the masses of all components: water, stirrer, and reservoir. Make sure the water fills at least half of the reservoir of the calorimeter. Heat the water up to about 50◦C. Measure the volume of the thermometer by use of the graduated cylinder. 70 Experimental Physics I 4. 5. 6. 7. 8. 9. Put ice into water in a beaker for a while, and bring the temperature of the ice down to 0◦C. Take the ice out of the water in the beaker, dry it, and then measure its mass quickly by the scale. Then put the ice into the reservoir, and cover the lid promptly. Measure the temperature of the system every 30 seconds. Plot a graph of temperature vs. time. A typical graph of temperature vs. time for the system is shown in Figure. Extrapolate from the graph as shown to get T1, T2, and their uncertainties. Use these values in Equation 3 to determine Lf , and compare it to the expected value. 2.4 HEAT OF VAPORIZATION OF WATER Water in its liquid form has an unusually high boiling point temperature, a value close to 100°C. As a result of the network of hydrogen bonding present between water molecules, a high input of energy is required to transform one gram of liquid water into water vapor, an energy requirement called the heat of vaporization. Water has a heat of vaporization value of 40.65 kJ/mol. A considerable amount of heat energy (586 calories) is required to accomplish this change in water. This process occurs on the surface of water. As liquid water heats up, hydrogen bonding makes it difficult to separate the water molecules from each other, which is required for it to enter its gaseous phase (steam). As a result, water acts as a heat sink, or heat reservoir, and requires much more heat to boil than does a liquid such as ethanol (grain alcohol), whose hydrogen bonding with other ethanol molecules Heat 71 is weaker than water’s hydrogen bonding. Eventually, as water reaches its boiling point of 100° Celsius (212° Fahrenheit), the heat is able to break the hydrogen bonds between the water molecules, and the kinetic energy (motion) between the water molecules allows them to escape from the liquid as a gas. Even when below its boiling point, water’s individual molecules acquire enough energy from each other such that some surface water molecules can escape and vaporize; this process is known as evaporation. The fact that hydrogen bonds need to be broken for water to evaporate means that a substantial amount of energy is used in the process. As the water evaporates, energy is taken up by the process, cooling the environment where the evaporation is taking place. In many living organisms, including humans, the evaporation of sweat, which is 90 percent water, allows the organism to cool so that homeostasis of body temperature can be maintained. The (latent) heat of vaporization (∆Hvap) also known as the enthalpy of vaporization or evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance, to transform a given quantity of the substance into a gas. The enthalpy of vaporization is a function of the pressure at which that transformation takes place. The heat of vaporization diminishes with increasing temperature and it vanishes completely at a certain point called the critical temperature (Critical temperature for water: 373.946 °C or 705.103 °F, Critical pressure: 220.6 bar = 22.06 MPa = 3200 psi ). 72 Experimental Physics I Heat of vaporization for liquid water at saturation pressure at temperatures from 0 to 374 °C: Temperature Vapor pressure Heat of vaporization, ∆Hvap [°C] [kPa] [100*bar] [J/mol] [kJ/kg] [Wh/kg] [Btu(IT)/lbm] 0.01 0.61165 45054 2500.9 694.69 1075.2 2 0.70599 44970 2496.2 693.39 1073.2 4 0.81355 44883 2491.4 692.06 1071.1 10 1.2282 44627 2477.2 688.11 1065.0 14 1.5990 44456 2467.7 685.47 1060.9 18 2.0647 44287 2458.3 682.86 1056.9 20 2.3393 44200 2453.5 681.53 1054.8 Heat 73 25 3.1699 43988 2441.7 678.25 1049.7 30 4.2470 43774 2429.8 674.94 1044.6 34 5.3251 43602 2420.3 672.31 1040.5 40 7.3849 43345 2406.0 668.33 1034.4 44 9.1124 43172 2396.4 665.67 1030.3 50 12.352 42911 2381.9 661.64 1024.0 54 15.022 42738 2372.3 658.97 1019.9 60 19.946 42475 2357.7 654.92 1013.6 70 31.201 42030 2333.0 648.06 1003.0 80 47.414 41579 2308.0 641.11 992.26 90 70.182 41120 2282.5 634.03 981.30 96 87.771 40839 2266.9 629.69 974.59 100 101.42 40650 2256.4 626.78 970.08 110 143.38 40167 2229.6 619.33 958.56 120 198.67 39671 2202.1 611.69 946.73 140 361.54 38630 2144.3 595.64 921.88 160 618.23 37508 2082.0 578.33 895.10 180 1002.8 36286 2014.2 559.50 865.95 200 1554.9 34944 1939.7 538.81 833.92 220 2319.6 33462 1857.4 515.94 798.54 240 3346.9 31804 1765.4 490.39 758.99 260 4692.3 29934 1661.6 461.56 714.36 280 6416.6 27798 1543.0 428.61 663.37 300 8587.9 25304 1404.6 390.17 603.87 320 11284 22310 1238.4 344.00 532.42 340 14601 18507 1027.3 285.36 441.66 360 18666 12967 719.8 199.9 309.5 373.946 22064 0 0.0 0.0 0.0 Heat of vaporization for liquid water at saturation pressure at temperatures from 0 to 705 °F: Temperature Vapor pressure Heat of vaporization, ∆Hvap [°F] [psi] [Btu(IT)/mol] [Btu(IT)/lbm] [cal/g] [kJ/kg] 32.2 0.0891 42.70 1075.2 597.33 2500.9 40 0.1219 42.52 1070.7 594.82 2490.4 50 0.1783 42.30 1065.0 591.67 2477.2 74 Experimental Physics I 60 0.2564 42.07 1059.4 588.54 2464.1 70 0.3632 41.85 1053.7 585.39 2450.9 80 0.5073 41.62 1048.0 582.25 2437.7 90 0.6990 41.40 1042.4 579.09 2424.5 100 0.9506 41.17 1036.7 575.92 2411.3 110 1.277 40.95 1030.9 572.74 2398.0 120 1.695 40.72 1025.2 569.55 2384.6 130 2.226 40.49 1019.4 566.34 2371.2 140 2.893 40.26 1013.6 563.13 2357.7 150 3.723 40.02 1007.7 559.86 2344.0 160 4.747 39.79 1001.8 556.58 2330.3 170 6.000 39.55 995.87 553.26 2316.4 180 7.519 39.31 989.85 549.92 2302.4 190 9.350 39.07 983.76 546.54 2288.2 200 11.54 38.83 977.60 543.11 2273.9 210 14.14 38.58 971.35 539.64 2259.4 212 14.71 38.53 970.08 538.93 2256.4 220 17.20 38.33 965.02 536.12 2244.6 240 24.99 37.81 952.05 528.92 2214.5 260 35.45 37.28 938.64 521.46 2183.3 280 49.22 36.73 924.71 513.73 2150.9 300 66.6 36.15 910.21 505.67 2117.1 350 135 34.59 870.97 483.87 2025.9 400 247 32.82 826.41 459.12 1922.2 450 422 30.78 774.93 430.51 1802.5 500 680 28.37 714.36 396.87 1661.6 550 1044 25.48 641.56 356.42 1492.3 600 1541 21.93 552.09 306.72 1284.2 625 1849 19.41 488.64 271.46 1136.6 650 2205 16.95 426.81 237.11 992.8 675 2615 13.47 339.22 188.46 789.0 705.103 3196 0.00 0.00 0.00 0.0 The heat of vaporization (here denoted as Lv) is defined as the heat added (or given off) when unit mass undergoes isobaric phase Heat 75 transformation in any closed two-phase, one-component liquid/vapor system. In engineering and meteorology, Lv is used in a restricted sense to mean the heat of vaporization of the two-phase liquid water/water vapor system. Although much of the subsequent discussion focuses on Lv for this specific system as an example, the concepts covered are universally applicable to all fluid/vapor systems. As illustrated below, for the liquid water/water vapor system, Lv represents the heat gained when unit mass of water in the system evaporates in the isobaric phase transformation H2O (liquid) → H2O (vapor). For the reverse phase change H2O (vapor) → H2O (liquid) i.e. condensation, Lv is lost from the system. This seemingly simple phase transition H2O (liquid) ↔ H2O (vapor) is the fundamental driving process of the earth’s hydrological cycle, the working principle of the steam engine that ushered humanity into the industrial revolution along with its (often negative) social and environmental pollution consequences, and the physical mechanism that maintains the body temperature of plants and warm-blooded animals. In general, the state of any closed two-phase, one-component system is defined by the state variables temperature (T in o K), saturation vapor pressure (P in Pascal), and volume (V in m3 ). The behavior of any such system (generally termed as PVT systems) is usually represented as a family of experimental constant temperature curves (isotherms) on a P-V coordinate plane called an Amagat-Andrews diagram. The general shape of these experimental isotherms is illustrated below. 76 Experimental Physics I For the liquid water/water vapor system the liquid and vapor phases co-exist in equilibrium only at P-V coordinates between 2 and 3 along an isotherm, provided that T1 is above the triple point temperature of water (0.01 o C, i.e. the temperature at which ice, liquid water, and water vapor can coexist in equilibrium), and below the critical temperature (374 o C, i.e. the temperature above which it is impossible to produce condensation by increasing the pressure). Between 2 and 1 the system can exist as vapor only, and as liquid between 3 and 4. Thus the liquidvapor phase transition at a given temperature can only take place at constant pressure or vice-versa. Consequently, as shown in Figure 8, isobaric liquid vaporization and condensation is necessarily an isothermal process, implying that the triple point saturation vapor pressure is fixed (it is 611 Pa), and so is the saturation vapor pressure at the critical temperature (it is 2.21 x 107 Pa = 218.2 atm). Similar isotherms and parameters exist for all liquid/vapor systems. This observed behavior of closed two-phase, one-component systems is of course predicted by Gibb’s Phase Rule namely, F + N = C + 2 where F = degrees of freedom i.e. the smallest number of intensive variables (such as pressure, temperature, concentration of components in each phase) that must be specified to completely describe the state of the system, N = number of phases i.e. distinct subsystems of uniform chemical composition and physical properties, and C = the number of components i.e. the number of independent chemical constituents meaning those constituents whose concentration can be varied independently in the different phases. In a liquid / vapor system P = 2, and C =1, and therefore F =1 implying only one intensive variable is needed to specify the state of the system. Therefore, temperature and pressure cannot be fixed independently. For the liquid water/water vapor system, this means physically that at a given temperature between the triple point and critical temperature, water vapor will evaporate or condense to achieve the equilibrium saturation vapor pressure as would be evidenced in a complete Amagat-Andrews diagram for water. Heat 77 The earth’s atmosphere and oceans can be considered as a vast closed two phase liquid water/moist air system. Consequently for most practical engineering and meteorological applications one is interested in the heat of vaporization of the liquid water/moist air system rather than a pure liquid water/water vapor system. Fortunately, the presence of the other gases (collectively called dry air) in the liquid water/moist air system has negligible effect on the saturation vapor pressure. The reason is that the dry air component in the liquid water/moist air system remains unchanged and is always in the gaseous state during phase transition at temperatures and pressures of practical interest. Therefore it can be considered as a closed sub-system as opposed to the open liquid water and water vapor sub-systems. Consequently, results obtained from an analysis of the thermodynamics of the pure system are applicable to the natural liquid water/moist air system. Energy conservation required under the first law of thermodynamics implies that heat (Q) exchanged reversibly with the surroundings between equilibrium states of any closed twophase PVT system is consumed by any internal energy change (ΔU) of the liquid and vapor phases associated with the mass change from one phase to the other, and any mechanical work (± PΔV) realized as the volume of the system increases (positive work) or decreases (negative work). Stated mathematically Q = ΔU + PΔV or in differential form δQ = dU + pdV. Here P is the saturation vapor pressure (Pvap). Since entropy (S) is defined as Q/T, then δQ = TdS. The first law can therefore be restated in terms of exact differentials as TdS = dU + PdV. Dividing by dV at constant T and rearranging gives dU/ dV = T (dS/dV) – P. Using the Maxwell relation (∂S/∂V)T = (∂P/∂T)V dS/ dV can be replaced (for fixed T) by dP/dT. The equation becomes dU/dV = T (dP/dT) – P. At a given pressure and temperature, the internal energy of the system (U in Joules) can be partitioned as mwuw + mvap uvap where mw, mvap and uw, uvap represent the masses and the specific internal energies (internal energy per unit mass in J kg-1) of the water and water vapor in the system. Similarly, the volume (V in m3 ) of the system can be partitioned as mwvw + mvap vvap where vw, vvap represent the specific volume (volume per unit mass in J kg-1) of the water and water vapor in the system. If, as illustrated in Figure 8, the system internal energy changes by ΔU from U to U + ΔU as a result of Lv Joules of heat absorption to convert unit mass of water to water vapor, then U + ΔU = (mw – 1)uw + (mw + 1)uvap and therefore ΔU = (uw – uvap). Similar reasoning shows that, if the volume changes from V to V + ΔV in the process, then ΔV = (vvap – vw). The 78 Experimental Physics I mechanical work due to volume change is PΔV where P (the saturation vapor pressure) is a constant at a fixed temperature. Therefore, the heat absorbed (or released) by the system for isobaric phase transition of unit mass in the liquid water/water vapor system (Lv by definition) = ΔU + P ΔV. Dividing by ΔV gives ΔU/ΔV = (Lv/ΔV) - P. Substituting ΔV = (vvap – vw) gives ΔU/ΔV = [Lv / (vvap – vw)] – P or in differential form dU/dV = [Lv / (vvap – vw)] – P. [It should be noted that since enthalpy (H) is defined as H = U + PV, then ΔU + P ΔV = ΔH, and therefore Lv is the same as the specific enthalpy change (Δh = ΔH per unit mass) for phase transition of unit mass in the liquid water/water vapor system]. Combining the results for dU/dV from the two previous paragraphs gives T (dP/dT) – P = [Lv / (vvap – vw)] – P and therefore T (dP/dT) = Lv / (vvap – vw) which can be rearranged to obtain the general forms of the Clapeyron equation dP/dT = Lv/ [T(vvap – vw)] = Δh / [T(vvap – vw)] or Lv = [T (vvap – vw)] dP/dT. The Clapeyron equation can be used to obtain Lv at a given temperature T for any liquid provided one can obtain values of (vvap – vw) and an accurate representation of dP/dT. Values of (vvap – vw) can be obtained from tabulated measurements. Alternatively, since vvap >> vw at the low pressures (vvap – vw) can be taken as equal to vvap. Assuming further that at low pressures water vapor behavior closely approximates that of an ideal gas, then vvap = RT/P where R is the specific gas constant for water = 8.314/0.018 = 461.9 J kg-1 o K-1. The Clapeyron equation becomes Lv = (RT2 /P) dP/dT and this form is referred to as the ClausiusClapeyron equation. 2.4.1 Convective Heat Transfer Heat energy transferred between a surface and a moving fluid with different temperatures - is known as convection. In reality this is a combination of diffusion and bulk motion of molecules. Near the surface the fluid velocity is low, and diffusion dominates. At distance from the surface, bulk motion increases the influence and dominates. Heat 79 Convective heat transfer can be • • • forced or assisted convection natural or free convection Conductive Heat Transfer Forced or Assisted Convection Forced convection occurs when a fluid flow is induced by an external force, such as a pump, fan or a mixer. Natural or Free Convection Natural convection is caused by buoyancy forces due to density differences caused by temperature variations in the fluid. At heating the density change in the boundary layer will cause the fluid to rise and be replaced by cooler fluid that also will heat and rise. This continues phenomena is called free or natural convection. Boiling or condensing processes are also referred to as a convective heat transfer processes. • The heat transfer per unit surface through convection was first described by Newton and the relation is known as the Newton’s Law of Cooling. 80 Experimental Physics I The equation for convection can be expressed as: q = hc A dT (1) where q = heat transferred per unit time (W, Btu/hr) A = heat transfer area of the surface (m2, ft2) hc = convective heat transfer coefficient of the process (W/(m2oC, Btu/ (ft h oF)) 2 dT = temperature difference between the surface and the bulk fluid ( C, F) o Heat Transfer Coefficients - Units • • • • 1 W/(m2K) = 0.85984 kcal/(h m2 oC) = 0.1761 Btu/(ft2 h oF) 1 Btu/(ft2 h oF) = 5.678 W/(m2 K) = 4.882 kcal/(h m2 oC) 1 kcal/(h m2 oC) = 1.163 W/(m2K) = 0.205 Btu/(ft2 h oF) Overall Heat Transfer Coefficients Convective Heat Transfer Coefficients Convective heat transfer coefficients - hc - depends on type of media, if its gas or liquid, and flow properties such as velocity, viscosity and other flow and temperature dependent properties. Typical convective heat transfer coefficients for some common fluid flow applications: • • • • • • • Free Convection - air, gases and dry vapors : 0.5 - 1000 (W/ (m2K)) Free Convection - water and liquids: 50 - 3000 (W/(m2K)) Forced Convection - air, gases and dry vapors: 10 - 1000 (W/ (m2K)) Forced Convection - water and liquids: 50 - 10000 (W/(m2K)) Forced Convection - liquid metals: 5000 - 40000 (W/(m2K)) Boiling Water : 3.000 - 100.000 (W/(m2K)) Condensing Water Vapor: 5.000 - 100.000 (W/(m2K)) Convective Heat Transfer Coefficient for Air The convective heat transfer coefficient for air flow can be approximated to Heat 81 hc = 10.45 - v + 10 v1/2 (2) where hc = heat transfer coefficient (kCal/m2h°C) v = relative speed between object surface and air (m/s) Since 1 kcal/m2h°C = 1.16 W/m2°C - (2) can be modified to hcW = 12.12 - 1.16 v + 11.6 v1/2 (2b) where hcW = heat transfer coefficient (W/m2°C) Example - Convective Heat Transfer A fluid flows over a plane surface 1 m by 1 m. The surface temperature is 50oC, the fluid temperature is 20oC and the convective heat transfer coefficient is 2000 W/m2oC. The convective heat transfer between the hotter surface and the colder air can be calculated as q = (2000 W/(m2oC)) ((1 m) (1 m)) ((50 oC) - (20 oC)) = 60000 (W) = 60 (kW) 82 Experimental Physics I Convective Heat Transfer Chart 2.4.2 Mechanism of Forced Convection Convection heat transfer is complicated since it involves fluid motion as well as heat conduction. The fluid motion enhances heat transfer (the higher the velocity the higher the heat transfer rate). The rate of convection heat transfer is expressed by Newton’s law of cooling: The convective heat transfer coefficient h strongly depends on the fluid properties and roughness of the solid surface, and the type of the fluid flow (laminar or turbulent). Heat 83 Figure 8: Forced convection. It is assumed that the velocity of the fluid is zero at the wall, this assumption is called no‐ slip condition. As a result, the heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid is motionless. Thus, The convection heat transfer coefficient, in general, varies along the flow direction. The mean or average convection heat transfer coefficient for a surface is determined by (properly) averaging the local heat transfer coefficient over the entire surface. Velocity Boundary Layer Consider the flow of a fluid over a flat plate, the velocity and the temperature of the fluid approaching the plate is uniform at U∞ and T∞. The fluid can be considered as adjacent layers on top of each others. Figure 9: Velocity boundary layer. 