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11 VIBRATIONS AND WAVES The world around us is filled with waves. Some of them we can see or hear, but many more our senses of sight or hearing cannot detect. In the submicroscopic world, atoms and molecules are made up of electrons, protons, neutrons, and mesons that move around as waves within their boundaries. Appropriately stimulated, these same atoms and molecules emit waves we call γ rays, X rays, light waves, heat waves, and radio waves. In our world of macroscopic bodies, water waves and sound waves are produced by moving masses of considerable size. Earthquakes produce waves as the result of sudden shifts in land masses. Water waves are produced by the wind or ships as they pass by. Sound waves are the result of quick movements of objects in the air. Any motion that repeats itself in equal intervals of time is called periodic motion. The swinging of a clock pendulum, the vibrations of the prongs of a tuning fork, and a mass dancing from the lower end of a coiled spring are but three examples. These particular motions and many others like them that occur in nature are referred to as simple harmonic motion (SHM). PART TWO Wave Optics 216 FUNDAMENTALS OF OPTICS 11.1 SIMPLE HARMONIC MOTION Simple harmonic motion is defined as the projection on any diameter of a graph point moving in a circle with uniform speed. The motion is illustrated in Fig. 11A. The graph point p moves around the circle of radius a with a uniform speed v. If at every instant of time a normal is drawn to the diameter AB, the intercept P, called the mass point, moves with SHM. Moving back and forth along the line AB, the mass point is continually changing speed vx. Starting from rest at the end points A or B, the speed increases until it reaches C. From there it slows down again coming to rest at the other end of its path. The return of the mass point is a repetition of this motion in reverse. The displacement of an object undergoing SHM is defined as the distance from its equilibrium position C to the point P. It will be seen in Fig. 11A that the displacement x varies in magnitude from zero up to its maximum value a, which is the radius of the circle of reference. The maximum displacement a is called the amplitude, and the time required to make one complete vibration is called the period. If a vibration starts at B, it is completed when the mass point P moves across to A and back again to B. If it starts at C and moves to B and back to C, only half a vibration has been completed. The amplitude a is measured in meters, or a fraction thereof, while the period is measured in seconds. The frequency of vibration is defined as the number of complete vibrations per second. If a particular vibrating body completes one vibration in s, the period T = 1 s and it will make one complete vibration in 1 s. If a body makes 10 vibrations in 1 s, its period will be T = 1/10 s. In other words, the frequency of vibration ν and the period T are reciprocals of each other: frequency = 1/period period = 1/frequency In algebraic symbols, ν = 1/T T = 1/v (11a) If the vibration of a body is described in terms of the graph point p, moving in a circle, the frequency is given by the number of revolutions per second, or cycles per second 1 cycle/second = 1 vibration/second (11b) now called the hertz* (Hz) 1 vib/s = 1 Hz (11c) * Heinrich Rudolf Hertz (1857-1894), German physicist, was born at Hamburg. He studied physics under Helmholtz in Berlin, at whose suggestion he first became interested in Maxwell's electromagnetic theory. His researches with electromagnetic waves which made his name famous were carried out at Karlsruhe Polytechnic between 1885 and 1889. As professor of physics at the University of Bonn, after 1889, he experimented with electrical discharges through gases and narrowly missed the discovery of X rays described by Röntgen a few years later. By his premature death, science lost one of its most promising disciples. 217 VIBRATIONS AND WAVES FIGURE 11A Simple harmonic motion along a straight line AB. 11.2 THE THEORY OF SIMPLE HARMONIC MOTION At this point we present the theory of SHM and derive an equation for the period of vibrating bodies. In Fig. 11B we see that the displacement x is given by x = a cos θ (11d) As the graph point p moves with constant speed v, the radius vector a rotates with constant angular speed ω, so that the angle θ changes at a constant rate x = a cos ωt (11d) The graph point p, moving with a speed v, travels once around the circle of reference, a distance equal to 2πa, in the time of one period T. We now use the relation in mechanics that time equals distance divided by speed, and obtain T = 2πa/v (11e) To obtain the angular speed ω of the graph point in terms of the period, we have T = 2π/ω or ω = 2π/T (11f) An object moving in a circle with uniform speed v has a centripetal acceleration toward the center, given by a_c = v²/a (11g) Since this acceleration a_c continually changes the direction of the motion, its component a_x along the diameter, or x axis, changes in magnitude and is given by a_x = a_c cos θ Substituting in Eq. (11g), we find a_x = -v²/a cos θ 218 FUNDAMENTALS OF OPTICS FIGURE 11B The acceleration ax of any mass moving with simple harmonic motion is toward a position of equilibrium C. From the right triangle CPp, cos θ = x/a, direct substitution gives a_x = v²x/a . or a_x = v²/a² x We now multiply both sides of the equation by a²/a_x x², take the square root of both sides of the equation, and obtain a²/v² = x/a_x, and a/v = √x/a_x For a/v in Eq. (11e) we now substitute √x/a_x and obtain for the period of any SHM the relation T = 2π √x/a_x (11h) If the displacement is to the right of C, its value is +x, and if the acceleration is to the left, its value is -a_x. Conversely, when the displacement is to the left of C, we have -x, and the acceleration is to the right, or +a_x. This is the reason for writing T = 2π √-x/a_x (11i) 11.3 STRETCHING OF A COILED SPRING As an illustration of the relationships generally applied to vibrating sources, we consider in some detail the stretching of a coiled spring, followed by its vibration with SHM when the stretching force is suddenly released (see Fig. 11C). As a laboratory experiment, one end of a meterstick is placed at marker Q. A force of 2.0 newtons (N) is applied to the spring, stretching it a distance of 1.25 cm. VIBRATIONS AND WAVES 219 FIGURE 11C An experiment for measuring the distance x a coiled spring S stretches for different values of the applied force. When a total force of 4.0 N is applied, the total stretch is 2.50 cm. By applying forces of 6.0, 8.0, and 10.0 N, respectively, the total distances recorded are as shown in Table 11A. Plotting these data on graph paper produces a straight line, as shown in Fig. 11D. Properly interpreted, this graph means that the applied force F and the displacement of the spring x are directly proportional to each other, and we can write F ∝ x or F = kx The proportionality constant k is the slope of the straight line and is a direct measure of the stiffness of the spring. The experimental value of k in this experiment is calculated as follows: k = F / x = 10 N / 0.0625 m = 160 N/m (11j) The stiffer the spring, the larger its stretch constant k. Within the limits of this experiment, the spring exerts an equal and opposite force – F, as the reaction to the applied force + F. For the spring, – F = kx, and we can write F = – kx (11k) Table 11A RECORDED DATA FOR STRETCHING A COILED SPRING F N x m 0 0 2 0.0125 4 0.0250 6 0.0375 8 0.0500 10 0.0625 220 FUNDAMENTALS OF OPTICS FIGURE 11D Experimental results on the stretching of a coiled spring as shown in Fig. 11C. This is a demonstration of Hooke's law. The fact that we obtain a straight line graph in Fig. 11D shows that the stretching of a spring obeys Hooke's law. * This is typical of nearly all elastic bodies as long as the body is not permanently deformed, indicating that the forces applied had been carried beyond the elastic limit. Since the work done in stretching the spring is given by the force multiplied by the distance and the force here varies linearly with the distance, Work = ∫F dx (11l) As can be seen in Fig. 11E, the average force is given by ½F. This, multiplied by the distance x through which it acts, gives the area under the curve, which is the work done† W = ½Fx (11m) If we now replace F by its equivalent value kx from Eq. (11j), we obtain W = ½kx² (11n) * Robert Hooke (1635-1703), English experimental physicist, is known principally for his contributions to the wave theory of light, universal gravitation, and atmospheric pressure. He originated many physical ideas but perfected few of them. Hooke's scientific achievements would undoubtedly have received greater acclaim if his efforts had been confined to fewer subjects. He had an irritable temper and made many virulent attacks on Newton and other men of science, claiming that work published by them was due to him. † In most elementary physics texts it is shown that the area under the curve of a graph, where F is plotted against x, is equal to the total work done. VIBRATIONS AND WAVES 221 FIGURE 11E The work done and the energy stored in stretching a spring are given by the area under the graph line F = kx. This relation shows that if the stretch of a spring increases twofold, the energy required, or stored, is increased fourfold, and increasing the displacement threefold increases the energy ninefold. 