84 Experimental Physics I Assuming no‐slip condition at the wall, the velocity of the fluid layer at the wall is zero. The motionless layer slows down the particles of the neighboring fluid layers as a result of friction between the two adjacent layers. The presence of the plate is felt up to some distance from the plate beyond which the fluid velocity U∞ remains unchanged. This region is called velocity boundary layer. Boundary layer region is the region where the viscous effects and the velocity changes are significant and the inviscid region is the region in which the frictional effects are negligible and the velocity remains essentially constant. The friction between two adjacent layers between two layers acts similar to a drag force (friction force). The drag force per unit area is called the shear stress: where μ is the dynamic viscosity of the fluid kg/m.s or N.s/m2. Viscosity is a measure of fluid resistance to flow, and is a strong function of temperature. The surface shear stress can also be determined from: where Cf is the friction coefficient or the drag coefficient which is determined experimentally in most cases. The drag force is calculated from: The flow in boundary layer starts as smooth and streamlined which is called laminar flow. At some distance from the leading edge, the flow turns chaotic, which is called turbulent and it is characterized by velocity fluctuations and highly disordered motion. The transition from laminar to turbulent flow occurs over some region which is called transition region. Heat 85 The velocity profile in the laminar region is approximately parabolic, and becomes flatter in turbulent flow. The turbulent region can be considered of three regions: laminar sublayer (where viscous effects are dominant), buffer layer (where both laminar and turbulent effects exist), and turbulent layer. The intense mixing of the fluid in turbulent flow enhances heat and momentum transfer between fluid particles, which in turn increases the friction force and the convection heat transfer coefficient. Non‐dimensional Groups In convection, it is a common practice to non‐dimensionalize the governing equations and combine the variables which group together into dimensionless numbers (groups). Nusselt number: non‐dimensional heat transfer coefficient where δ is the characteristic length, i.e. D for the tube and L for the flat plate. Nusselt number represents the enhancement of heat transfer through a fluid as a result of convection relative to conduction across the same fluid layer. Reynolds number: ratio of inertia forces to viscous forces in the fluid At large Re numbers, the inertia forces, which are proportional to the density and the velocity of the fluid, are large relative to the viscous forces; thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid (turbulent regime). The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number. For flat plate the critical Re is experimentally determined to be approximately Re critical = 5 x105 . Prandtl number: is a measure of relative thickness of the velocity and thermal boundary layer 86 Experimental Physics I where fluid properties are: Thermal Boundary Layer Similar to velocity boundary layer, a thermal boundary layer develops when a fluid at specific temperature flows over a surface which is at different temperature. Figure 10: Thermal boundary layer. The thickness of the thermal boundary layer δt is defined as the distance at which: The relative thickness of the velocity and the thermal boundary layers is described by the Prandtl number. For low Prandtl number fluids, i.e. liquid metals, heat diffuses much faster than momentum flow (remember Pr = ν/α<< 1) and the velocity boundary layer is fully contained within the thermal boundary layer. On the other hand, for high Prandtl number fluids, i.e. oils, heat diffuses much slower than the momentum and the thermal boundary layer is contained within the velocity boundary layer. Heat 87 Flow Over Flat Plate The friction and heat transfer coefficient for a flat plate can be determined by solving the conservation of mass, momentum, and energy equations (either approximately or numerically). They can also be measured experimentally. It is found that the Nusselt number can be expressed as: where C, m, and n are constants and L is the length of the flat plate. The properties of the fluid are usually evaluated at the film temperature defined as: Laminar Flow The local friction coefficient and the Nusselt number at the location x for laminar flow over a flat plate are where x is the distant from the leading edge of the plate and Rex = ρV∞x / μ. The averaged friction coefficient and the Nusselt number over the entire isothermal plate for laminar regime are: Taking the critical Reynolds number to be 5 x105 , the length of the plate xcr over which the flow is laminar can be determined from 88 Experimental Physics I Turbulent Flow The local friction coefficient and the Nusselt number at location x for turbulent flow over a flat isothermal plate are: The averaged friction coefficient and Nusselt number over the isothermal plate in turbulent region are: Combined Laminar and Turbulent Flow If the plate is sufficiently long for the flow to become turbulent (and not long enough to disregard the laminar flow region), we should use the average values for friction coefficient and the Nusselt number. where the critical Reynolds number is assumed to be 5x105 . After performing the integrals and simplifications, one obtains: Heat 89 The above relationships have been obtained for the case of isothermal surfaces, but could also be used approximately for the case of non‐isothermal surfaces. In such cases assume the surface temperature be constant at some average value. For isoflux (uniform heat flux) plates, the local Nusselt number for laminar and turbulent flow can be found from: Note the isoflux relationships give values that are 36% higher for laminar and 4% for turbulent flows relative to isothermal plate case. Example 1 Engine oil at 60°C flows over a 5 m long flat plate whose temperature is 20°C with a velocity of 2 m/s. Determine the total drag force and the rate of heat transfer per unit width of the entire plate. We assume the critical Reynolds number is 5x105 . The properties of the oil at the film temperature are: The Re number for the plate is: 90 Experimental Physics I which is less than the critical Re. Thus we have laminar flow. The friction coefficient and the drag force can be found from: The Nusselt number is determined from: Flow across Cylinders and Spheres The characteristic length for a circular tube or sphere is the external diameter, D, and the Reynolds number is defined: The critical Re for the flow across spheres or tubes is 2x105 . The approaching fluid to the cylinder (a sphere) will branch out and encircle the body, forming a boundary layer. Figure 11: Typical flow patterns over sphere and streamlined body and drag forces. Heat 91 At low Re (Re < 4) numbers the fluid completely wraps around the body. At higher Re numbers, the fluid is too fast to remain attached to the surface as it approaches the top of the cylinder. Thus, the boundary layer detaches from the surface, forming a wake behind the body. This point is called the separation point. To reduce the drag coefficient, streamlined bodies are more suitable, e.g. airplanes are built to resemble birds and submarine to resemble fish, Fig. 11. In flow past cylinder or spheres, flow separation occurs around 80° for laminar flow and 140° for turbulent flow. where frontal area of a cylinder is AN = L×D, and for a sphere is AN = πD2 / 4. The drag force acting on a body is caused by two effects: the friction drag (due to the shear stress at the surface) and the pressure drag which is due to pressure differential between the front and rear side of the body. As a result of transition to turbulent flow, which moves the separation point further to the rear of the body, a large reduction in the drag coefficient occurs. As a result, the surface of golf balls is intentionally roughened to induce turbulent at a lower Re number, see Fig. 12. Figure 12: Roughened golf ball reduces CD. 92 Experimental Physics I The average heat transfer coefficient for cross‐flow over a cylinder can be found from the correlation presented by Churchill and Bernstein: where fluid properties are evaluated at the film temperature Tf = (Ts + T∞) / 2. For flow over a sphere, Whitaker recommended the following: which is valid for 3.5 < Re < 80,000 and 0.7 < Pr < 380. The fluid properties are evaluated at the free‐stream temperature T∞, except for μs which is evaluated at surface temperature. Example 2 The decorative plastic film on a copper sphere of 10‐mm diameter is cured in an oven at 75°C. Upon removal from the oven, the sphere is subjected to an air stream at 1 atm and 23°C having a velocity of 10 m/s, estimate how long it will take to cool the sphere to 35°C. Assumptions: 1. Negligible thermal resistance and capacitance for the plastic layer. 2. Spatially isothermal sphere. 3. Negligible Radiation. Heat 93 The time required to complete the cooling process may be obtained from the results for a lumped capacitance. Whitaker relationship can be used to find h for the flow over sphere: where Re = ρVD / μ = 6510. Hence, The required time for cooling is then 2.4.3 Forced Convection Heat Transfer The fluid motion is, sustained by a difference of pressure created by an external device such as a pump or fan, the term of “forced convection” is used. On the other hand, if the fluid motion is predominantly sustained by the presence of a thermally induced density gradient, then the term of “natural convection” is used. Heat transfer by convection occurs as the result of a moving fluid encountering a fixed surface. The moving fluid carries the heat and deposits it on the surface or draws it out of the surface. There are two types of convection. In forced convection, the fluid is being driven or forced along by some mechanism other than thermal gradients at the surface. In free convection, the fluid is moved along by thermal gradients or temperature differences at the surface. Convection obeys Newton’s law of cooling given by 94 Experimental Physics I q in this case is the heat flux per unit area at the wall. The symbol h is identified as the film heat transfer coefficient. It has units of W/ m2 K or Btu/h/ft2 /R. The thermal conductivity, is a function of only the material and its temperature, h, the film heat transfer coefficient, depends on the properties of the fluid, the temperature of the fluid, and the flow characteristics. Multiple correlations have been determined for calculating an appropriate h for most materials and flow situations. In Eqs. 1a and 1b, the wall temperature is designated by Tw and T∞ is the temperature of fluid far from the wall at free-stream condition. To understand better the heat exchange between a solid and fluid, consider a heated wall over which a fluid flows as sketched in Fig.12. In this figure, U∞ is the velocity of the fluid under free-stream condition and far away from the wall as well. For a given stream velocity, the velocity of the fluid decreases as we get closer to the wall. This is due to the viscous effects of the flowing fluid. On the wall, because of the adherence (nonslip) condition the velocity of the fluid is zero. The region in which the velocity of the fluid varies from the free-stream value to zero is called “velocity boundary layer.” Similarly, the region in which the fluid temperature varies from its freestream value to that on the wall is called the “thermal boundary layer,” and both these boundary layers are defined in previous chapters. Since the velocity of the fluid at the wall is zero, the heat must be transferred by conduction at that point. Thus, we calculate the heat transfer by using the Fourier’s heat conduction law, with thermal conductivity of the fluid corresponding to the wall temperature and the fluid temperature gradient at the wall. Figure 13: Convection heat transfer to a flow over a heated wall. Heat 95 The question at this point is that: since the heat flows by conduction in this layer, why do we speak of convection heat transfer and need to consider the velocity of the fluid. The short answer to this question is that the temperature gradient of the fluid on the wall is highly dependent on the flow velocity of the free-stream. As this velocity increases, the distance from the wall we travel to reach fret stream temperature decreases. In other words, the thickness of velocity and thermal boundary layers on the wall decreases. The consequence of this decrease is to increase the temperature gradient of the fluid at the wall, i.e., an increase in the rate of heat transferred from the wall to the fluid. The effect of increasing fret stream velocity on the fluid velocity and temperature profiles close to the wall is illustrated in Fig. 8. Note also that the temperature gradient of the fluid on the wall increases with increasing free-stream velocity. Newton’s experiments end up finding the heat flux on the wall is based on Eqs. 1a and 1b. Table 1 gives the orders of magnitude of convective heat transfer coefficients. We need to remind you that most flow that are occurring in practical applications are turbulent. As we know so far, that turbulent flow characterization is based on disorderly displacement of individual volumes of fluid within the flow. From the above discussion, we conclude that the basic laws of heat conduction must be coupled with those of fluid motion to describe, mathematically, the process of convection. The mathematical treatment of the resulting system of differential equations is very complex. Therefore, for engineering applications, the convection will be treated by an ingenious combination of mathematical techniques, empirical evidence, and experimentation. Velocity, temperature, pressure, and other properties change continuously in time at every point of turbulent flow and the governing equations of mass, energy. Applies here and are valid for turbulent flow as well as laminar flow even in transient mode. Therefore, we have to take the note of the fact, that all the quantities such as velocity, pressure, and temperature in these equations are instantaneous values. In this chapter for time being, we concentrate on problems of heat transfer related to laminar forced convection flow in pipes and ducts. For example, the rate of heat transfer can become high at the location of reattachment of the upstream flow on to the surface of the step, as is also the case at the leading edge of a cylinder in cross-flow, but the detailed mechanisms remain incompletely understood and research continues. 96 Experimental Physics I Table 2: Order of magnitude of convective heat transfer coefficients EXERCISE Answer the following questions: 1. What is thermal linear expansion? 2. Explain area thermal expansion coefficient. 3. What are the types of calorimeter? 4. What do you understand by the specific heat and heat capacity? 5. Discuss about the heat of fusion formula. MULTIPLE CHOICE QUESTIONS Tick the correct answer: 1. What is the value of coefficient of volume expansion for an ideal gas? a. (ΔV/V)/Δ T b. 2*coefficient of linear expansion c. 1/3 * coefficient of linear expansion d. (V/ΔV)/ Δ T 2. What is the relation between coefficient of area expansion(A) and coefficient of volume expansion(B)? a. B = 1.5A b. A = 1.5B c. A = 3B d. B = 3A 3. The latent heat of fusion of ice is: a. The sensible heat required for melting ice b. 335 kJ per kg Heat 4. 5. 97 c. 335 kJ per ton d. The latent heat required for evaporation of ice Melting point is also known as a. fusion point b. constant point c. boiling point d. freezing point Latent heat of fusion of ice is a. 4.36 × 105 Jkg-1 b. 2.36 × 105 Jkg-1 c. 1.36 × 105 Jkg-1 d. 3.36 × 105 Jkg-1 ANSWERS 1. (a) 2. (a) 3. (b) 4. (a) 5. (d) REFERENCES 1. 2. 3. 4. Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and Engineers - Volume 1 Mechanics/Oscillations and Waves/ Thermodynamics. New York, NY: Worth Publishers. pp. 666–670. ISBN 978-1-4292-0132-2. Papini, Jon J.; Dyre, Jeppe C.; Christensen, Tage (2012-11-29). “Cooling by Heating---Demonstrating the Significance of the Longitudinal Specific Heat”. Physical Review X. 2 (4): 041015. arXiv:1206.6007. Bibcode:2012PhRvX...2d1015P. Bönisch, Matthias; Panigrahi, Ajit; Stoica, Mihai; Calin, Mariana; Ahrens, Eike; Zehetbauer, Michael; Skrotzki, Werner; Eckert, Jürgen (10 November 2017). “Giant thermal expansion and α-precipitation pathways in Ti-alloys”. Nature Communications. 8 (1): 1429. Bibcode:2017NatCo...8.1429B. Measurement and Prediction of Solubility of Paracetamol in WaterIsopropanol Solution. Part 2. Prediction H. Hojjati and S. Rohani Org. Process Res. Dev.; 2006; 10(6) pp 1110–1118; (Article) Waves and Sound 99 CHAPTER 3 WAVES AND SOUND OBJECTIVES After reading this chapter, you should be able to: • Explain the wave motion / vibrating strings • Describe the properties of sound • Define the characteristics of sound waves • discuss about basic concepts of vibration • Focus on vibration measurement • Explain the standing waves on strings • Understanding to study longitudinal sound waves created in an air column of variable length 3.1 WAVE MOTION / VIBRATING STRINGS The physical phenomenon of sound is a disturbance of matter that is transmitted from its source outward. Hearing is the perception of sound, just as seeing is the perception of visible light. On the atomic scale, sound is a disturbance of atoms that is far more ordered than their thermal motions. In many instances, sound is a periodic wave, and the 100 Experimental Physics I atoms undergo simple harmonic motion. Thus, sound waves can induce oscillations and resonance effects ((Figure)). Figure 1: This glass has been shattered by a high-intensity sound wave of the same frequency as the resonant frequency of the glass. A speaker produces a sound wave by oscillating a cone, causing vibrations of air molecules. In (Figure), a speaker vibrates at a constant frequency and amplitude, producing vibrations in the surrounding air molecules. As the speaker oscillates back and forth, it transfers energy to the air, mostly as thermal energy. But a small part of the speaker’s energy goes into compressing and expanding the surrounding air, creating slightly higher and lower local pressures. These compressions (high-pressure regions) and rarefactions (low-pressure regions) move out as longitudinal pressure waves having the same frequency as the speaker—they are the disturbance that is a sound wave. (Sound waves in air and most fluids are longitudinal, because fluids have almost no shear strength. In solids, sound waves can be both transverse and longitudinal.) (Figure 2) (a) shows the compressions and rarefactions, and also shows a graph of gauge pressure versus distance from a speaker. As the speaker moves in the positive x-direction, it pushes air molecules, displacing them from their equilibrium positions. As the speaker moves in the negative x-direction, the air molecules move back toward their equilibrium positions due to a restoring force. The air molecules oscillate in simple harmonic motion about their equilibrium positions, as shown in part (b). Note that sound waves in air are longitudinal, and in the figure, the wave propagates in the positive x-direction and the molecules oscillate parallel to the direction in which the wave propagates. Waves and Sound 101 Figure 2: (a) A vibrating cone of a speaker, moving in the positive x-direction, compresses the air in front of it and expands the air behind it. As the speaker oscillates, it creates another compression and rarefaction as those on the right move away from the speaker. After many vibrations, a series of compressions and rarefactions moves out from the speaker as a sound wave. The red graph shows the gauge pressure of the air versus the distance from the speaker. Pressures vary only slightly from atmospheric pressure for ordinary sounds. 3.1.1 Types of Sound There are many different types of sound including, audible, inaudible, unpleasant, pleasant, soft, loud, noise and music. You’re likely to find the sounds produced by a piano player soft, audible, and musical. And while the sound of road construction early on Saturday morning is also audible, it certainly isn’t pleasant or soft. Other sounds, such as a dog whistle, are inaudible to the human ear. This is because dog whistles produce sound waves that are below the human hearing range of 20 Hz to 20,000 Hz. Waves below 20 Hz are called infrasonic waves (infrasound), while higher frequencies above 20,000 Hz are known as ultrasonic waves (ultrasound). Infrasonic Waves (Infrasound) Infrasonic waves have frequencies below 20 Hz, which makes them inaudible to the human ear. Scientists use infrasound to detect earthquakes and volcanic eruptions, to map rock and petroleum formations Note Gauge pressure is modeled with a sine function, where the crests of the function line up with the compressions and the troughs line up with the rarefactions. Sound waves can also be modeled using the displacement of the air molecules. The blue graph shows the displacement of the air molecules versus the position from the speaker and is modeled with a cosine function. Notice that the displacement is zero for the molecules in their equilibrium position and are centered at the compressions and rarefactions. 102 Experimental Physics I underground, and to study activity in the human heart. Despite our inability to hear infrasound, many animals use infrasonic waves to communicate in nature. Whales, hippos, rhinos, giraffes, elephants, and alligators all use infrasound to communicate across impressive distances – sometimes hundreds of miles! Ultrasonic Waves (Ultrasound) Sound waves that have frequencies higher than 20,000 Hz produce ultrasound. Because ultrasound occurs at frequencies outside the human hearing range, it is inaudible to the human ear. Ultrasound is most often used by medical specialists who use sonograms to examine their patients’ internal organs. Some lesser-known applications of ultrasound include navigation, imaging, sample mixing, communication, and testing. In nature, bats emit ultrasonic waves to locate prey and avoid obstacles. How is Sound Produced? Sound is produced when an object vibrates, creating a pressure wave. This pressure wave causes particles in the surrounding medium (air, water, or solid) to have vibrational motion. As the particles vibrate, they move nearby particles, transmitting the sound further through the medium. The human ear detects sound waves when vibrating air particles vibrate small parts within the ear. In many ways, sound waves are similar to light waves. They both originate from a definite source and can be distributed or scattered using various means. Unlike light, sound waves can only travel through a medium, such as air, glass, or metal. This means there’s no sound in space! Figure 3: Sound waves. Waves and Sound 103 How Does Sound Travel? Mediums Before we discuss how sound travels, it’s important to understand what a medium is and how it affects sound. We know that sound can travel through gases, liquids, and solids. But how do these affect its movement? Sound moves most quickly through solids, because its molecules are densely packed together. This enables sound waves to rapidly transfer vibrations from one molecule to another. Sound moves similarly through water, but its velocity is over four times faster than it is in air. The velocity of sound waves moving through air can be further reduced by high wind speeds that dissipate the sound wave’s energy. Mediums and the Speed of Sound The speed of sound is dependent on the type of medium the sound waves travel through. In dry air at 20°C, the speed of sound is 343 m/s! In room temperature seawater, sound waves travel at about 1531 m/s! When physicists observe a disturbance that expands faster than the local speed of sound, it’s called a shockwave. When supersonic aircraft fly overhead, a local shockwave can be observed! Generally, sound waves travel faster in warmer conditions. As the ocean warms from global climate, how do you think this will affect the speed of sound waves in the ocean? 3.1.2 Propagation of Sound Waves When an object vibrates, it creates kinetic energy that is transmitted by molecules in the medium. As the vibrating sound wave comes in contact with air particles passes its kinetic energy to nearby molecules. As these energized molecules begin to move, they energize other molecules that repeat the process. Imagine a slinky moving down a staircase. When falling down a stair, the slinky’s motion begins by expanding. As the first ring expands forward, it pulls the rings behind it forward, causing a compression wave. This push and pull chain reaction causes each ring of the slinky’s coil to be displaced from its original position, gradually transporting the original energy from the first coil to the last. The compressions and rarefactions of sound waves are similar to the slinky’s pushing and pulling of its coils. 104 Experimental Physics I Compression & Rarefaction Sound waves are composed of compression and rarefaction patterns. Compression happens when molecules are densely packed together. Alternatively, rarefaction happens when molecules are distanced from one another. As sound travels through a medium, its energy causes the molecules to move, creating an alternating compression and rarefaction pattern. It is important to realize that molecules do not move with the sound wave. As the wave passes, the molecules become energized and move from their original positions. After a molecule passes its energy to nearby molecules, the molecule’s motion diminishes until it is affected by another passing wave. The wave’s energy transfer is what causes compression and rarefaction. During compression there is high pressure, and during rarefaction there is low pressure. Different sounds produce different patterns of high- and low-pressure changes, which allows them to be identified. The wavelength of a sound wave is made up of one compression and one rarefaction. Figure 4: Different sounds produce different patterns of high- and low-pressure changes. Sound waves lose energy as they travel through a medium, which explains why you cannot hear people talking far away, but you can hear them whispering nearby. As sound waves move through space, they are reflected by mediums, such as walls, pillars, and rocks. This sound reflection is better known as an echo. If you’ve ever been inside a cave or canyon, you’ve probably heard your echo carry much farther than usual. This is due to the large rock walls reflecting your sound off one another. 