11.4 VIBRATING SPRING All bodies in nature are elastic, some more so than others. If a distorting force is applied to change the shape of a body and its shape is not permanently altered, upon release of the force it will be set in vibration. This property is demonstrated in Fig. 11F by a mass m suspended from the lower end of a spring. In diagram (a) a force F has been applied to stretch the spring a distance a. Upon release, the mass moves up and down with SHM. In diagram (c), m is at its highest point and the spring is shown compressed. The amplitude of the vibration is determined by the distance the spring is stretched from its equilibrium position, and the period of vibration T is given by T = 2π √(m/k) (11o) where k is the stiffness of the spring and m is the mass of the vibrating body. This equation shows that if a stiffer spring is used, k being in the denominator, the period is decreased and the vibration frequency is increased. If the mass m is increased, the period is increased and the frequency is decreased. Since the stretching of the spring obeys Hooke's law, we can apply Eq. (11k). Using the force equation from mechanics, F = ma and replacing F in Eq. (11k) by ma, we obtain ma = – kx or –x/a = m/k (11p) Hence by the replacement of –x/a by m/k in Eq. (11i) we obtain Eq. (11o). 222 FUNDAMENTALS OF OPTICS FIGURE 11F A mass m suspended from a coiled spring is shown in three positions as it vibrates up and down with simple harmonic motion. EXAMPLE 1 If a 4.0-kg mass is suspended from the lower end of a coiled spring, as shown in Fig. 11F, it stretches a distance of 18.0 cm. If the spring is then extended farther and released, it will be set vibrating up and down with SHM. Find (a) the spring constant k, (b) the period T, (c) the frequency v, and (d) the total energy stored in the vibrating system. SOLUTION The given quantities in the mks system of units are m = 4.0 kg, x = 0.180 m; the acceleration due to gravity is g = 9.80 m/s^2. (a) We can use Eq. (11k), solve for the value of k, and substitute the appropriate values: k = -F/x = 4.0 x 9.80/0.180 = 217.8 N/m (b) We can use Eq. (11o), and upon direct substitution of the known values obtain T = 2π sqrt(m/k) = 2π sqrt(4.0 kg / 217.8 N/m) = 0.852 s
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11 VIBRATIONS AND WAVES The world around us is filled with waves. Some of them we can see or hear, but many more our senses of sight or hearing cannot detect. In the submicroscopic world, atoms and molecules are made up of electrons, protons, neutrons, and mesons that move around as waves within their boundaries. Appropriately stimulated, these same atoms and molecules emit waves we call γ rays, X rays, light waves, heat waves, and radio waves. In our world of macroscopic bodies, water waves and sound waves are produced by moving masses of considerable size. Earthquakes produce waves as the result of sudden shifts in land masses. Water waves are produced by the wind or ships as they pass by. Sound waves are the result of quick movements of objects in the air. Any motion that repeats itself in equal intervals of time is called periodic motion. The swinging of a clock pendulum, the vibrations of the prongs of a tuning fork, and a mass dancing from the lower end of a coiled spring are but three examples. These particular motions and many others like them that occur in nature are referred to as simple harmonic motion (SHM). PART TWO Wave Optics 216 FUNDAMENTALS OF OPTICS 11.1 SIMPLE HARMONIC MOTION Simple harmonic motion is defined as the projection on any diameter of a graph point moving in a circle with uniform speed. The motion is illustrated in Fig. 11A. The graph point p moves around the circle of radius a with a uniform speed v. If at every instant of time a normal is drawn to the diameter AB, the intercept P, called the mass point, moves with SHM. Moving back and forth along the line AB, the mass point is continually changing speed vx. Starting from rest at the end points A or B, the speed increases until it reaches C. From there it slows down again coming to rest at the other end of its path. The return of the mass point is a repetition of this motion in reverse. The displacement of an object undergoing SHM is defined as the distance from its equilibrium position C to the point P. It will be seen in Fig. 11A that the displacement x varies in magnitude from zero up to its maximum value a, which is the radius of the circle of reference. The maximum displacement a is called the amplitude, and the time required to make one complete vibration is called the period. If a vibration starts at B, it is completed when the mass point P moves across to A and back again to B. If it starts at C and moves to B and back to C, only half a vibration has been completed. The amplitude a is measured in meters, or a fraction thereof, while the period is measured in seconds. The frequency of vibration is defined as the number of complete vibrations per second. If a particular vibrating body completes one vibration in s, the period T = 1 s and it will make one complete vibration in 1 s. If a body makes 10 vibrations in 1 s, its period will be T = 1/10 s. In other words, the frequency of vibration ν and the period T are reciprocals of each other: frequency = 1/period period = 1/frequency In algebraic symbols, ν = 1/T T = 1/v (11a) If the vibration of a body is described in terms of the graph point p, moving in a circle, the frequency is given by the number of revolutions per second, or cycles per second 1 cycle/second = 1 vibration/second (11b) now called the hertz* (Hz) 1 vib/s = 1 Hz (11c) * Heinrich Rudolf Hertz (1857-1894), German physicist, was born at Hamburg. He studied physics under Helmholtz in Berlin, at whose suggestion he first became interested in Maxwell's electromagnetic theory. His researches with electromagnetic waves which made his name famous were carried out at Karlsruhe Polytechnic between 1885 and 1889. As professor of physics at the University of Bonn, after 1889, he experimented with electrical discharges through gases and narrowly missed the discovery of X rays described by Röntgen a few years later. By his premature death, science lost one of its most promising disciples. 217 VIBRATIONS AND WAVES FIGURE 11A Simple harmonic motion along a straight line AB. 11.2 THE THEORY OF SIMPLE HARMONIC MOTION At this point we present the theory of SHM and derive an equation for the period of vibrating bodies. In Fig. 11B we see that the displacement x is given by x = a cos θ (11d) As the graph point p moves with constant speed v, the radius vector a rotates with constant angular speed ω, so that the angle θ changes at a constant rate x = a cos ωt (11d) The graph point p, moving with a speed v, travels once around the circle of reference, a distance equal to 2πa, in the time of one period T. We now use the relation in mechanics that time equals distance divided by speed, and obtain T = 2πa/v (11e) To obtain the angular speed ω of the graph point in terms of the period, we have T = 2π/ω or ω = 2π/T (11f) An object moving in a circle with uniform speed v has a centripetal acceleration toward the center, given by a_c = v²/a (11g) Since this acceleration a_c continually changes the direction of the motion, its component a_x along the diameter, or x axis, changes in magnitude and is given by a_x = a_c cos θ Substituting in Eq. (11g), we find a_x = -v²/a cos θ 218 FUNDAMENTALS OF OPTICS FIGURE 11B The acceleration ax of any mass moving with simple harmonic motion is toward a position of equilibrium C. From the right triangle CPp, cos θ = x/a, direct substitution gives a_x = v²x/a . or a_x = v²/a² x We now multiply both sides of the equation by a²/a_x x², take the square root of both sides of the equation, and obtain a²/v² = x/a_x, and a/v = √x/a_x For a/v in Eq. (11e) we now substitute √x/a_x and obtain for the period of any SHM the relation T = 2π √x/a_x (11h) If the displacement is to the right of C, its value is +x, and if the acceleration is to the left, its value is -a_x. Conversely, when the displacement is to the left of C, we have -x, and the acceleration is to the right, or +a_x. This is the reason for writing T = 2π √-x/a_x (11i) 11.3 STRETCHING OF A COILED SPRING As an illustration of the relationships generally applied to vibrating sources, we consider in some detail the stretching of a coiled spring, followed by its vibration with SHM when the stretching force is suddenly released (see Fig. 11C). As a laboratory experiment, one end of a meterstick is placed at marker Q. A force of 2.0 newtons (N) is applied to the spring, stretching it a distance of 1.25 cm. VIBRATIONS AND WAVES 219 FIGURE 11C An experiment for measuring the distance x a coiled spring S stretches for different values of the applied force. When a total force of 4.0 N is applied, the total stretch is 2.50 cm. By applying forces of 6.0, 8.0, and 10.0 N, respectively, the total distances recorded are as shown in Table 11A. Plotting these data on graph paper produces a straight line, as shown in Fig. 11D. Properly interpreted, this graph means that the applied force F and the displacement of the spring x are directly proportional to each other, and we can write F ∝ x or F = kx The proportionality constant k is the slope