3.1.3 Types of Waves So, what type of wave is sound? Sound waves fall into three categories: longitudinal waves, mechanical waves, and pressure waves. Keep reading to find out what qualifies them as such. Waves and Sound 105 Longitudinal Sound Waves A longitudinal wave is a wave in which the motion of the medium’s particles is parallel to the direction of the energy transport. Sound waves in air and fluids are longitudinal waves, because the particles that transport the sound vibrate parallel to the direction of the sound wave’s travel. If you push a slinky back and forth, the coils move in a parallel fashion (back and forth). Similarly, when a tuning fork is struck, the direction of the sound wave is parallel to the motion of the air particles. Mechanical Sound Waves A mechanical wave is a wave that depends on the oscillation of matter, meaning that it transfers energy through a medium to propagate. These waves require an initial energy input that then travels through the medium until the initial energy is effectively transferred. Examples of mechanical waves in nature include water waves, sound waves, seismic waves and internal water waves, which occur due to density differences in a body of water. There are three types of mechanical waves: transverse waves, longitudinal waves, and surface waves. Why is sound a mechanical wave? Sound waves move through air by displacing air particles in a chain reaction. As one particle is displaced from its equilibrium position, it pushes or pulls on neighboring molecules, causing them to be displaced from their equilibrium. As particles continue to displace one another with mechanical vibrations, the disturbance is transported throughout the medium. These particleto-particle, mechanical vibrations of sound conductance qualify sound waves as mechanical waves. Sound energy, or energy associated with the vibrations created by a vibrating source, requires a medium to travel, which makes sound energy a mechanical wave. Wireless Sound Sensor The Wireless Sound Sensor features two key sensors in one portable package: a sound wave sensor for measuring relative changes in sound pressure and a sound level sensor with both dBA- and dBC-weighted scales. With live data reporting and a wide range of displays (FFT, scope, digits), the Wireless Sound Sensor’s simple design makes it easy to use for introductory sound explorations, while its onboard memory and robust software features support higher-level investigations into the science of sound. 106 Experimental Physics I Pressure Sound Waves A pressure wave, or compression wave, has a regular pattern of high- and low-pressure regions. Because sound waves consist of compressions and rarefactions, their regions fluctuate between low and high-pressure patterns. For this reason, sound waves are considered to be pressure waves. For example, as the human ear receives sound waves from the surrounding environment, it detects rarefactions as lowpressure periods and compressions as high-pressure periods. Transverse Waves Transverse waves move with oscillations that are perpendicular to the direction of the wave. Sound waves are not transverse waves because their oscillations are parallel to the direction of the energy transport; however sound waves can become transverse waves under very specific circumstances. Transverse waves, or shear waves, travel at slower speeds than longitudinal waves, and transverse sound waves can only be created in solids. Ocean waves are the most common example of transverse waves in nature. A more tangible example can be demonstrated by wiggling one side of a string up and down, while the other end is anchored (see standing waves video below). Still a little confused? Check out the visual comparison of transverse and longitudinal waves below. Figure 5: Visual comparison of longitudinal and transverse waves. How to Create Standing Waves With PASCO’s String Vibrator, Sine Wave Generator, and Strobe System, students can create, manipulate and measure standing waves in real time. The Sine Wave Generator and String Vibrator work together to propagate a sine wave through the rope, while the Strobe System can be used to “freeze” waves in time. Create clearly defined nodes, illuminate Waves and Sound 107 standing waves, and investigate the quantum nature of waves in realtime with this modern investigative approach. 3.2 PROPERTIES OF SOUND What makes music different from noise? A bird’s call is more melodic than a car alarm. And, we can usually tell the difference between ambulance and police sirens - but how do we do this? We use the four properties of sound: pitch, dynamics (loudness or softness), timbre (tone color), and duration. 3.2.1 Frequency (Pitch) Pitch is the quality that enables us to judge sounds as being “higher” and “lower. It provides a method for organizing sounds based on a frequency-based scale. Pitch can be interpreted as the musical term for frequency, though they are not exactly the same. A high-pitched sound causes molecules to rapidly oscillate, while a low-pitched sound causes slower oscillation. Pitch can only be determined when a sound has a frequency that is clear and consistent enough to differentiate it from noise. Because pitch is primarily based on a listener’s perception, it is not an objective physical property of sound. 3.2.2 Amplitude (Dynamics) The amplitude of a sound wave determines it relative loudness. In music, the loudness of a note is called its dynamic level. In physics, we measure the amplitude of sound waves in decibels (dB), which do not correspond with dynamic levels. Higher amplitudes correspond with louder sounds, while shorter amplitudes correspond with quieter sounds. Despite this, studies have shown that humans perceive sounds at very low and very high frequencies to be softer than sounds in the middle frequencies, even when they have the same amplitude. 3.2.3 Timbre (Tone Color) Timbre refers to the tone color, or “feel” of the sound. Sounds with various timbres produce different wave shapes, which affect our interpretation of the sound. The sound produced by a piano has a different tone color than the sound from a guitar. In physics, we refer to this as the timbre of a sound. It’s what allows humans to quickly identify sounds (e.g. a cat’s meow, running water, the sound of a friend’s voice). 108 Experimental Physics I 3.2.4 Duration (Tempo/Rhythm) In music, duration is the amount of time that a pitch, or tone, lasts. They can be described as long, short, or as taking some amount of time. The duration of a note or tone influences the timbre and rhythm of a sound. A classical piano piece will tend to have notes with a longer duration than the notes played by a keyboardist at a pop concert. In physics, the duration of a sound or tone begins once the sound registers and ends after it cannot be detected. Creating Music with the 4 Properties of Sound Musicians manipulate the four properties of sound to make repeating patterns that form a song. Duration is the length of time a musical sound lasts. When you strum a guitar, the duration of the sound is stopped when you quiet the strings. Pitch is the relative highness or lowness that is heard in a sound and is determined by the frequency of sound vibrations. Faster vibrations produce a higher pitch than slower vibrations. The thicker strings of the guitar produce slower vibrations, creating a deeper pitch, while the thinner strings produce faster vibrations and a higher pitch. A sound with a definite pitch, or specific frequency, is called a tone. Tones have specific frequencies that reach the ear at equal time intervals, such as 320 cycles per second. When two tones have different pitches, they sound dissimilar, and the difference between their pitches is called an interval. Musicians frequently use an interval called an octave, which allows two tones of varying pitches to share a similar sound. Dynamics refers to a sound’s degree of loudness or softness and is related to the amplitude of the vibration that produces the sound. The harder a guitar string is plucked, the louder the sound will be. Tone color, or timbre, describes the overall feel of an instrument’s produced sound. If we were to describe a trumpet’s tone color, we may refer to it as bright or brilliant. When we consider a cello, we may say it has a rich tone color. Each instrument offers its own tone color, and new tone colors can be created by layering instruments together. Furthermore, modern music styles like EDM have introduced new tone styles, which were unavailable prior to digital music creation. What Makes Sound Music or Noise? Acousticians, or scientists who study sound acoustics, have studied how different sound types, primarily noise and music, affect humans. Randomized, unpleasant sound waves are often referred to as noise. Alternatively, constructed patterns of sound waves are known as music. Waves and Sound 109 Studies have shown that the human body responds differently to noise and music, which may explain why road construction on a Saturday morning makes us more tense than a pianist’s song. Acoustics Acoustics is an interdisciplinary science that studies mechanical waves, including vibration, sound, infrasound and ultrasound in various environments, such as solids, liquids and gases. Professionals in acoustics can range from acoustical engineers, who investigate new applications for sound in technology, to audio engineers, who focus on recording and manipulating sound, to acousticians, who are scientists concerned with the science of sound. 3.3 CHARACTERISTICS OF SOUND WAVES There are five main characteristics of sound waves: wavelength, amplitude, frequency, time period, and velocity. The wavelength of a sound wave indicates the distance that wave travels before it repeats itself. The wavelength itself is a longitudinal wave that shows the compressions and rarefactions of the sound wave. The amplitude of a wave defines the maximum displacement of the particles disturbed by the sound wave as it passes through a medium. A large amplitude indicates a large sound wave. The frequency of a sound wave indicates the number of sound waves produced each second. Low-frequency sounds produce sound waves less often than high-frequency sounds. The time period of a sound wave is the amount of time required to create a complete wave cycle. Each vibration from the sound source produces a wave’s worth of sound. Each complete wave cycle begins with a trough and ends at the start of the next trough. Lastly, the velocity of a sound wave tells us how fast the wave is moving and is expressed as meters per second. Figure 6: A wave cycle occurs between two troughs. 110 Experimental Physics I 3.3.1 Units of Sound When we measure sound, there are four different measurement units available to us. The first unit is called the decibel (dB). The decibel is a logarithmic ratio of the sound pressure compared to a reference pressure. The next most frequently used unit is the hertz (Hz). The hertz is a measure of sound frequency. Hertz and decibels are widely used to describe and measure sounds, but phon and sone are also used. A sone is the perceived loudness of a sound and a phon is the unit of loudness for pure tones. Additionally, the phon refers to subjective loudness, while the sone is the perceived loudness. 3.3.2 Sound Wave Graphs Explained Sound waves can be described by graphing either displacement or density. Displacement-time graphs represent how far the particles are from their original places and indicates which direction they’ve moved. Particles that show up on the zero line in a particle displacement graph didn’t move at all from their normal position. These seemingly motionless particles experience more compressions and rarefactions than other particles. Since pressure and density are related, a pressure versus time graph will display the same information as a density versus time graph. These graphs indicate where the particles are compressed and where they are very expanded. Unlike displacement graphs, particles along the zero line in a density graph are never squished or pulled apart. Instead, they are the particles that move back and forth the most. Figure 7: Sound waves can be described by graphing either displacement or density. Waves and Sound 111 3.3.3 Sound Pressure Sound pressure describes the local pressure deviation from the ambient atmospheric pressure as a sound wave travels. It’s important to recognize that sound pressure and air pressure are not the same concept. Overall, the speed of sound is not influenced by air pressure. As sound waves pass from the sound source through the air, they alter the pressure experienced by air nearby particles. Sound Level Sound level is a comparison of the sound wave’s pressure relative to the reference point. Sound level is measured in decibels, with higher decibels correlating to higher sound levels. Some sound instruments measure sound level in dBc, which is the power ratio (decibels) of a signal to its carrier signal. Other sound instruments measure the relative loudness of sounds as perceived by the human ear using a-weighted decibels, known as dBa. When dBa is used, sounds at low frequencies have their decibel values reduced and compared to unweighted decibels. Figure 8: Sound Level is a comparison of the sound wave’s pressure relative to the reference point. A dBc meter measures high and low frequencies, while a dBa meter measures mid-level frequencies. 3.3.4 Sound Intensity Sound intensity is the power per unit area carried by a sound wave. The more intense the sound is, the larger the amplitude oscillations will be. As sound intensity increases, the pressure exerted by the sound waves on nearby objects also increases. Decibels are used to measure the ratio of a given intensity (I) to the threshold of hearing intensity, which typically has a value of 1000 Hz for the human ear. 112 Experimental Physics I Figure 9: Sound Intensity is the power per unit area carried by a sound wave. The more intense the sound is, the larger the amplitude oscillations will be. As sound intensity increases, the pressure exerted by the sound waves on nearby objects also increases. 3.3.5 Sound Intensity in an Air Column An air column is a large, hollow tube that is open on one side and closed on the other. The conditions created by an air column are especially useful for investigating sound characteristics such as intensity and resonance. Check out the video below to see how air columns can be used to investigate nodes, antinodes and resonance. 3.4 BASIC CONCEPTS OF VIBRATION Any motion that repeats itself after an interval of time is called vibration or oscillation. The swinging of a pendulum and the motion of a plucked string are typical examples of vibration. The theory of vibration deals with the study of oscillatory motions of bodies and the forces associated with them. A vibratory system, in general, includes a means for storing potential energy (spring or elasticity), a means for storing kinetic energy (mass or inertia), and a means by which energy is gradually lost (damper). The vibration of a system involves the transfer of its potential energy to kinetic energy and of kinetic energy to potential energy, alternately. If the system is damped, some energy is dissipated in each cycle of vibration and must be replaced by an external source if a state of steady vibration is to be maintained. As an example, consider the vibration of the simple pendulum shown in Figure 10. Let the bob of mass m be released after being given an angular displacement. At position 1 the velocity of the bob and hence its kinetic energy is zero. But it has a potential energy of magnitude mgl(1 − cos θ) with respect to the datum position 2. Waves and Sound 113 Since the gravitational force mg induces a torque about the point O, the bob starts swinging to the left from position 1. This gives the bob certain angular acceleration in the clockwise direction, and by the time it reaches position 2, all of its potential energy will be converted into kinetic energy. Figure 10: A simple pendulum. Hence the bob will not stop in position 2 but will continue to swing to position 3. However, as it passes the mean position 2, a counterclockwise torque due to gravity starts acting on the bob and causes the bob to decelerate. The velocity of the bob reduces to zero at the left extreme position. By this time, all the kinetic energy of the bob will be converted to potential energy. Again due to the gravity torque, the bob continues to attain a counterclockwise velocity. Hence the bob starts swinging back with progressively increasing velocity and passes the mean position again. This process keeps repeating, and the pendulum will have oscillatory motion. Note The magnitude of oscillation gradually decreases and the pendulum ultimately stops due to the resistance (damping) offered by the surrounding medium (air). This means that some energy is dissipated in each cycle of vibration due to damping by the air. 114 Experimental Physics I 3.4.1 Time Response Vibrations can be analyzed either in the time domain or in the frequency domain. Both free and forced vibrations may have to be analyzed. Free and natural vibrations occur in systems because of the presence of two forms of energy storage that are interchangeable. When the stored energy is repeatedly exchanged between these two forms, the resulting time response of the system is oscillatory. In a mechanical system, natural vibrations can occur because kinetic energy, which is manifested as velocities of mass (inertia) elements, may be converted into potential energy (which can be classified into two basic types-elastic potential energy resulting from the deformation of spring-like elements and gravitation potential energy resulting from the elevation of mass elements under the Earth’s gravitational pull) and back to kinetic energy, repetitively, during motion. An oscillatory excitation (forcing function) is able to make a “dynamic” system respond with an oscillatory motion (at the same frequency as the forcing excitation) even in the absence of the two forms of energy storage. Such motions are forced responses rather than natural or free responses. Nevertheless, clear analogies exist with electrical and fluid systems as well as mixed systems such as electromechanical systems. Natural oscillations of electrical signals occur in circuits because of the presence of electrostatic energy (electric charge storage in capacitor-like elements) and electromagnetic energy (magnetic fields in inductor-like elements). Fluid systems can exhibit natural oscillatory responses as they possess two forms of energy. However, purely thermal systems do not produce natural oscillations because they, as far as we know, have only one type of energy. Mechanical vibrations can occur as both free (natural) and forced responses in numerous practical situations. Some of these vibrations are desirable and useful, and some others are undesirable and should be avoided or suppressed. The sound that is generated when a string of a guitar is plucked is a free vibration whereas the sound of a violin is a mixture of both free and forced vibrations. These sounds are generally pleasant and desirable. The response of an automobile as it hits a road bump is an undesirable free vibration. The vibrations that are felt while operating a concrete drill are required for the drilling process itself, but are undesirable forced vibrations for the human who operates the drill. In the design and development of a mechanical system, regardless of whether it is intended for generating desirable vibrations or for operating without Waves and Sound 115 vibrations, an analytical model of the system can serve a very useful purpose. The model will represent the dynamic system, and may be analyzed and modified more quickly and cost effectively than one could build and test a physical prototype. Similarly, in the control or suppression of vibrations, it is possible Note to design, develop, and evaluate vibration isolators and control schemes through A frequencyanalytical means before they are physically domain model is a implemented. It follows that analytical set of input-output models are useful in the analysis, control, and transfer functions evaluation of vibrations in dynamic systems, with respect to the and also in the design and development of independent variable frequency. dynamic systems for achieving the desired performance when operating in vibratory environments. An analytical model of a mechanical system is a set of equations, and may be developed either by the Newtonian approach, where Newton’s second law is explicitly applied to each inertia element, or by the Lagrangian or Hamiltonian approach, which is based on the concepts of energy (kinetic and potential). The time response describes how the system moves (responds) as a function of time. The frequency response describes the way the system moves when excited by a harmonic (sinusoidal) forcing input, and is a function of the frequency of excitation. 