of the straight line and is a direct measure of the stiffness of the spring. The experimental value of k in this experiment is calculated as follows: k = F / x = 10 N / 0.0625 m = 160 N/m (11j) The stiffer the spring, the larger its stretch constant k. Within the limits of this experiment, the spring exerts an equal and opposite force – F, as the reaction to the applied force + F. For the spring, – F = kx, and we can write F = – kx (11k) Table 11A RECORDED DATA FOR STRETCHING A COILED SPRING F N x m 0 0 2 0.0125 4 0.0250 6 0.0375 8 0.0500 10 0.0625 220 FUNDAMENTALS OF OPTICS FIGURE 11D Experimental results on the stretching of a coiled spring as shown in Fig. 11C. This is a demonstration of Hooke's law. The fact that we obtain a straight line graph in Fig. 11D shows that the stretching of a spring obeys Hooke's law. * This is typical of nearly all elastic bodies as long as the body is not permanently deformed, indicating that the forces applied had been carried beyond the elastic limit. Since the work done in stretching the spring is given by the force multiplied by the distance and the force here varies linearly with the distance, Work = ∫F dx (11l) As can be seen in Fig. 11E, the average force is given by ½F. This, multiplied by the distance x through which it acts, gives the area under the curve, which is the work done† W = ½Fx (11m) If we now replace F by its equivalent value kx from Eq. (11j), we obtain W = ½kx² (11n) * Robert Hooke (1635-1703), English experimental physicist, is known principally for his contributions to the wave theory of light, universal gravitation, and atmospheric pressure. He originated many physical ideas but perfected few of them. Hooke's scientific achievements would undoubtedly have received greater acclaim if his efforts had been confined to fewer subjects. He had an irritable temper and made many virulent attacks on Newton and other men of science, claiming that work published by them was due to him. † In most elementary physics texts it is shown that the area under the curve of a graph, where F is plotted against x, is equal to the total work done. VIBRATIONS AND WAVES 221 FIGURE 11E The work done and the energy stored in stretching a spring are given by the area under the graph line F = kx. This relation shows that if the stretch of a spring increases twofold, the energy required, or stored, is increased fourfold, and increasing the displacement threefold increases the energy ninefold. 11.4 VIBRATING SPRING All bodies in nature are elastic, some more so than others. If a distorting force is applied to change the shape of a body and its shape is not permanently altered, upon release of the force it will be set in vibration. This property is demonstrated in Fig. 11F by a mass m suspended from the lower end of a spring. In diagram (a) a force F has been applied to stretch the spring a distance a. Upon release, the mass moves up and down with SHM. In diagram (c), m is at its highest point and the spring is shown compressed. The amplitude of the vibration is determined by the distance the spring is stretched from its equilibrium position, and the period of vibration T is given by T = 2π √(m/k) (11o) where k is the stiffness of the spring and m is the mass of the vibrating body. This equation shows that if a stiffer spring is used, k being in the denominator, the period is decreased and the vibration frequency is increased. If the mass m is increased, the period is increased and the frequency is decreased. Since the stretching of the spring obeys Hooke's law, we can apply Eq. (11k). Using the force equation from mechanics, F = ma and replacing F in Eq. (11k) by ma, we obtain ma = – kx or –x/a = m/k (11p) Hence by the replacement of –x/a by m/k in Eq. (11i) we obtain Eq. (11o). 222 FUNDAMENTALS OF OPTICS FIGURE 11F A mass m suspended from a coiled spring is shown in three positions as it vibrates up and down with simple harmonic motion. EXAMPLE 1 If a 4.0-kg mass is suspended from the lower end of a coiled spring, as shown in Fig. 11F, it stretches a distance of 18.0 cm. If the spring is then extended farther and released, it will be set vibrating up and down with SHM. Find (a) the spring constant k, (b) the period T, (c) the frequency v, and (d) the total energy stored in the vibrating system. SOLUTION The given quantities in the mks system of units are m = 4.0 kg, x = 0.180 m; the acceleration due to gravity is g = 9.80 m/s^2. (a) We can use Eq. (11k), solve for the value of k, and substitute the appropriate values: k = -F/x = 4.0 x 9.80/0.180 = 217.8 N/m (b) We can use Eq. (11o), and upon direct substitution of the known values obtain T = 2π sqrt(m/k) = 2π sqrt(4.0 kg / 217.8 N/m) = 0.852 s