3.4.2 Classification of Vibration Vibration can be classified in several ways. Some of the important classifications are as follows. Free and Forced Vibrations • Free Vibration. If a system, after an initial disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration. No external force acts on the system. The oscillation of a simple pendulum is an example of free vibration. Free vibrations are oscillations where the total energy stays the same over time. This means that the amplitude of the vibration stays the same. This is a theoretical idea because in real systems the energy is dissipated to the surroundings over 116 Experimental Physics I • time and the amplitude decays away to zero, this dissipation of energy is called damping. Forced Vibration. If a system is subjected to an external force (often, a repeating type of force), the resulting vibration is known as forced vibration. The oscillation that arises in machines such as diesel engines is an example of forced vibration. If the frequency of the external force coincides with one of the natural frequencies of the system, a condition known as resonance occurs, and the system undergoes dangerously large oscillations. Failures of such structures as buildings, bridges, turbines, and airplane wings have been associated with the occurrence of resonance. Free vibrations can be defined as oscillations about a system’s equilibrium position that occur in the absence of an external excitation. Examples of free vibrations are oscillations of a pendulum about a vertical equilibrium position and a motion of a vehicle suspension system after the vehicle encounters a pothole. The oscillations in free vibrations will eventually die out over time. Let’s consider a mass attached to a spring, k and a damper, c. The mass is pulled to the right in horizontal direction and then released. Oscillations occur about its equilibrium position until it stops. Figure 11: Free vibration: free body diagram. Using Newton’s Second Law, force equilibrium in horizontal direction (x-direction) is given by: + cx + kx = mx 0 (1) Forced vibrations occur when work is being done on the system while the vibrations/oscillations occur. Examples of free vibrations include a motion caused by an unbalanced rotating component, a motion of reciprocating pistons in engine. Diagram of forced vibration can be represented by a spring-mass-damper system with external force F(t) = Waves and Sound 117 F0 sinωt. Oscillations occur about its equilibrium position. Oscillations do not stop because of applied external force. Figure 12: Forced vibration: free body diagram. Force equilibrium in horizontal direction (x-direction) is given by: mx + cx + kx= F0 sin ωt (2) Comparing the equations of free vibration and forced vibrations, free vibrations have no external force term, while forced vibrations have external force term, F0 sinωt. Forced vibrations are commonly observed in washing machines due to unbalanced mass, fans due to broken blade, compressors due to reciprocating action of piston etc. Undamped and Damped Vibration If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is known as undamped vibration. If any energy is lost in this way, however, it is called damped vibration. In many physical systems, the amount of damping is so small that it can be disregarded for most engineering purposes. However, consideration of damping becomes extremely important in analyzing vibratory systems near resonance. Damped and undamped vibration refer to two different types of vibrations. The main difference between damped and undamped vibration is that undamped vibration refer to vibrations where energy of the vibrating object does not get dissipated to surroundings over time, whereas damped vibration refers to vibrations where the vibrating object loses its energy to the surroundings. Undamped Vibration In undamped vibrations, no resistive force acts on the vibrating object. As the object oscillates, the energy in the object is continuously 118 Experimental Physics I transformed from kinetic energy to potential energy and back again, and the sum of kinetic and potential energy remains a constant value. In practice, it’s extremely difficult to find undamped vibrations. For instance, even an object vibrating in air would lose energy over time due to air resistance. Let us consider an object undergoing simple harmonic motion. Here, the objet experiences a restoring force towards the equilibrium point, and the size of this force is proportional to displacement. If the displacement of the object is given by x, then for an object with mass m in simple harmonic motion, we can write: m d2 x = −kx (3) dt 2 This is a differential equation. A solution to this equation can be written in the form: = x A cos(ωt + φ) Here, ω = k m (4) If vibration is undamped, the object continues to oscillate sinusoidally. Damped Vibration In damped vibrations, external resistive forces act on the vibrating object. The object loses energy due to resistance and as a result, the amplitude of vibrations decreases exponentially. We can model the damping force to be directly proportional to the speed of the object at the time. If the constant of proportionality for the damping force is , then we can write: m d2 x dx = −kx − b 2 dt (5) dt The solution to this differential equation can be given in the form: −b t = x A e 2m cos(ω′t + φ) (6) Waves and Sound 119 2 ω Here, the= k b − m 2m (7) We can write this as: 2 ω = b 1− 2 mk (8) Writing the equation in this form is useful because the quantity b can be used to determine the nature of a particular oscillation. 2 mk Often, this quantity is called the damping coefficient ζ ,i.e. ζ = b . 2 mk If ζ =1, then we have critical damping. Under this condition, the oscillating object returns to its equilibrium position as soon as possible without completing any more oscillations. When ζ <1, we have underdamping. In this case, the object continues to oscillate, but with an ever-reducing amplitude. For ζ >1 the resistive forces are very strong. The object would not oscillate again, but the object is slowed down so much, that it goes towards the equilibrium much more slowly compared to an object that is critically damped. Over damping is the name given to this type of scenario. When ζ =0, there is no resistive force and the object is undamped. Theoretically, the object continues to carry out simple harmonic motion without any reduction in amplitude. The graph below illustrates how the displacement of the object changes under these three different conditions: Figure 13: Damping under resistive forces with different damping constants. 120 Experimental Physics I Linear and Nonlinear Vibration If all the basic components of a vibratory system the spring, the mass, and the damper behave linearly, the resulting vibration is known as linear vibration. If, however, any of the basic components behave nonlinearly, the vibration is called nonlinear vibration. The differential equations that govern the behavior of linear and nonlinear vibratory systems are linear and nonlinear, respectively. If the vibration is linear, the principle of superposition holds, and the mathematical techniques of analysis are well developed. For nonlinear Important vibration, the superposition principle is not If the value valid, and techniques of analysis are less or magnitude of the well known. Since all vibratory systems excitation (force or tend to behave nonlinearly with increasing motion) acting on a amplitude of oscillation, a knowledge of vibratory system is nonlinear vibration is desirable in dealing known at any given with practical vibratory systems. time, the excitation Deterministic and Random Vibration is called deterministic. The resulting vibration is known as deterministic vibration. In some cases, the excitation is nondeterministic or random; the value of the excitation at a given time cannot be predicted. In these cases, a large collection of records of the excitation may exhibit some statistical regularity. It is possible to estimate averages such as the mean and mean square values of the excitation. Examples of random excitations are wind velocity, road roughness, and ground motion during earthquakes. If the excitation is random, the resulting vibration is called random vibration. In this case the vibratory response of the system is also random; it can be described only in terms of statistical quantities. Figure 14 shows examples of deterministic and random excitations. Figure 14: Deterministic and random excitations. Waves and Sound 121 3.4.3 Vibration Analysis Procedure A vibratory system is a dynamic one for which the variables such as the excitations (inputs) and responses (outputs) are time dependent. The response of a vibrating system generally depends on the initial conditions as well as the external excitations. Most practical vibrating systems are very complex, and it is impossible to consider all the details for a mathematical analysis. Only the most important features are considered in the analysis to predict the behavior of the system under specified input conditions. Often the overall behavior of the system can be determined by considering even a simple model of the complex physical system. Thus, the analysis of a vibrating system usually involves mathematical modeling, derivation of the governing equations, solution of the equations, and interpretation of the results. Step 1: Mathematical Modeling. The purpose of mathematical modeling is to represent all the important features of the system for the purpose of deriving the mathematical (or analytical) equations governing the system s behavior. The mathematical model should include enough details to allow describing the system in terms of equations without making it too complex. The mathematical model may be linear or nonlinear, depending on the behavior of the system s components. Linear models permit quick solutions and are simple to handle; however, nonlinear models sometimes reveal certain characteristics of the system that cannot be predicted using linear models. Thus a great deal of engineering judgment is needed to come up with a suitable mathematical model of a vibrating system. (a) 122 Experimental Physics I (b) (c) Figure 15: Modeling of a forging hammer. Sometimes the mathematical model is gradually improved to obtain more accurate results. In this approach, first a very crude or elementary model is used to get a quick insight into the overall behavior of the system. Subsequently, the model is refined by including more components and/or details so that the behavior of the system can be observed more closely. To illustrate the procedure of refinement used in mathematical modeling, consider the forging hammer shown in Figure 15 (a). It consists of a frame, a falling weight known as the tup, an anvil, and a foundation block. The anvil is a massive steel block on which material is forged into desired shape by the repeated blows of the tup. The anvil is usually mounted on an elastic pad to reduce the transmission of vibration to the foundation block and the frame. For a first approximation, the frame, anvil, elastic pad, foundation block, and soil are modeled as a single degree of freedom system as shown in Figure 15 (b). For a refined approximation, the weights of the frame and anvil and the foundation Waves and Sound 123 block are represented separately with a two-degree-of-freedom model as shown in Figure 15 (c). Further refinement of the model can be made by considering eccentric impacts of the tup, which cause each of the masses shown in Figure 15 (c) to have both vertical and rocking (rotation) motions in the plane of the paper. Step 2: Derivation of Governing Equations. Once the mathematical model is available, we use the principles of dynamics and derive the equations that describe the vibration of the system. The equations of motion can be derived conveniently by drawing the free-body diagrams of all the masses involved. The free-body diagram of a mass can be obtained by isolating the mass and indicating all externally applied forces, the reactive forces, and the inertia forces. The equations of motion of a vibrating system are usually in the form of a set of ordinary differential equations for a discrete system and partial differential equations for a continuous system. The equations may be linear or nonlinear, depending on the behavior of the components of the system. Several approaches are commonly used to derive the governing equations. Among them are Newton s second law of motion, D Alembert s principle, and the principle of conservation of energy. Step 3: Solution of the Governing Equations. The equations of motion must be solved to find the response of the vibrating system. Depending on the nature of the problem, we can use one of the following techniques for finding the solution: standard methods of solving differential equations, Laplace transform methods, matrix methods, and numerical methods. If the governing equations are nonlinear, they can seldom be solved in closed form. Furthermore, the solution of partial differential equations is far more involved than that of ordinary differential equations. Numerical methods involving computers can be used to solve the equations. However, it will be difficult to draw general conclusions about the behavior of the system using computer results. Step 4: Interpretation of the Results. The solution of the governing equations gives the displacements, velocities, and accelerations of the various masses of the system. These results must be interpreted with a clear view of the purpose of the analysis and the possible design implications of the results. 3.5 VIBRATION MEASUREMENT Transducers are available for the direct measurement of instantaneous acceleration, velocity, displacement and surface strain. 124 Experimental Physics I In noise-control applications, the most commonly measured quantity is acceleration, as this is often the most convenient to measure. However, the quantity that is most useful is vibration velocity, as its square is related directly to the structural vibration energy, which in turn is often related directly to the radiated sound power. Also, most machines and radiating surfaces have a flatter velocity spectrum than acceleration spectrum, which means that the use of velocity signals is an advantage in frequency analysis as it allows the maximum amount of information to be obtained using an octave or third-octave filter, or spectrum analyzer with a limited dynamic range. For single frequencies or narrow bands of noise, the displacement, d, velocity, ν, and acceleration, a, are related by the frequency, ω(rad/s), as dω2=νω=a. In terms of phase angle, velocity leads displacement by 90º and acceleration leads velocity by 90°. For narrow band or broadband signals, velocity can also be derived from acceleration measurements using electronic integrating circuits. On the other hand, deriving velocity and acceleration signals by differentiating displacement signals is generally not practical due primarily to the limited dynamic range of displacement transducers and secondarily to the cost of differentiating electronics. One alternative, which is rarely used in noise control, is to bond strain gauges to the surface to measure vibration levels. However, this technique will not be discussed further here. 3.5.1 Acceleration Transducers Vibratory motion for noise-control purposes is most commonly measured with an accelerometer attached to the vibrating surface. The accelerometer most generally used consists of a small piezoelectric crystal, loaded with a small weight and designed to have a natural resonance frequency well above the anticipated excitation frequency range. Where this condition may not be satisfied and consequently a problem may exist involving excitation of the accelerometer resonance, mechanical filters are available which, when placed between the accelerometer base and the measurement surface, minimize the effect of the accelerometer resonance at the expense of the high-frequency response. This results in loss of accuracy at lower frequencies, effectively shifting the ±3 dB error point down in frequency by a factor of five. However, the transverse sensitivity at higher frequencies is also much reduced by use of a mechanical filter, which in some cases is a significant benefit. Sometimes Waves and Sound 125 it may also be possible to filter out the accelerometer resonance response using an electrical filter on the output of the amplifier, but this could effectively reduce the dynamic range of the measurements due to the limited dynamic range of the amplifier. Figure 16: Single degree of freedom: (a) forced mass, rigid bas; (b) vibrating base. The mass-loaded piezoelectric crystal accelerometer may be thought of as a one degree- of-freedom system driven at the base, such as that of case (b) of Figure 16. The crystal, which may be loaded in compression or shear, provides the stiffness and system damping as well as a small contribution to the inertial mass, while the load provides the major part of the system inertial mass. As may readily be shown, the response of such a system driven well below resonance is controlled by the system (crystal) stiffness. Within the frequency design range, motion of the base of the accelerometer imparts in-phase motion to the weight on the crystal, resulting in small stresses in the crystal. The latter stresses are detected as induced charge on the crystal by means of some very highimpedance voltage detection circuit, like that provided by an ordinary sound level meter or a charge amplifier. Let the displacement, d, of the motion to be measured be: d=D0ejωt(9) Referring to Figure 16 (b), the amplitude, D, of displacement response of the accelerometer mass is as follows: D=D0X2/Z (10) 126 Experimental Physics I where |Z|=[(1−X2)2+(2Xζ)2]1/2, and X=ƒ/ƒ0(11) In the above equations, X is the ratio of the driving frequency to the resonance frequency of the accelerometer, ζ is the damping ratio of the accelerometer and Z is the modulus of the impedance seen by the accelerometer mass, which represents the reciprocal of a magnification factor. The voltage generated by the accelerometer will be proportional to D and, as shown by Equation (10), to the acceleration D0X2 divided by the modulus of the impedance, |Z|. If a vibratory motion is periodic it will generally have overtones. Alternatively, if it is not periodic, the response may be thought of as a continuum of overtones. In any case, if distortion in the measured acceleration is to be minimal, then it is necessary that the magnification factor be essentially constant over the frequency range of interest. In this case the accelerometer mass displacement amplitude, D (and the associated voltage which is proportional to the mass displacement), associated with any component of acceleration will be proportional to the amplitude of the component, and there will then be no distortion. However, as the magnification factor, 1/|Z|, is a function of frequency ratio, X, it can only be approximately constant by design over some prescribed range and some distortion will always be present. The percent amplitude distortion is defined as: Amplitude distortion=[(1/|Z|)−1)]×100% (12) To minimize distortion, the accelerometer should have a damping ratio of between 0.6 and 0.7, giving a useful frequency range of 0<X<0.6. Where voltage amplification is used, the sensitivity of an accelerometer is dependent upon the length of cable between the accelerometer and its amplifier. Any motion of the connecting cable can result in spurious signals. The voltage amplifier must have a very high input impedance to measure low frequency vibration and not significantly load the accelerometer electrically because the amplifier decreases the electrical time constant of the accelerometer and effectively reduces its sensitivity. Commercially available high impedance voltage amplifiers allow accurate measurement down to about 20 Hz, but are rarely used due to the above-mentioned problems. Alternatively, charge amplifiers (which, unfortunately, are relatively expensive) are usually preferred, as they have a very high Waves and Sound 127 input impedance and thus do not load the accelerometer output; they allow measurement of acceleration down to frequencies of 0.2 Hz; they are insensitive to cable lengths up to 500 m and they are relatively insensitive to cable movement. Many charge amplifiers also have the capability of integrating acceleration signals to produce signals proportional to velocity or displacement. This facility should be used with care, particularly at low frequencies, as phase errors and high levels of electronic noise may be present, especially if double integration is used to obtain a displacement signal. Some accelerometers have in-built charge amplifiers and thus have a low impedance voltage output, which is easily amplified using standard low impedance voltage amplifiers. Figure 17: Basic configuration of strain Gage Acceleration Transducer. The minimum vibration level that can be measured by an accelerometer is dependent upon its sensitivity and can be as low as 10−4 m/s2. The maximum level is dependent upon size and can be as high as 106 m/s2 for small shock accelerometers. Most commercially available accelerometers at least cover the range 10−2 to 5×104 m/s2. This range is then extended at one end or the other, depending upon accelerometer type. The transverse sensitivity of an accelerometer is its maximum sensitivity to motion in a direction at right angles to its main axis. The maximum value is usually quoted on calibration charts and should be less than 5% of the axial sensitivity. Clearly, readings can be significantly affected if the transverse vibration amplitude at the measurement location is an order of magnitude larger than the axial amplitude. The frequency response of an accelerometer is regarded as essentially flat over the frequency range for which its electrical output is proportional to within ±5% of its mechanical input. The upper limit is generally just less than one third of the resonance frequency. The 128 Experimental Physics I resonance frequency is dependent upon accelerometer size and may be as low as 2,500 Hz or as high as 180 kHz. In general, accelerometers with higher resonance frequencies are smaller in size and less sensitive. When choosing an accelerometer, some compromise must always be made between its weight and sensitivity. Small accelerometers are more convenient to use; they can measure higher frequencies and are less likely to mass load a test structure and affect its vibration characteristics. However, they have low sensitivity, which puts a lower limit on the acceleration amplitude that can be measured. Accelerometers range in weight from miniature 0.65 grams for high-level vibration amplitude (up to a frequency of 18 kHz) on light weight structures, to 500 grams for low level ground vibration measurement (up to a frequency of 700 Hz). Thus, prior to choosing an accelerometer, it is necessary to know approximately the range of vibration amplitudes and frequencies to be expected as well as detailed accelerometer characteristics, including the effect of various types of amplifiers. Sources of Measurement Error Temperatures above 100ºC can result in small reversible changes in accelerometer sensitivity up to 12% at 200ºC. If the accelerometer base temperature is kept low using a heat sink and mica washer with forced air cooling, then the sensitivity will change by less than 12% when mounted on surfaces having temperatures up to 400ºC. Accelerometers cannot generally be used on surfaces characterized by temperatures in excess of 400ºC. Strain variation in the base structure on which an accelerometer is mounted may generate spurious signals. Such effects are reduced using a shear type accelerometer and are virtually negligible for piezoresistive accelerometers. Magnetic fields have a negligible effect on an accelerometer output, but intense electric fields can have a strong effect. The effect of intense electric fields can be minimized by using a differential pre-amplifier with two outputs from the same accelerometer (one from each side of the piezoelectric crystal with the accelerometer casing as a common earth) in such a way that voltages common to the two outputs are cancelled. This arrangement is generally necessary when using accelerometers near large generators or alternators. Waves and Sound 129 If the test object is connected to ground, the accelerometer must be electrically isolated from it, otherwise an earth loop may result, and producing a high level 50 Hz hum in the resulting acceleration signal. Sources of Error in the Measurement of Transients If the accelerometer charge amplifier lower limiting frequency is insufficiently low for a particular transient or very low frequency acceleration waveform, then the phenomenon of leakage will occur. This results in the waveform output by the charge amplifier not being the same as the acceleration waveform and errors in the peak measurement of the waveform will occur. To avoid this problem, the lower limiting frequency of the preamplifier should be less than 0.008/T for a square wave transient and less than 0.05/T for a half-sine transient, where T is the period of the transient in seconds. Thus, for a square wave type of pulse of duration 100 ms, the lower limiting frequency set on the charge amplifier should be 0.1 Hz. Another phenomenon, called zero shift, that can occur when any type of pulse is measured is that the charge amplifier output at the end of the pulse could be negative or positive, but not zero and can take a considerable time longer (up to 1000 times longer than the pulse duration) to decay to zero. Thus, large errors can occur if integration networks are used in these cases. The problem is worst when the accelerometers are being used to measure transient acceleration levels close to their maximum capability. A mechanical filter placed between the accelerometer and the structure on which it is mounted can reduce the effects of zero shift. The phenomenon of ringing can occur when the transient acceleration that is being measured contains frequencies above the useful measurement range of the accelerometer and its mounting configuration. The accelerometer mounted resonance frequency should not be less than 10/T, where T is the length of the transient in seconds. The effect of ringing is to distort the charge amplifier output waveform and cause errors in the measurement. The effects of ringing can be minimized by using a mechanical filter between the accelerometer and the structure on which it is mounted. Another phenomenon, called zero shift, that can occur when any type of pulse is measured is that the charge amplifier output at the end of the pulse could be negative or positive, but not zero and can take a considerable time longer (up to 1000 times longer than the pulse 130 Experimental Physics I duration) to decay to zero. Thus, large errors can occur if integration networks are used in these cases. The problem is worst when the accelerometers are being used to measure transient acceleration levels close to their maximum capability. A mechanical filter placed between the accelerometer and the structure on which it is mounted can reduce the effects of zero shift. The phenomenon of ringing can occur when the transient acceleration that is being measured contains frequencies above the useful measurement range of the accelerometer and its mounting configuration. The accelerometer mounted resonance frequency should not be less than 10/T, where T is the length of the transient in seconds. The effect of ringing is to distort the charge amplifier output waveform and cause errors in the measurement. The effects of ringing can be minimized by using a mechanical filter between the accelerometer and the structure on which it is mounted. Accelerometer Calibration In normal use, accelerometers may be subjected to violent treatment, such as dropping, which can alter their characteristics. Thus, the sensitivity should be periodically checked by mounting the accelerometer on a shaker table which either produces a known value of acceleration at some reference frequency or on which a reference accelerometer of known calibration may be mounted for comparison. Accelerometer Mounting Generally, the measurement of acceleration at low to middle frequencies poses few mechanical attachment problems. Use of a magnetic base usually limits the upper frequency bound to about 2 kHz. Beeswax may be used on surfaces that are cooler than 30ºC, for frequencies below 10 kHz. Thus, for the successful measurement of acceleration at high frequencies, some care is required to ensure (1) that the accelerometer attachment is firm, and (2) that the mass loading provided by the accelerometer is negligible. With respect to the former it is suggested that the manufacturer’s recommendation for attachment be carefully followed. With respect to the latter the following is offered as a guide. Let the mass of the accelerometer be ma grams. When the mass, ma, satisfies the appropriate equation which follows, the measured vibration level will be at most 3 dB below the unloaded level due to the mass loading by the accelerometer. For thin plates: and for massive structures: Waves and Sound 131 ma≤3.7×10−4 (ρcLh2/ƒ) (grams)(13) ma≤0.013 (ρcL2 Da/ƒ2) (grams)(14) In the above equations ρ is the material density (kg/m3), h is the plate thickness (mm), Da is the accelerometer diameter (mm), ƒ is the frequency (Hz) and cL is the ongitudinal speed of sound (m/s). For the purposes of Equations (13) and (14) it will be sufficient to approximate cL as E ρ As a general guide, the accelerometer mass should be less than 10% of the dynamic mass (or modal mass) of the vibrating structure to which it is attached. The effect of the accelerometer mass on any resonance frequency, ƒs, of a structure is given by: fm = fs ms ms + ma (15) where ƒm is the resonance frequency with the accelerometer attached, ma is the mass of the accelerometer and ms is the dynamic mass of the structure (often approximated as the mass in the vicinity of the accelerometer). One possible means of accurately determining a structural resonance frequency would be to measure it with a number of different weights placed between the accelerometer and the structure, plot measured resonance frequency versus added mass and extrapolate linearly to zero added mass. If mass loading is a problem, an alternative to an accelerometer is to use a laser Doppler velocimeter, especially if the frequency range of interest is below a few kHz. Piezo-resistive Accelerometers An alternative type of accelerometer is the piezo-resistive type, which relies upon the measurement of resistance change in a piezo-resistive element (such as a strain gauge) subjected to stress. Piezo-resistive accelerometers are less common than piezoelectric accelerometers and generally are less sensitive by an order of magnitude for the same size and frequency response. Piezo-resistive accelerometers are capable of measuring down to d.c. (or zero frequency), are easily calibrated, and can be used effectively with low impedance voltage amplifiers. 132 Experimental Physics I However, they require a stable d.c. power supply to excite the piezoresistive element (or elements). 3.5.2 Velocity Transducers Measurement of velocity provides an estimate of the energy associated with structural vibration; thus, a velocity measurement is often a useful parameter to quantify sound radiation. Velocity transducers are generally of three types. The least common is the non-contacting magnetic type consisting of a cylindrical permanent magnetic on which is wound an insulated coil. As this type of transducer is only suitable for relative velocity measurements between two surfaces or structures, its applicability to noise control is limited; thus, it will not be discussed further. The most common type of velocity transducer consists of a moving coil surrounding a permanent magnet. Inductive electromotive force (EMF) is induced in the coil when it is vibrated. This EMF (or voltage signal) is proportional to the velocity of the coil with respect to the permanent magnet. In the 10 Hz to 1 kHz frequency range, for which the transducers are suitable, the permanent magnet remains virtually stationary and the resulting voltage is directly proportional to the velocity of the surface on which it is mounted. Outside this frequency range the electrical output of the velocity transducer is not proportional to velocity. This type of velocity transducer is designed to have a low natural frequency (below its useful frequency range); thus it is generally quite heavy and can significantly mass load light structures. Some care is needed in mounting but this is not as critical as for accelerometers due to the relatively low upper frequency limit characterizing the basic transducer. The preceding two types of velocity transducer generally cover the dynamic range of 1 to 100 mm/s. Some extend down to 0.1 mm/s while others extend up to 250 mm/s. Sensitivities are generally high, of the order of 20 mV/mm s−1. Low impedance, inexpensive voltage amplifiers are suitable for amplifying the signal. Temperatures during operation or storage should not exceed 120ºC. A third type of velocity transducer is the laser Doppler velocimeter. Currently available instrumentation has a dynamic range typically of 80 dB or more. Instruments can usually be adjusted using different processing modules so that the minimum and maximum Waves and Sound 133 measurable levels can be varied, while maintaining the same dynamic range. Instruments are available that can measure velocities up to 20 m/s and down to 1 μm/sec (although not with the same Important processing electronics). 3.5.3 Instrumentation Systems Around 1830, Michael Faraday established that the reactions at each of the two electrode–electrolyte interfaces provide the “seat of emf” for the voltaic cell, that is, these reactions drive the current and are not an endless source of energy as was initially thought. The instrumentation system which is used in conjunction with the transducers just described, depends upon the level of sophistication desired. Overall or octave band vibration levels can be recorded in the field using a simple vibration meter. If more detailed analysis is required, a portable spectrum analyzer can be used. Alternatively, if it is preferable to do the data analysis in the laboratory, samples of the data can be recorded using a high quality FM or DAC tape recorder and replayed through the spectrum analyzer. This latter method has the advantage of enabling one to re-analyze data in different ways and with different frequency resolutions, which is useful when diagnosing a particular vibration problem. Units of Vibration It is often convenient to express vibration amplitudes in decibels. The International Standards Organization has recommended that the following units and reference levels be used for acceleration and velocity. Velocity is measured as a root mean square (r.m.s.) quantity in millimeters per second and the level reference is one nanometer per second (10−6 mm/s) The velocity level expression, Lν, is: Lν=20 log10(ν/νref); νref=10−6 mms−1(16) Acceleration is measured as an r.m.s. quantity in meters per second2 (m/s) and the level reference is one micrometer per second squared (10−6 m/s). The acceleration level expression, La, is: La=20 log10(a/aref) aref=10−6ms−2(17) Although there is no standard for displacement, it is customary to measure it as a peak to peak quantity in micrometers (μm) and use a level reference of one picometer (10−6mm). 134 Experimental Physics I The displacement level expression, Ld, is: Ld=20 log10(d/dref) dref=10−6 μm(18) When vibratory force is measured in dB, the standard reference quantity is 1 μN. The force level expression is then: Lƒ=20 log10(F/Fref) Fref=1 μN(19) Wave Function The wave function is the most fundamental concept of quantum mechanics. It was first introduced into the theory by analogy; the behavior of microscopic particles likes wave, and thus a wave function is used to describe them. Schrödinger originally regarded the wave function as a description of real physical wave. But this view met serious objections and was soon replaced by Born’s probability interpretation, which becomes the standard interpretation of the wave function today. According to this interpretation, the wave function is a probability amplitude, and the square of its absolute value represents the probability density for a particle to be measured in certain locations. However, the standard interpretation is still unsatisfying when applying to a fundamental theory because of resorting to measurement. In view of this problem, some alternative realistic interpretations of the wave function have been proposed and widely studied. There are in general two possible ways to interpret the wave function of a single quantum system in a realistic interpretation. One view is to take the wave function as a physical entity simultaneously distributing in space such as a field, and it is assumed by de Broglie-Bohm theory, many-worlds interpretation and dynamical collapse theories etc. For example, in de Broglie-Bohm theory the wave function is generally taken as an objective physical field, called Ψ-field, though there are various views on exactly what field the wave function is. The other view is to take the wave function as a description of some kind of ergodic motion of a particle (or corpuscle), and it is assumed by stochastic interpretation etc. The essential difference between a field and the ergodic motion of a particle lies in the property of simultaneity. The field exists throughout space simultaneously, whereas the ergodic motion of a particle exists throughout space in an essentially local way; the particle is still in one position at each instant, and it is only during a time interval that the ergodic motion of the particle spreads throughout space. Waves and Sound 135 It is widely expected that the correct realistic interpretation of the wave function can only be determined by future precise experiments. We will argue that the above two interpretations of the wave function can in fact be tested by analyzing the mass and charge density distributions of a quantum system, and the former has already been excluded by experimental observations. Moreover, a further analysis can also determine which kind of ergodic motion of particles the wave function describes. The plan of this paper is as follows. We first argue that a quantum system with mass m and charge Q, which is described by the wave function ψ (x,t), has effective mass and charge density distributions in space respectively. This argument is strengthened by showing that the result is also a consequence of protective measurement. We argue that the field explanation of the wave function entails the existence of an electrostatic self-interaction for the wave function of a charged quantum system, as the charge density will be distributed in space simultaneously for a physical field. This contradicts the predictions of quantum mechanics as well as experimental observations. Thus we conclude that the wave function cannot be a description of a physical field. This leads us to the second view that interprets the wave function as a description of the ergodic motion of particles. It is argued that the classical ergodic models, which assume continuous motion of particles, cannot be consistent with quantum mechanics, and thus they have been excluded. Further investigates the possibility that the wave function is a description of the quantum motion of particles, which is random and discontinuous in nature. It is shown that this new interpretation of the wave function provides a natural realistic alternative to the orthodox interpretation, and its implications for other realistic interpretations of quantum mechanics. How do Mass and Charge Distribute for a Single Quantum System? The mass and charge of a charged classical system always localize in a definite position in space at each moment. For a charged quantum system described by the wave function ψ (x,t), how do its mass and charge distribute in space then? We can measure the total mass and charge of the quantum system by gravitational and electromagnetic interactions and find them in some region of space. Thus the mass and 136 Experimental Physics I charge of a quantum system must also exist in space with a certain distributions if assuming a realistic view. Although the mass and charge distributions of a single quantum system seem meaningless according to the probability interpretation of the wave function, it should have a physical meaning in a realistic interpretation of the wave function such as de Broglie-Bohm theory. As we think, the Schrödinger equation of a charged quantum system under an external electromagnetic potential already provides an important clue. The equation is (20) where m and Q is respectively the mass and charge of the system, ϕ and A are the electromagnetic potential, is Planck’s constant divided by 2π , c is the speed of light. The electrostatic interaction term Qϕψ (x,t) in the equation seems to indicate that the charge of the quantum system distributes throughout the whole region where its wave function ψ (x,t) is not zero. If the charge does not distribute in some regions where the wave function is nonzero, then there will not exist any electrostatic interaction there. But the term Qϕψ (x,t) implies that there exists an electrostatic interaction in all regions where the wave function is nonzero. Thus it seems that the charge of the quantum system should distribute throughout the whole region where its wave function is not zero. Furthermore, since the integral is the total charge of the system, the charge density distribution in space will be . Similarly, the mass density can be obtained from the Schrödinger equation of a quantum system with mass m under an external gravitational potential V G: (21) The gravitational interaction term in the equation also indicates that the (passive gravitational) mass of the quantum system distributes throughout the whole region where its wave function ψ (x,t) is not zero, and the mass density distribution in space is . The above result can be more readily understood when the wave function is a complete realistic description of a single quantum system Waves and Sound 137 as in many-worlds interpretation and dynamical collapse theories. If the mass and charge of a quantum system does not distribute as above in terms of its wave function ψ (x,t) , then other supplement quantities will be needed to describe the mass and charge distributions of the system in space, while this obviously contradicts the premise that the wave function is a complete description. In fact, the dynamical collapse theories such as GRW theory already admit the existence of mass density. In addition, which takes the wave function as an incomplete description and admits supplement hidden variables, there are also some arguments for the above mass and charge density explanation. It was argued that since the Ψ-field depends on the parameters such as mass and charge, it may be said to be massive and charged. Brown, argued that properties sometimes attributed to the “particle” aspect of a neutron, for example. Mass and magnetic moment, cannot straightforwardly be regarded as localized at the hypothetical position of the particle in Bohm’s theory. They also argued that it is hard to understand how the Aharonov-Bohm effect is possible if that the charge of the electron which couples with the electromagnetic vector-potential is not co-present in the regions on all sides of the confined magnetic field accessible to the electron. One may object that de Broglie-Bohm theory and many-worlds interpretation seemingly never admit the above mass density explanation, and no existing interpretation of quantum mechanics including dynamical collapse theories endows charge density to the wave function either. As we think, however, protective measurement provides a more convincing argument for the existence of mass and charge density distributions. The wave function of a single quantum system, especially its mass and charge density, can be directly measured by protective measurement. 3.5.4 Protective Measurement Different from the conventional measurement, protective measurement aims at measuring the wave function of a single quantum system by repeated measurements that do not destroy its state. The general method is to let the measured system be in a non-degenerate eigenstate of the whole Hamiltonian using a suitable interaction, and then make the measurement adiabatically so that the wave function of the system neither changes nor becomes entangled with the measuring device appreciably. The suitable interaction is called the protection. 138 Experimental Physics I As a typical example of protective measurement, we consider a quantum system in a discrete nondegenerate energy eigenstate ψ(x). The protection is natural for this situation, and no additional protective interaction is needed. The interaction Hamiltonian for measuring the value of an observable A in the state is: (22) where P denotes the momentum of the pointer of the measuring device, which initial state is taken to be a Gaussian wave packet centered around zero. The time-dependent coupling g(t) is normalized to , where T is the total measuring time. In conventional von Neumann measurements, the interaction HI is of short duration and so strong that it dominates the rest of the Hamiltonian. As a result, the time evolution will lead to an entangled state: eigenstates of A with eigenvalues αi are entangled with measuring device states αi in which the pointer is shifted by these values . Due to the collapse of the wave function, the measurement result can only be one of the eigenvalues of observable A , say αi , with a certain probability pi . The expectation value of A is then obtained as the statistical average of eigenvalues for an ensemble of identical systems, namely . By contrast, protective measurements are extremely slow measurements. We let g(t) =1/T for most of the time T and assume that g(t) goes to zero gradually before and after the period T. In the limit T → ∞ , we can obtain an adiabatic process in which the system cannot make a transition from one energy eigenstate to another, and the interaction Hamiltonian does not change the energy eigenstate. As a result, the corresponding time evolution shifts the pointer by the expectation value < A > . This result strongly contrasts with the conventional measurement in which the pointer shifts by one of the eigenvalues of A. It should be stressed that T → ∞ is only an ideal situation, and a protective measurement can never be performed on a single quantum system with absolute certainty because of the tiny unavoidable entanglement. For example, for any given values of P and T, the energy shift of the above eigenstate, given by first-order perturbation theory, is Waves and Sound 139 (23) We can only obtain the exact expectation value < A > with a probability very close to one, and the measurement result can also be the expectation value < A >⊥ , with a probability proportional to 1/T2 , where ⊥ refers to the normalized state in the subspace normal to the initial state ψ(x) as picked out by first-order perturbation theory. Therefore, an ensemble, which may be considerably small, is still needed for protective measurements. Although a protective measurement can never be performed on a single quantum system with absolute certainty, the measurement is distinct from the standard one: in no stage of the measurement we obtain the eigenvalues of the measured variable. Each system in the small ensemble contributes the shift of the pointer proportional not to one of the eigenvalues, but to the expectation value. This essential novel point has been repeatedly stressed by the inventors of protective measurement. As we know, in the orthodox interpretation of quantum mechanics, the expectation values of variables are not considered as physical properties of a single system, as only one of the eigenvalues is observed in the outcome of the standard measuring procedure and the expectation value can only be defined as a statistical average of the eigenvalues. However, for protective measurements, we obtain the expectation value directly for a single system and not as a statistical average of eigenvalues for an ensemble. Since the expectation value of a variable can be directly measured for a single system, it must be a physical characteristic of a single system, not of an ensemble. This is a definite conclusion we can reach by the analysis of protective measurement. In the following we will show that the mass and charge density can be measured by protective measurement as expectation values of certain variable for a single quantum system, and thus it is the physical property of the system. Consider again a quantum system in a discrete nondegenerate energy eigenstate ψ(x). The interaction Hamiltonian for measuring the value of an observable An in the state assumes the same form as equation (24): (24) where An is a normalized projection operator on small regions Vn having volume n vn, which can be written as follows: 140 Experimental Physics I (25) Then a protective measurement of An will yield the following result: (26) It is the average of the density |ψ (x)|2 over the small region Vn . When vn →0 and after performing measurements in sufficiently many regions Vn we can find the whole density distribution |ψ (x)|2 . For a charged system with charge Q the density |ψ (x)|2 times the charge yields the effective charge density Q |ψ (x)|2. In particular, an appropriate adiabatic measurement of the Gauss flux out of a certain region will yield the value of the total charge inside this region, namely the integral of the effective charge density Q |ψ (x)|2 over this region. Similarly, we can measure the effective mass density of the system in principle by an appropriate adiabatic measurement of the flux of its gravitational field. Therefore, protective measurement shows that the mass and charge of a single quantum system described by the wave function ψ(x) is indeed distributed throughout space with effective mass density m |ψ (x)|2 and effective charge density Q |ψ (x)|2 respectively. For instance, in the double-slit experiment of an electron, a protective measurement of the charge density will show that there is a charge of e/2 in each of the slits when the electron is passing the slits. Although protective measurement strongly suggests a realistic interpretation of the wave function, it does not directly tell us what the wave function is. It may describe a physical wave or field, as suggested by the inventors of protective measurement. It is also possible that the wave function describes some kind of ergodic motion of particles, though this view was rejected by Aharonov and Vaidman. Correspondingly, the mass and charge density may result from a physical field or the ergodic motion of a particle. These two explanations are essentially different in that a field exists throughout space simultaneously, whereas the ergodic motion of a particle exists throughout space in an essentially local way. Why the Wave Function is not a Physical Field? If the wave function is a physical field, then its mass and charge density will simultaneously distribute in space. This has two disaster Waves and Sound 141 consequences at least. One is that charge will not be quantized; the total charge inside a very small region can be much smaller than an elementary charge for a single quantum system. This obviously contradicts the common expectation that charge should be quantized. But maybe our expectation needs to be revised. So this consequence is not fatal for the field explanation of the wave function. The other is that the wave function will not satisfy the superposition principle. For example, for the wave function of a single electron, different spatial parts of the wave function will have gravitational and electrostatic interactions, as these parts have mass and charge simultaneously. Let’s analyze the second consequence in more detail. Interestingly, the so-called Schrödinger-Newton equation, which was proposed for other purposes, just describes the gravitational self-interaction of the wave function. The equation for a single quantum system can be written as: (27) where m is the mass of the quantum system, V is an external potential, and G is Newton’s gravitational constant. Some experimental schemes have been also proposed to test its physical validity. As we will see, although such gravitational self-interactions cannot yet be excluded by experiments the existence of electrostatic self-interaction already contradicts experimental observations. (28) If there is also an electrostatic self-interaction, then the equation for a free quantum system with mass m and charge Q will be where k is the Coulomb constant. Note that the gravitational self-interaction is an attractive force, while the electrostatic self-interaction is a repulsive force. It has been shown that the measure of the potential strength of a gravitational self-interaction is for a free particle with mass m. This quantity represents the strength of the influence of selfinteraction on the normal evolution of the wave function; when ε2 ≈1 the influence will be significant. Similarly, for a free charged particle with charge Q, the measure of the potential strength of the electrostatic 142 Experimental Physics I self-interaction is . As a typical example, for a free electron with charge e, the potential strength of the electrostatic self-interaction will be . This indicates that the electrostatic self-interaction will have significant influence on the evolution of the wave function of a free electron. If such an interaction indeed exists, it should have been detected by precise experiments on charged microscopic particles. As another example, consider the electron in the hydrogen atom. Since the potential of its electrostatic self-interaction is of the same order as the Coulomb potential produced by the nucleus, the energy levels of hydrogen atoms will be significantly different from those predicted by quantum mechanics and confirmed by experimental observations. Therefore, the electrostatic self-interaction cannot exist for the wave function of a charged quantum system. Since the field explanation of the wave function entails the existence of such electrostatic self-interactions, it cannot be right, i.e. the wave function cannot be a description of a physical field. One may object to the above argument with the example of classical electromagnetic field. Electromagnetic field is a field, but it has no self-interaction. Thus a field does not require the existence of self-interaction. However, this is a common misunderstanding. The crux of the matter is that the non-existence of electromagnetic selfinteraction results from the fact that electromagnetic field itself has no charge. If the electromagnetic field had charge, then there would also exist electromagnetic self-interaction due to the nature of field, namely the simultaneous existence of its properties in space. In fact, although electromagnetic field has no electromagnetic self-interaction, it does have gravitational self-interaction; the simultaneous existence of energy densities in different spatial locations for an electromagnetic field must generate a gravitational interaction, though the interaction is too weak to be detected by current technology. One may further object that the superposition principle in quantum mechanics already prohibits the existence of the above self-interactions. But this is just the key point we use to argue against the field explanation of the wave function. Let’s state the argument more explicitly. If the wave function of a charged quantum system is a physical field, then the different spatial parts of this field will have gravitational and electrostatic interactions. But the superposition principle in quantum mechanics, which has been verified within astonishing precision, does Waves and Sound 143 not permit the existence of the remarkable electrostatic self-interaction. Therefore, the field explanation of the wave function is already refuted by the superposition principle of quantum mechanics. Why Classical Ergodic Models Fail If the wave function is not a description of a physical field, then exactly what does the wave function describe? This naturally leads us to the second view that takes the wave function as a description of some sort of ergodic motion of particles. On this view, the effective mass and charge density are formed by time average of the motion of a charged particle, and they distribute in different locations at different moments. At every instant, there is only a localized particle with mass and charge. Thus there will not exist any self-interaction for the wave function, and this view can be consistent with quantum mechanics and experimental observations. In fact, if the mass and charge density does not exist in different regions simultaneously as the field explanation holds, they can only be formed by the ergodic motion of a particle. As a result, the wave function must be a description of some sort of ergodic motion of particles. There are indeed some realistic interpretations of quantum mechanics that attempt to explain the wave function in terms of the ergodic motion of particles. A well-known example is the stochastic interpretation of quantum mechanics. Nelson derived the Schrödinger equation from Newtonian mechanics via the hypothesis that every particle of mass m is subject to a Brownian motion with diffusion coefficient h / 2m and no friction. In more technical terms, the quantum mechanical process is claimed to be equivalent to a classical Markovian diffusion process. On this interpretation, particles have continuous trajectories but no velocities, and the wave function is a statistical average description of their ergodic motion. However, it has been pointed out that the classical stochastic interpretations are inconsistent with quantum mechanics argued that the Schrödinger equation is not equivalent to a Markovian process, and the various correlation functions used in quantum mechanics do not have the properties of the correlations of a classical stochastic process. One must add by hand a quantization condition, as in the old quantum theory, in order to recover the Schrödinger equation, and thus the Schrödinger equation and the Madelung hydrodynamic equations are not equivalent. There is an empirical difference between the predictions of quantum mechanics and his stochastic mechanics when considering 144 Experimental Physics I quantum entanglement and nonlocality. In addition, it can be generally argued that the classical ergodic models that assume continuous motion of particles cannot be consistent with quantum mechanics. In order to see this let’s examine whether the continuous motion of particles can generate the charge density at the level of time average. Consider an electron in a one-dimensional box in an energy eigenstate such as the first excited state. Its wave function has a node at the center of the box, where its charge density is zero. The electron performs a very fast motion in the box. At a particular time the charge density is either zero (if the electron is not there) or singular (if the electron is inside the infinitesimally small region including the space point in question). But during a time interval, the motion of the electron will generate a charge density cloud with an effective charge density. The question is whether this density can assume the same form as e |ψ (x) |2. Since the effective charge density is proportional to the amount of time the electron spends in a given position, the electron must be in the left half of the box half of the time and in the right half of the box half of the time. But it can spend no time at the center of the box where the effective charge density is zero; in other words, it must move at infinite velocity at the center. Certainly, the appearance of infinite velocity or velocity faster than light may be not a fatal problem, as our discussion is entirely in the context of non-relativistic quantum mechanics and especially the infinite potential assumed in the example is also an ideal situation. However, it is hard to understand why the electron speeds up at the node and where the infinite energy required for the acceleration comes from. Moreover, the sudden acceleration of the electron near the node will result in large radiation, which is inconsistent with both the predictions of quantum mechanics and experimental observations. Maybe one can also assume in an ad hoc way that the accelerating electron does not radiate here in order to make the model be consistent with quantum mechanics and experimental observations. Let’s further consider a superposition of two energy eigenstate respectively limited in two widely-separated boxes. In this example, even if one assumes that the electron can move with an infinite velocity, it cannot move from one box to another due to the limitation of box walls. Therefore, any sort of continuous motion cannot generate the charge density e |ψ (x) |2 at the level of time average. One may still object that this is merely an artificial result of the idealization of infinite potential. But even in this ideal example, the model should also be able to generate the charge density by means of some sort of ergodic motion Waves and Sound 145 of the electron. In fact, there is a very similar situation in double-slit experiment. The wave function of a single electron passes through two channels that are well separated in space. The wave function disappears outside the channels for all practical purposes, and the electron can only move inside the channels (otherwise the electron will be detected outside the channels, which contradicts experimental observations). Again, a classical ergodic model cannot explain this experiment. There is a general objection to all classical ergodic models. Any classical ergodic model will inevitably introduce a finite ergodic time parameter, which is needed to generate the effective mass and charge density, because it must take a finite time for the particle to continuously move throughout all regions where the wave function is not zero. However, it can be argued that no finite time scale is permitted to exist for the ergodic motion. First of all, the existence of a finite time scale, denoted by Tc, is inconsistent with the standard quantum theory, as there is no such a time constant in the theory. Next, if there exists a time scale Tc , then when the measuring time T of protective measurement is shorter than Tc (i.e. T < Tc ), the measurement result will be not the expectation value of a variable such as charge density, as no whole time average can be obtained. This also contradicts the prediction of protective measurement. As an extreme example, consider a spatial superposition state ψ L +ψ R, where ψ L and ψ R are Gaussian wave packets and their centers are well separated in space. When T < Tc, the particle has no enough time to move throughout the whole regions including both L and R. Then the result of a protective position measurement will be not the expectation value of ψ L +ψ R, but the eigenvalue corresponding to ψL or ψR. Moreover, the results distribution is also different from that predicted by protective measurement. When T < Tc , the distribution of position measurement results will concentrate near L and R, while according to protective measurement, the distribution should concentrate near the midpoint between L and R. In fact, for protective measurement, during any period of time the pointer of the measuring device always shifts by an amount proportional to the expectation value of the measured variable, rather than to one of its eigenvalues. Thus the expectation value can be associated with any short period of time. Certainly, pointer shifts during short time intervals are practically unobservable since they are much smaller than the uncertainty, and only when the total shift accumulated during the whole period of measurement is much larger than the width of the initial distribution it becomes observable. 146 Experimental Physics I Therefore, we conclude that the continuous ergodic motion of particles cannot generate the effective mass and charge density measurable by protective measurement, and the classical ergodic models cannot be consistent with quantum mechanics. As a result, the wave function cannot be a description of the continuous ergodic motion of particles. The Wave Function as a Description of Quantum Motion of Particles The failure of classical ergodic models does not exclude all possible ergodic motion of particles. We will argue that another different kind of motion – random discontinuous motion can naturally generate the effective mass and charge density measurable by protective measurement and what the wave function describes is probably such quantum motion of particles, which is essentially discontinuous and random. If the motion of a particle is not continuous but discontinuous and random, and the probability density of the particle being in certain positions is proportional to the square of the absolute value of its wave function at every instant, then the particle can readily move throughout all possible regions where the wave function is nonzero during an arbitrarily short time interval near a given instant. This will solve the problems plagued by the classical ergodic models. The discontinuous ergodic motion requires no existence of a finite time scale. A particle undergoing discontinuous motion can also move from one region to another spatially separated region, no matter whether there is an infinite potential wall between them. Besides, discontinuous motion can also solve the problems of infinite velocity and accelerating radiation. The reason is that no classical velocity and acceleration can be defined for discontinuous motion, and energy and momentum will require new definition and new understanding as in quantum mechanics. Thus it seems that the discontinuous ergodic motion of particles can in principle generate the effective mass and charge density measurable by protective measurement, and thus the wave function is probably a description of such random discontinuous motion of particles. In some sense, the above interpretation of the wave function seems to be an inevitable consequence of protective measurement. According to protective measurement, a charged quantum system has effective Waves and Sound 147 mass and charge density distributing in space, proportional to the square of the absolute value of its wave function. There are two possible ways to explain the existence of the mass and charge density; one is to take the wave function as a description of some kind of physical field, and the mass and charge density of this field exists throughout space simultaneously, the other is to take the wave function as a description of some sort of ergodic motion of particles, and the effective mass and charge density formed by such motion exists throughout space in an essentially local way. The first view has been rejected because it entails the existence of a remarkable electrostatic self-interaction that contradicts experimental observations. Thus the wave function can only be a description of some sort of ergodic motion of particles. Since the classical ergodic models have also been excluded, the ergodic motion of particles cannot be continuous and must be essentially discontinuous. Besides, when considering the randomness of the results of conventional quantum measurement, such motion must be also random. Therefore, what the wave function describes can only be random discontinuous motion of particles. The wave function is a (complete) description for the motion of particles, we can reach the random discontinuous motion in a more direct way, independent of the analysis of protective measurement. If the wave function ψ(x,t) is a description of the state of motion for a single particle, then the quantity |ψ (x,t)|2 dx not only gives the probability of the particle being found in an infinitesimal space interval dx near position x at instant t, but also gives the objective probability of the particle being there. This accords with the well-accepted assumption that the probability distribution of the measurement outcomes of a property is the same as the actual distribution of the property in the measured state. Then at instant t the particle may appear in any location where the probability density |ψ (x,t)|2 is nonzero, and during an infinitesimal time interval near instant t , the particle will move throughout the whole space where the wave function ψ(x,t) spreads, though it is still in one position at each instant. Moreover, the density distribution of its positions is equal to the probability density |ψ (x,t)|2. Obviously, this kind of motion is essentially random and discontinuous. 148 Experimental Physics I Figure 18: The description of RDM of a single particle. The strict mathematical description of random discontinuous motion (RDM henceforth) can be obtained by using the measure theory. Consider the motion state of a single particle in finite intervals Δt and Δx near a space-time point (ti , xj ) as shown in Fig. 17. For RDM, the position of the particle forms a random discontinuous point set in the whole space for the time interval Δt near the instant ti. Accordingly, there is a local discontinuous point A random discontinuous point set can be defined as a set of points (t, x) in continuous space and time, for which the function x(t) is discontinuous and random at all instants. The definition of a discontinuous function is as follows. Suppose A is an open set in ℜ (say an interval A = (a,b) , or A = ℜ ), and f : A→ ℜ is a function. Then f is discontinuous at x ∈ A, if f is not continuous at x. Note that a function f: A→ ℜ is continuous if and only if for every x ∈ A and every real number ε > 0 , there set in the space interval Δx near the position xj. The local discontinuous point set represents the motion state of the particle in the finite intervals Δt and Δx near the space-time point (ti, xj ). It is projection in the t-axis, namely the corresponding dense instant set in the time interval Δt . Let W be the discontinuous trajectory or worldset of the particle and Q be the square region [xj, xj + Δx ] × [ti , ti + Δt ]. The dense instant set can be denoted by π t(W IQ) ⊂ ℜ, where πt is the projection on the t-axis. According to the measure theory, we can define the Lebesgue measure: (29) Since the sum of the measures of all such dense instant sets in the time interval Δt is equal to the length of the continuous time interval Δt, we have: Waves and Sound 149 (30) Then we can define the measure density: (31) The limit exists for a random discontinuous point set. This provides a strict mathematical description of the point distribution situation for the above local discontinuous point set. We call this measure density position measure density. Since the local discontinuous point set represents the motion state of the particle, the position measure density ρ(x,t) will be a descriptive quantity for the RDM of a single particle. It represents the relative frequency of the particle appearing in an infinitesimal space interval dx near position x during an infinitesimal interval dt near instant t. From equation (31) we can see that ρ(x,t) satisfies the normalization relation, namely . Furthermore, we can define the position measure flux density j(x,t) through the relation j(x,t) = ρ(x,t) ν (x,t) , where ν (x,t) is the velocity of the local discontinuous point set. It describes the change of the position measure density with time. Due to the conservation of measure, ρ(x,t) and j(x,t) satisfy the following equation: (32) The position measure density ρ(x,t) and the position measure flux density j(x,t) provide a complete description for the RDM of a single particle. It is very natural to extend the description of the motion of a single particle to the motion of many particles. For the RDM state of N particles, we can define a joint position measure density . This represents the relative probability of the situation in which particle 1 is in position x1, particle 2 is in position x2,… , and particle N is in position xN. In a similar way, we can define the joint position measure flux density . It satisfies the joint measure conservation equation. (33) 150 Experimental Physics I When these N particles are independent, the joint position measure density can be reduced to the direct product of the position measure density of each particle, namely . It is worth noting that the joint position measure density and the joint position measure flux density are not defined in the threedimensional real space, but defined in the 3N-dimensional configuration space. As we will see later, these two quantities can further constitute the wave function, and the many-body wave functions thus defined also live on the 3N-dimensional configuration space. With respect to the RDM of a particle, the motion of the particle is completely discontinuous and random. The probability density for the particle to appear at position x at instant t is ρ(x,t) . There is no evolution law for the position state of a particle, and the trajectory function x(t) is random and discontinuous at every instant. However, the discontinuity of RDM is absorbed into the motion state of a particle, which is defined during an infinitesimal time interval, by the descriptive quantities of position measure density ρ(x,t) and position measure flux density j(x,t). Therefore, the evolution law for the motion state of a particle will contain no discontinuities and can be a continuous equation. By assuming that the nonrelativistic evolution equation of RDM is the Schrödinger equation, the wave function ψ(x,t) can be uniquely expressed by the position measure density ρ(x,t) and the position measure flux density j(x,t): (34) where . Since ρ(x,t) and j(x,t) provide a complete description of the RDM of a single particle, the wave function ψ(x,t) also provides a complete description of the RDM of a single particle. The new interpretation of the wave function in terms of RDM of particles can be taken as a natural realistic alternative to the orthodox view. According to the standard probability interpretation of the wave function, the square of the absolute value of a N-particle wave function, which can be denoted by , gives the probability of particle 1 being found in the infinitesimal interval dx1 near position x1 and particle 2 being found in the infinitesimal interval dx2 near position x2, …, and particle N being found in the infinitesimal interval dxN near position xN. By contrast, according to the new interpretation, the square Waves and Sound 151 of the absolute value of the wave function not only gives the probability of a particle being found in certain locations, but also gives the objective probability of the particle being there. For example, also represents the objective probability of particle 1 being in the infinitesimal interval dx1 near position x1 and particle 2 being in the infinitesimal interval dx2 near position 2 x2, … , and particle N being in the infinitesimal interval dxN near position xN. Certainly, the transition process from “being” to “being found”, which is closely related to the notorious quantum measurement problem, also needs to be explained. Properties of the Wave Function The physical interpretation of the wave function in terms of a probability implied that a function needs to satisfy some conditions, before it is an acceptable wave function. We will now look at some more properties of the solutions to the Schr¨odinger equation. For simplicity we will restrict our attention to one-dimensional spatial problems, although most results are applicable to higher dimensional problems. The wave function is the solution of the eigenvalue equation where is the Hamiltonian operator and has the form Over and above the conditions we will now show that, for finite dψ potentials, the first derivative of the wave function, dx should also be single valued and continuous. To this end we rewrite the Schrödinger equation in the form dψ A discontinuity in dx would mean that the LHS of this expression would become infinite, which for finite V is unacceptable. 152 Experimental Physics I We will analyze the one-dimensional Schrödinger equation in slightly greater detail, which will allow us to draw some other general conclusions about the wave function. Consider a potential of the form given in the figure below. We will consider four different regions and will focus our attention on the Schrödinger equation written in the form given in equation. E < Vmin In this region V −E is always positive, so that the second derivative of the wave function, which represents the concavity, has the same sign as the wave function. This may be a good time to step aside and remind ourselves some simple things about how one sketches functions from a knowledge of the derivatives. The derivative of a function y = f(x), is the rate at which y changes with respect to x. It defines the slope of the function’s graph at x and provides an estimate of how much y changes when x is changed by a small amount. If a function has a derivative over an interval, then it is continuous over the interval and its graph over the interval is unbroken. Even more information about the graph of a differentiable function can be obtained from a knowledge of where the derivative is positive, negative, and zero. Suppose that over an interval a function y = f(x) has derivative at every point x, then f increases in the interval if f ′(x) > 0 and decreases if ( ) . Moreover, if f ′ changes continuously from positive to negative as we move from left to right through a point c, then the value of f at c is a local maximum of f. But, a horizontal tangent does not imply a maximum or a minimum. Take the function y = x3 which crosses the horizontal tangent at x = 0 but keeps on rising. If you look at the function closely, however, we find that the portions of the curves in the region (−∞, 0) and (0, ∞) turn in different ways. To the left of the origin the curve turns to the right, while to the right of the origin it turns to the left. In other words, the left portion “bends downward” while the right portion “bends upward.” More rigorously we would say that the curve is concave down on the interval (−∞, 0) and concave up on (0, ∞). The direction of concavity is determined by the sign of f ′′ being concave down if f ′′ < 0 and concave up if f ′′ > 0. A point on the curve where the concavity changes from up to down or vice versa is called a point of inflection, f ′′ = 0. f′ x < 0 Waves and Sound 153 Figure 19: A typical one-dimensional potential showing a minimum at x0. The potential has two classical turning points at x1 and x2 when the Vmin < E < V− and a turning point x3 when V− < E < V+. Coming back to the problem at hand, from the discussion in the last paragraph we now know that function will be concave upwards when ψ > 0 and concave downwards when ψ < 0 as is represented in Fig. 20. Since V − E is positive for all x, it is clear that a solution which remains finite everywhere cannot be found. Figure 20: The curvature of the wavefunction when the energy is below Vmin is of the same sign as the wavefunction and is hence concave upwards if ψ is > 0 and concave downwards if ψ is < 0. In fact, |ψ(x)| grows without limit as x → +∞ or/and as x → −∞. The best we could do is to select from among the two linearly independent solutions a function which approaches the x-axis asymptotically either on the left or on the right, but then this solution will necessarily ‘blow up’ on the other side. We conclude therefore that there is no acceptable solution of the Schr¨odinger equation when E < V (x) for all x. classically also this is not possible since this would correspond to negative kinetic energy. Vmin < E < V− This region can be broken up into three parts: left of the classical turning point x1, between the two turning points, and to the right of the turning point x2. The classical turning points are those points at which 154 Experimental Physics I E = V, that is points where the kinetic energy becomes zero. Classically at these points the particle would stop and turn around and no classical motion is possible to the left of x1 and to the right of x2. In the first and the third parts, V − E > 0, and hence the curvature of the wavefunction is similar to that in region 0 and is represented by Fig. 21. This implies that we cannot have oscillatory solutions in these two parts. Between the two turning points V − E < 0 and hence the curvature is opposite to the sign of the wavefunction. The function is concave downwards when ψ > 0 and concave upwards when ψ < 0. The curvature in this region is represented in Fig. 21. Thus in this part the wavefunction will exhibit oscillatory behavior. For some values of the energy, say E’, the oscillatory solution between the two turning points will connect smoothly with the asymptotic solutions to the left of x1 and to the right of x2 as in the middle part of Figure 21. For values of the energy either greater or lesser than E’ the solutions are diverged to +∞ or −∞ when x → ∞. It is important to realize that this smooth connection to give physically acceptable solutions called eigenfunctions only happens for some values of energy, which are called the eigenvalues. The number of such energies for which reasonable solutions exist will depend on the characteristics of the potential. Finally, we observe that we have obtained a fundamental result: the quantization of the bound state energies, the determination of which appears in Schrödinger’s mechanics as an eigenvalue problem. Figure 21: For values of the energy greater or less than the eigenvalue E’ the solutions of the Schrodinger equation give physically unreasonable results. Note that there may be a number of E’s for which physically acceptable results are obtained. V − < E < V+ Waves and Sound 155 In this energy range we note there is only one classical turning point, at x = x3. No classical motion is allowed for x > x3 or else it would be reflected at x3. The classical particle is unbound since it can move in an infinite region of the x-axis. Quantum mechanically, we see that for x < x3 there are two linearly independent solutions (since we are solving a second order differential equation) both of which are oscillatory (reasons evident by looking at Fig. 21), one corresponding to the particle moving to the left and the other to it moving to the right. For values of x > x3 also there will be two solutions, one which tends to zero as x → ∞ and the other which is unbounded as x approaches ∞. Obviously the only physically acceptable solution is the former. The continuity conditions of ψ and dψ/dx at x3 yields two conditions which determine a unique linear combination of the two independent oscillatory solutions in the region x < x3. It is important to note that in this region ’smooth matching’ at x = x3 is possible for every energy. We conclude that in this interval the allowed energies form a continuum. E > V+ In this region there are no classical turning points because V − E is negative for all values of x. For reasons discussed with regard to Fig. 21 we expect the solutions to be oscillatory. There are two linearly independent solutions corresponding to the two directions that the particle could be moving for every value of the energy. Thus in this region the energy values are continuous and doubly degenerate. Infinite Potential Energy So far our discussions have been restricted to the case where the potential energy was finite. We would like to look finally at the situation when the potential energy becomes infinite. Indeed, because a particle cannot have infinite potential energy, it cannot penetrate into a region of space where V = ∞. We focus our attention on the Schr¨odinger equation written in the form given in equation 1. Both E and ψ on the RHS are finite while the LHS has one term that is infinite. This equality can only hold when ψ is identically zero, which implies that the wavefunction goes to zero when the potential becomes infinite. We also note that since V exhibits a second order discontinuity (infinite jump) across the border, d2ψ/dx2 will also exhibit this discontinuity, so that dψ/dx will be discontinuous there. One last thing that we will look at is the relation that exists between the energy of a state and the number of nodes. Even without carrying 156 Experimental Physics I out detailed calculations, it is often possible to make general conclusions concerning the ordering of the states by considering their nodal structure. We consider a one-dimensional potential, one-particle system with the Hamiltonian where and V (x) is some function of x depending only upon the spatial coordinates. If the potential supports the existence of bound states we can show that: • • The ground state wave function does not change sign (has no nodes) The bound state with n sign changes has a lower energy than the state with n + 1 sign changes. Consider first the functions ψ0 and ψ1, which are the Eigen states of , with zero and one node respectively and energies E0 and E1 respectively. Let the boundaries of the system be a and b and let ψ1 undergo a zero-crossing at c. Consider the region a < x < c so that both ψ0 and ψ1 are positive. From the eigenvalue equation we then have Taking the difference of these two expressions we get The Hermiticity of implies that the term in parenthesis is zero (the integrand is generally not zero). To estimate the sign of E1 − E0, multiply by ψ1ψ0, then integrate from a to c. We then obtain: where Waves and Sound 157 Consider the first term in B. Integrating this by parts we get Combining these results we obtain Because we obtain that is, Thus the node less wavefunction has a lower energy than the wave function with one node. A similar proof shows that ψ0 has a lower energy than any wave function with more than one node. Hence the ground state of the system has no nodal points. 158 Experimental Physics I Similarly, the above proof can be applied to the comparison of En and En+1, that is, the energies for wave functions having n and n + 1 nodes, respectively. The result is that and hence the eigenstates of a system have energies increasing in the same sequence as the number of nodes. Formal Properties of Wave Functions It is very useful to have a broader, more precise framework for analyzing quantum properties. This is done in terms of Hilbert spaces. Hilbert Space of Linear Vectors Hilbert space is a generalization of Euclidian space to finite or infinite number of dimensions and real or complex vectors. It is employed for the mathematical description of quantum mechanics. The description of quantum states as vectors in Hilbert space gives the model an algebraic structure with means to measure distances, lengths and angles. Furthermore, it guarantees the use of the techniques of calculus on functions in H. Every Hilbert space H has an orthonormal basis. This means that any vector a in an n-dimensional Hilbert space can be constructed from n linear independent basis vectors ei with a normalized absolute value of 1. The basis is required to be complete with no points in space missing. There exists an inner product in H, which associates a pair of vectors a and b with a real or complex number. The inner product is linear in its first argument and turns into its complex conjugate when the argument order is reversed. The inner product of a vector a with itself is always larger than 0 and vanishes if a is the null vector. A norm is defined for a Hilbert space, which allows for the measurement of a vector’s length, the distance between points, and Waves and Sound 159 the angle between two vectors. Hilbert space is therefore both an inner product space and a complete metric space. For real numbers the inner product is the multiplication of two numbers, for complex numbers the inner product is the multiplication of one number z1 with the complex conjugate of the second number, z2*. In Euclidian space this operation is the dot product. In Euclidian geometry, the three Cartesian coordinates ex, ey and ez span a three dimensional Hilbert space. In quantum mechanics the n eigenstates of a quantum mechanical system span an n-dimensional Hilbert space, where the dimensionality of the Hilbert space is equal to the number of eigenstates. n may be infinite. For isolated spins the dimensionality of the Hilbert space is finite, 2 in the case of single spins 1/2 and 4 in the case of two coupled spins. Interactions with the surroundings, i.e. in spin relaxation, increases the number of possible states. Relaxation is therefore often treated in a semi-classical approximation. Each normalized basis vector corresponds to a pure eigenstate, which is fully populated. If the system is in a superposition state, the vector components are the coefficients in a linear combination of eigenstates. The absolute value of a vector must always be unity to fulfil the requirements of quantum mechanics that the described object exists. Coherences are superposition’s, in which the coherent states are equally populated and running in phase. The phase factor is generally a complex number. 160 Experimental Physics I The complex vector components of mixed states do not necessarily have a straightforward, graphical interpretation, but fulfil the eigenvalue equations for the respective cartesian operators. The metric of Hilbert space makes it possible to measure observables of the quantum mechanical state, as well as the probabilities with which they are measured. Liouville Space Since they work on vectors in a complex metric space, each operator has a representation as a square, hermitian matrix. Operators that work on vectors in Hilbert space can in turn be subject to transformations. The density operator for example, which gives the probability density for a state in an ensemble, evolves over time due to precession, coupling and relaxation. Transformations of the operators of Hilbert space form an analogous algebra called Liouville space with super operators working on the operators in H. The super operators also have a matrix representation that can be found by turning the n x n operator matrices column-wise into a single column vector. The super operators are n2 x n2 matrices. The application of a super operator on an operator vector in Liouville space corresponds to a two-sided matrix multiplication in Hilbert space. The Hamiltonian in Liouville space can be constructed by taking the commutator of the Hamiltonian in Hilbert space with the identity matrix. Super operators are denoted by a double hat. Vectors in Liouville space are written in analogy to the bra–ket notation with round brackets. The equation of motion in super operator space is called the Liouvillevon Neumann equation. Waves and Sound 161 Hermitian Operators Most operators in quantum mechanics are of a special kind called Hermitian. An operator is called Hermitian when it can always be flipped over to the other side if it appears in an inner product: f Ag = Af g always iff A is Hermitian That is the definition, but Hermitian operators have the following additional special properties: • • • They always have real eigenvalues, not involving i= −1 . (But the eigenfunctions, or eigenvectors if the operator is a matrix, might be complex.) Physical values such as position, momentum, and energy are ordinary real numbers since they are eigenvalues of Hermitian operators {N.3}. Their eigenfunctions can always be chosen so that they are normalized and mutually orthogonal, in other words, an orthonormal set. This tends to simplify the various mathematics a lot. Their eigenfunctions form a complete set. This means that any function can be written as some linear combination of the eigenfunctions. (There is a proof in derivation {D.8} for an important example. But see also {N.4}.) In practical terms, it means that you only need to look at the eigenfunctions to completely understand what the operator does. In the linear algebra of real matrices, Hermitian operators are simply symmetric matrices. A basic example is the inertia matrix of a solid body in Newtonian dynamics. The orthonormal eigenvectors of the inertia matrix give the directions of the principal axes of inertia of the body. An orthonormal complete set of eigenvectors or eigenfunctions is an example of a so-called “basis.” In general, a basis is a minimal set of vectors or functions that you can write all other vectors or functions in terms of. ˆˆ For example, the unit vectors i, j, and k̂ are a basis for normal threedimensional space. Every three-dimensional vector can be written as a linear combination of the three. The following properties of inner products involving Hermitian operators are often needed, so they are listed here: 162 Experimental Physics I ∗ If A is Hermitian : g Af = f Ag , f Af is real. The first says that you can swap f and g if you take the complex conjugate. (It is simply a reflection of the fact that if you change the sides in an inner product, you turn it into its complex conjugate. Normally, that puts the operator at the other side, but for a Hermitian operator, it does not make a difference.) The second is important because ordinary real numbers typically occupy a special place in the grand scheme of things. (The fact that the inner product is real merely reflects the fact that if a number is equal to its complex conjugate, it must be real; if there was an i in it, the number would change by a complex conjugate.) 3.6 STANDING WAVES ON STRINGS To observe standing waves on a stretched string and to verify the formula relating the wave speed to the tension and mass per length of the string. Figure 22: String vibrator, string, set of slotted weights, weight holder, pulley and clamp, meter stick, analytical balance. 3.6.1 Theory Standing waves are produced by the interference of two waves of the same wavelength, speed of propagation and amplitude, travelling in opposite directions through the same medium. If one end of a light, flexible string is attached to a vibrator and the other end passes over a fixed pulley to a weight holder, the waves travel down the string to the pulley and are then reflected, producing a reflected wave moving in the opposite direction. The vibration of the string is then a composite motion resulting from the combined effect of the two oppositely directed waves. If the proper relationship exists between the frequency, the length and the tension, a standing wave is produced and when the conditions are such as to make the amplitude of the standing wave a maximum, the Waves and Sound 163 system is said to be in resonance. A standing wave has points of zero displacement (due to destructive interference) and points of maximum displacement (due to constructive interference). The positions of no vibration are called nodes (N) and the positions of maximum vibration are called antinodes (A). The segment between two nodes is called a loop. Standing waves with one, two, three and four loops are given below. The solid line represents the form of the string at an instant of maximum displacement and the dotted line represents the configuration one half-period later when the displacements are reversed. In each case λ = 2l, where λ is the wavelength and l are the distance between two nearby nodes. For a string with both ends fixed the allowed wavelengths for standing waves can only take fixed values related to the length L of the string, as can be seen in the figure above. If one changes the tension in a vibrating string, the number of loops between the ends of the string change. As a result the distance between neighboring nodes changes, thus producing a change in wavelength. The speed of the wave can be obtained if the frequency f is known and the wave length λ has been measured: v = λf. The frequency is fixed by the string vibrator; the wavelength can only take on fixed values related to the length of the string, as shown in the figure above. Thus, standing waves can only exist for particular values of v that is controlled in this experiment by the tension of the string. The velocity of the wave is given by the Mercenne’s law: (35) 164 Experimental Physics I where m is the mass per length of the string and T is the tension. The tension of the string (in newtons) equals the total hanging mass M times the gravitational acceleration g = 9.8 m/s2 , that is, T = Mg. The main objective of the work is to compare the experimental value of the wave speed given by Eq. (1) and its theoretical value of Eq. (35). Experimental Setup A string is attached to a vibrator made of steel and then passed over a small pulley. The coil producing the alternating magnetic field acting on the vibrator is being fed by the standard ac current with a fixed frequency of 60 Hz. The attraction force exerted on the vibrator is proportional to the square of the magnetic field and thus of the electric current in the cirquit. As the result, the vibrator is vibrating at the double frequency, 120 Hz. The weight on the hanger has to be adjusted to achieve the tension T and thus the wave speed v at which a standing wave is clearly visible. As the range of the tension is limited, not all kinds of standing waves can be observed without changing the string length L. For instance, to observe lower overtones, the tension T can be increased or L can be decreased. The best way to measure the wave length λ is to measure the distance d between the end of the string at the pulley (where there is a node) and the node closest to the vibrator and count the number n of loops within this region. Then Keep in mind that there is no node directly at the vibrator, thus using L to obtain λ as shown in the figure above will result in errors. Procedure 1. Measure the length of the loose string (not the one attached to the vibrator) and then measure its mass using the analytical balance to the nearest milligram. Calculate the mass per unit lengh m for the string. 2. Suspend a light weight holder from the string and adjust the load until the string vibrates with maximum amplitude. Record the load in kilograms, including the weight of the holder. Measure λ as explained above and record it in the table together with the number of loops you observe. 3. Repeat the observations with load (and maybe the string length) adjusted to give other numbers of loops. Take measurements for at least three different numbers of loops. Waves and Sound 165 Data f= (frequency) Length= m Mass= kg m=mass/length= kg/m (length of loose string) (mass of loose string) (mass per length of string) Analysis 1) Calculate the mass per unit length and express it in kg/m. 2) Calculate the velocity of the wave on the string using equations (1) and (2) and compare the results. Calculate the percent discrepancy. Fill all of this in the data table. 3) In your conclusion discuss your results. 4) If the frequency of the vibrator were 240 Hz, calculate theoretically how much tension is necessary to produce a standing wave of two loops. Consider that the string is fixed at its two ends, has a length of 1 meter and has the same mass per length as determined before. 3.7 TO STUDY LONGITUDINAL SOUND WAVES CREATED IN AN AIR COLUMN OF VARIABLE LENGTH Sound is a mechanical wave that results from the back-and-forth vibration of the particles of the medium through which the sound wave is moving. If a sound wave is moving from left to right through air, then particles of air will be displaced both rightward and leftward as the energy of the sound wave passes through it. The motion of the particles is parallel (and anti-parallel) to the direction of the energy transport. This is what characterizes sound waves in air as longitudinal waves. 3.7.1 Compressions and Rarefactions A vibrating tuning fork is capable of creating such a longitudinal wave. As the tines of the fork vibrate back and forth, they push on neighboring 166 Experimental Physics I air particles. The forward motion of a tine pushes air molecules horizontally to the right and the backward retraction of the tine creates a lowpressure area allowing the air particles to move back to the left. Because of the longitudinal motion of the air particles, there are regions in the air where the air particles are compressed together and other regions where the air particles are spread apart. These regions are known as compressions and rarefactions respectively. The compressions are regions of high air pressure while the rarefactions are regions of low air pressure. The diagram below depicts a sound wave created by a tuning fork and propagated through the air in an open tube. The compressions and rarefactions are labeled. labeled. The wavelength of a wave is merely the distance that a disturbance travels along the medium in one complete wave cycle. Since a wave repeats its pattern once every wave cycle, the wavelength is sometimes referred to as the length of the repeating patterns - the length of one complete wave. For a transverse wave, this length is commonly measured from one wave crest to the next adjacent wave crest or from one wave trough to the next adjacent wave trough. Since a longitudinal wave does not contain crests and troughs, its wavelength must be measured differently. A longitudinal wave consists of a repeating pattern of compressions and rarefactions. Thus, the wavelength is commonly measured as the distance from one compression to the next adjacent compression or the distance from one rarefaction to the next adjacent rarefaction. Waves and Sound 167 What is a Pressure Wave? Since a sound wave consists of a repeating pattern of high-pressure and low-pressure regions moving through a medium, it is sometimes referred to as a pressure wave. If a detector, whether it is the human ear or a man-made instrument, were used to detect a sound wave, it would detect fluctuations in pressure as the sound wave impinges upon the detecting device. At one instant in time, the detector would detect a high pressure; this would correspond to the arrival of a compression at the detector site. At the next instant in time, the detector might detect normal pressure. And then finally a low pressure would be detected, corresponding to the arrival of a rarefaction at the detector site. The fluctuations in pressure as detected by the detector occur at periodic and regular time intervals. In fact, a plot of pressure versus time would appear as a sine curve. The peak points of the sine curve correspond to compressions; the low points correspond to rarefactions; and the “zero points” correspond to the pressure that the air would have if there were no disturbance moving through it. The diagram below depicts the correspondence between the longitudinal nature of a sound wave in air and the pressure-time fluctuations that it creates at a fixed detector location. The above diagram can be somewhat misleading if you are not careful. The representation of sound by a sine wave is merely an attempt to illustrate the sinusoidal nature of the pressure-time fluctuations. Do not conclude that sound is a transverse wave that has crests and troughs. Sound waves traveling through air are indeed longitudinal waves with compressions and rarefactions. As sound passes through air (or any fluid medium), the particles of air do not vibrate in a transverse manner. Do not be misled - sound waves traveling through air are longitudinal waves. 168 Experimental Physics I 3.7.2 Open-End Air Columns Many musical instruments consist of an air column enclosed inside of a hollow metal tube. Though the metal tube may be more than a meter in length, it is often curved upon itself one or more times in order to conserve space. If the end of the tube is uncovered such that the air at the end of the tube can freely vibrate when the sound wave reaches it, then the end is referred to as an open end. If both ends of the tube are uncovered or open, the musical instrument is said to contain an openend air column. A variety of instruments operate on the basis of openend air columns; examples include the flute and the recorder. Even some organ pipes serve as open-end air columns. 3.7.3 Standing Wave Patterns for the Harmonics As has already been mentioned, a musical instrument has a set of natural frequencies at which it vibrates at when a disturbance is introduced into it. These natural frequencies are known as the harmonics of the instrument; each harmonic is associated with a standing wave pattern. In Lesson 4 of Unit 10, a standing wave pattern was defined as a vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source in such a manner that specific points along the medium appear to be standing still. In the case of stringed instruments (discussed earlier), standing wave patterns were drawn to depict the amount of movement of the string at various locations along its length. Such patterns show nodes - points of no displacement or movement - at the two fixed ends of the string. In the case of air columns, a closed end in a column of air is analogous to the fixed end on a vibrating string. That is, at the closed end of an air column, air is not free to undergo movement and thus is forced into assuming the nodal positions of the standing wave pattern. Conversely, air is free to undergo its back-and-forth longitudinal motion at the open end of an air column; and as such, the standing wave patterns will depict antinodes at the open ends of air columns. So the basis for drawing the standing wave patterns for air columns is that vibrational antinodes will be present at any open end and vibrational nodes will be present at any closed end. If this principle is applied to open-end air columns, then the pattern for the fundamental frequency (the lowest frequency and longest wavelength pattern) will have antinodes at the two open ends and a single node in between. For Waves and Sound 169 this reason, the standing wave pattern for the fundamental frequency (or first harmonic) for an open-end air column looks like the diagram below. The distance between antinodes on a standing wave pattern is equivalent to one-half of a wavelength. A careful analysis of the diagram above shows that adjacent antinodes are positioned at the two ends of the air column. Thus, the length of the air column is equal to one-half of the wavelength for the first harmonic. The standing wave pattern for the second harmonic of an openend air column could be produced if another antinode and node was added to the pattern. This would result in a total of three antinodes and two nodes. This pattern is shown in the diagram below. Observe in the pattern that there is one full wave in the length of the air column. One full wave is twice the number of waves that were present in the first harmonic. For this reason, the frequency of the second harmonic is two times the frequency of the first harmonic. And finally, the standing wave pattern for the third harmonic of an open-end air column could be produced if still another antinode and node were added to the pattern. This would result in a total of four antinodes and three nodes. This pattern is shown in the diagram below. Observe in the pattern that there are one and one-half waves present in the length of the air column. One and one-half waves is three times the number of waves that were present in the first harmonic. For this reason, the frequency of the third harmonic is three times the frequency of the first harmonic. 170 Experimental Physics I 3.7.4 Length-Wavelength Relationships The process of adding another antinode and node to each consecutive harmonic in order to determine the pattern and the resulting length-wavelength relationship could be continued. If doing so, it is important to keep antinodes on the open ends of the air column and to maintain an alternating pattern of nodes and antinodes. When finished, the results should be consistent with the information in the table below. The relationships between the standing wave pattern for a given harmonic and the length-wavelength relationships for open end air columns are summarized in the table below. Harm. # # of Waves in Air Column # of Nodes # of Antinodes LengthWavelength Relationship 1 1/2 1 2 Wavelength = (2/1)*L 2 1 or 2/2 2 3 Wavelength = (2/2)*L 3 3/2 3 4 Wavelength = (2/3)*L 4 2 or 4/2 4 5 Wavelength = (2/4)*L 5 5/2 5 6 Wavelength = (2/5)*L Problem-Solving Scheme Now the aim of the above discussion is to internalize the mathematical relationships for open-end air columns in order to perform calculations predicting the length of air column required to produce a given natural frequency. And conversely, calculations can be performed to predict the natural frequencies produced by a known length of air column. Each of these calculations requires knowledge of the speed of a wave in air (which is approximately 340 m/s at room temperatures). The graphic below depicts the relationships between the key variables in such calculations. These relationships will be used to assist in the solution to problems involving standing waves in musical instruments. To demonstrate the use of the above problem-solving scheme, consider the following example problem and its detailed solution. Waves and Sound 171 EXERCISE Answer the following questions 1. Explain the wave motion. 2. Discuss about vibrating strings. 3. Describe the properties of sound. 4. Define the characteristics of sound waves. 5. Discuss about basic concepts of vibration. 6. Focus on vibration measurement. 7. Understanding an air column of variable length. MULTIPLE CHOICE QUESTIONS Tick the correct answer: 1. ________ is a wave motion of altering pressure. a. Compression b. Sound c. Vibration d. None is correct 2. Low frequencies produce bass tone. a. True b. False 3. High frequencies produce ________ tone. a. bass b. treble c. complex d. pure 4. A point of maximum pressure in a sound wave is called compression. a. True b. False 5. The minimum pressure in a sound wave is called ________. a. fraction b. rarefaction c. vibration d. wavelength 172 Experimental Physics I 6. Loudness is an intensity of a soundwave. a. True b. False 7. The formula to calculate the wavelength of sound wave is __________. a. λ=F/V b. λ = FV c. λ=V/F d. all are correct 8. The human ears detects sound waves by eardrum. a. True b. False 9. The velocity of sound wave in the air is __________. a. 1130 feet per seconds b. 344.3 feet per seconds c. 1280 feet per seconds d. a & b are correct 10. The unit of frequency wave is Hertz or cycles/sec. a. True b. False ANSWERS 1. (b) 2. (a) 3. (b) 4. (a) 5. (b) 6. (a) 7. (c) 8. (a) 9. (d) 10. (a) REFERENCES 1. 2. 3. Acoustical Society of America. “PACS 2010 Regular Edition— Acoustics Appendix”. Archived from the original on 14 May 2013. Retrieved 22 May 2013. “The Propagation of sound”. Archived from the original on 30 April 2015. Retrieved 26 June 2015. “What Does Sound Look Like?”. NPR. YouTube. Archived from the original on 10 April 2014. Retrieved 9 April 2014. Waves and Sound 4. 5. 6. 7. 8. 9. 173 Handel, S. (1995). Timbre perception and auditory object identification Archived 2020-01-10 at the Wayback Machine. Hearing, 425–461. “Scientists find upper limit for the speed of sound”. Archived from the original on 2020-10-09. Retrieved 2020-10-09. Trachenko, K.; Monserrat, B.; Pickard, C. J.; Brazhkin, V. V. (2020). “Speed of sound from fundamental physical constants”. Science Advances. 6 (41): eabc8662. arXiv:2004.04818. Bibcode:2020SciA....6.8662T. doi:10.1126/sciadv.abc8662. PMC 7546695. PMID 33036979. “The American Heritage Dictionary of the English Language” (Fourth ed.). Houghton Mifflin Company. 2000. Archived from the original on June 25, 2008. Retrieved May 20, 2010. Burton, R.L. (2015). The elements of music: what are they, and who cares? Archived 2020-05-10 at the Wayback Machine In J. Rosevear & S. Harding. (Eds.), ASME XXth National Conference proceedings. Paper presented at: Music: Educating for life: ASME XXth National Conference (pp. 22–28), Parkville, Victoria: The Australian Society for Music Education Inc. Nemiroff, R.; Bonnell, J., eds. (19 August 2007). “A Sonic Boom”. Astronomy Picture of the Day. NASA. Retrieved 26 June 2015. INDEX A Acousticians 108 Amplitude (Dynamics) 107 Angular acceleration 113 C Calorimeter 52, 53, 54, 55, 57, 58, 59, 68, 69, 96 Characteristics of Sound Waves 109 Chlamydomonas 3, 4, 5, 8, 16, 17, 36 Chromoplast 12 compressions and rarefactions 100, 101, 103, 106, 109, 110, 166, 167 Constant Flux Calorimeter 54 Convective heat transfer 79, 80, 81, 82, 95, 96 Convective heat transfer coefficients 80 D Damped vibration 117 Diffusion dominates 78 Displacement 112, 118, 119, 123, 124, 125, 126, 127, 133, 134 disturbance of matter 99 Double-membrane-bounded mitochondria 3 E Electromotive force (EMF) 132 equilibrium positions 100 F Flagellar membrane 7, 8, 10 Flagellum 4, 5, 6, 7, 9, 10, 11, 16, 17, 18 Fluid system 114 Forced convection 93, 95 Forced vibration 116 Free Convection 79, 80 Free vibration 114, 115, 117 G Gauge pressure 101 Green alga 3, 4, 5, 7, 16 H Heat Balance Calorimeter 54 Heat capacity 55 Heat Flow Calorimeter 53 I Infrasonic waves 101 Isotropic material 45 Isotropic materials 43, 44 176 Experimental Physics I K Kinesin proteins 5 Kinetic energy 112, 113, 114, 118 L Latent heat 40, 60, 61, 97 Linear expansion 40, 41, 42, 43, 48, 49, 50, 96 longitudinal. 100 low-pressure regions 100, 106, 167 M measure sound 110, 111 Mechanical Sound Waves 105 N Negative thermal expansion 46 O oscillating 100, 119 oscillation 105, 107, 112, 113, 115, 116, 117, 119, 120 P Particle-particle distance 41 Phaeophyceae 12, 28, 29 Phycology 1, 2, 35 Piezoelectric 124, 125, 128, 131 Plasma membrane 3, 4, 5, 8, 13, 16, 17, 30, 32 positive x-direction 100, 101 Power Compensation 54 Pressure Sound Waves 106 Prokaryotic cells 2 Propagation of Sound Waves 103 Properties of Sound 107, 108 Q Quantum mechanics 134, 135, 137, 139, 142, 143, 144, 146, 158, 159, 161 Quantum system 134, 135, 136, 137, 138, 139, 140, 141, 142, 146 S Schrödinger equation 136, 143, 150 simple harmonic motion 100, 118, 119 Sound Intensity 111, 112 sound travels 103, 104 sound waves 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 165, 167, 171, 172 Sound waves 100, 101, 102, 104, 105, 106, 110, 167 Surface shear stress 84 T Thermal expansion 40, 41 Thermal expansion coefficient 43, 44, 45, 96 Thermal linear expansion 40 Thermal stress 51 Thylakoids 3, 12, 13, 14, 15, 23, 24 transverse 100, 105, 106, 124, 127, 166, 167 U ultrasonic waves 101, 102 Index 177 V W vibrating air particles 102 Vibratory system 112, 120, 121 Volumetric coefficient 44, 45 Volumetric thermal expansion 44, 45 Wave function 134, 135, 136, 137, 138, 140, 141, 142, 143, 144, 145, 146, 147, 150, 151