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The pendulu of a clock is an eaple of oscillatory otion. Many kinds of oscillatory otion are sinusoidal in tie, or nearly so, and are referred to as being siple haronic otion. Real systes generally have at least soe friction, causing the otion to be daped. When an eternal sinusoidal force is eerted on a syste able to oscillate, resonance occurs if the driving force is at or near the natural frequency of vibration. Vibrations can give rise to waves such as water waves or waves traveling along a cord which travel outward fro their source. Fundaental Photographs Jonathan Nourok/PhotoEdit Many objects vibrate or oscillate an object on the end of a spring, a tuning fork, the balance wheel of an old watch, a pendulu, a plastic ruler held firly over the edge of a table and gently struck, the strings of a guitar or piano. Spiders detect prey by the vibrations of their webs; cars oscillate up and down when they hit a bup; buildings and bridges vibrate when heavy trucks pass or the wind is fierce. Indeed, because ost solids are elastic, they vibrate (at least briefly) when given an ipulse. Electrical oscillations occur in radio and television sets. At the atoic level, atos vibrate within a olecule, and the atos of a solid vibrate about their relatively fied positions. Because it is so coon in everyday life and occurs in so any areas of physics, oscillatory (or vibrational) otion is of great iportance. Mechanical vibrations are fully described on the basis of Newtonian echanics. Vibrations and wave otion are intiately related subjects. Waves whether ocean waves, waves on a string, earthquake waves, or sound waves in air have as their source a vibration. In the case of sound, not only is the source a vibrating object, but so is the detector the eardru or the ebrane of a icrophone. Indeed, when a wave travels through a ediu, the ediu vibrates (such as air for sound waves). In the second half of this Chapter, after we discuss vibrations, we will discuss siple waves such as those on water or on a string. Note: Sections arked with a star (*) are considered optional. 330
Siple Haronic Motion When an object vibrates or oscillates back and forth, over the sae path, each vibration taking the sae aount of tie, the otion is periodic. The siplest for of periodic otion is represented by an object oscillating on the end of a unifor coil spring. Because any other types of vibrational otion closely reseble this syste, we will look at it in detail. We assue that the ass of the spring can be ignored, and that the spring is ounted horizontally, as shown in Fig. a, so that the object of ass slides without friction on the horizontal surface. Any spring has a natural length at which it eerts no force on the ass. The position of the ass at this point is called the equilibriu position. If the ass is oved either to the left, which copresses the spring, or to the right, which stretches it, the spring eerts a force on the ass that acts in the direction of returning the ass to the equilibriu position; hence it is called a restoring force. We consider the coon situation where we can assue the agnitude of the restoring force F is directly proportional to the displaceent the spring has been stretched (Fig. b) or copressed (Fig. c) fro the equilibriu position: F = k. [force eerted by spring] () Note that the equilibriu position has been chosen at. Equation, which is often referred to as Hooke s law, is accurate as long as the spring is not copressed to the point where the coils are close to touching, or stretched beyond the elastic region. Equilibriu position (> 0) FIGURE A ass vibrating at the end of a unifor spring. (c) (< 0) The inus sign in Eq. indicates that the restoring force is always in the direction opposite to the displaceent. For eaple, if we choose the positive direction to the right in Fig., is positive when the spring is stretched, but the direction of the restoring force is to the left (negative direction). If the spring is copressed, is negative (to the left) but the force F acts toward the right (Fig. c). The proportionality constant k in Eq. is called the spring constant or spring stiffness constant. To stretch the spring a distance, one has to eert an (eternal) force on the free end of the spring at least equal to F = ±k. [eternal force on spring] The greater the value of k, the greater the force needed to stretch a spring a given distance. That is, the stiffer the spring, the greater the spring constant k. Note that the force F in Eq. is not a constant, but varies with position. Therefore the acceleration of the ass is not constant, so we cannot use the equations for constant acceleration. CAUTION Force and acceleration are not constant; the kineatic equations you ay be failiar with are not useful here 33
(c) (d) v = 0 = A v = 0 CAUTION For vertical spring, easure displaceent ( or y) fro the vertical equilibriu position = 0 v = +v a (e) = A (a. in positive direction) v = 0 = A = 0 v = v a (a. in negative direction) FIGURE Force on, and velocity of, a ass at different positions of its oscillation cycle on a frictionless surface. Let us eaine what happens when our unifor spring is initially copressed a distance = A, as shown in Fig. a, and then released. The spring eerts a force on the ass that pushes it toward the equilibriu position. But because the ass has been accelerated by the force, it passes the equilibriu position with considerable speed. Indeed, as the ass reaches the equilibriu position, the force on it decreases to zero, but its speed at this point is a aiu, v a, Fig. b. As the ass oves farther to the right, the force on it acts to slow it down, and it stops oentarily at = A, Fig. c. It then begins oving back in the opposite direction, accelerating until it passes the equilibriu point, Fig. d, and then slows down until it reaches zero speed at the original starting point, = A, Fig. e. It then repeats the otion, oving back and forth syetrically between = A and = A. EXERCISE A An object is oscillating back and forth. Which of the following stateents are true at soe tie during the course of the otion? The object can have zero velocity and, siultaneously, nonzero acceleration. The object can have zero velocity and, siultaneously, zero acceleration. (c) The object can have zero acceleration and, siultaneously, nonzero velocity. (d) The object can have nonzero velocity and nonzero acceleration siultaneously. To discuss vibrational otion, we need to define a few ters. The distance of the ass fro the equilibriu point at any oent is called the displaceent. The aiu displaceent the greatest distance fro the equilibriu point is called the aplitude, A. One cycle refers to the coplete to-and-fro otion fro soe initial point back to that sae point say, fro = A to = A and back to = A. The period, T, is defined as the tie required to coplete one cycle. Finally, the frequency, f, is the nuber of coplete cycles per second. Frequency is generally specified in hertz (Hz), where Hz = cycle per second As B. It is easy to see, fro their definitions, that frequency and period are inversely related: f = T and T = f ; for eaple, if the frequency is 5 cycles per second, then each cycle takes 5 s. The oscillation of a spring hung vertically is essentially the sae as that of a horizontal spring. Because of gravity, the length of a vertical spring with a ass on the end will be longer at equilibriu than when that sae spring is horizontal, as shown in Fig. 3. The spring is in equilibriu when F = 0 = g - k 0, so the spring stretches an etra aount 0 = g k to be in equilibriu. If is easured fro this new equilibriu position, Eq. can be used directly with the sae value of k. () FIGURE 3 Free spring, hung vertically. Mass attached to spring in new equilibriu position, which occurs when F = 0 = g - k 0. 0 F = k0 now easured fro here g B 33
EXAMPLE Car springs. When a faily of four with a total ass of 00 kg step into their 00-kg car, the car s springs copress 3.0 c. What is the spring constant of the car s springs (Fig. 4), assuing they act as a single spring? How far will the car lower if loaded with 300 kg rather than 00 kg? APPROACH We use Hooke s law. The etra force equal to the weight of the people, g, causes a 3.0-c displaceent. SOLUTION The added force of (00 kg)a9.8 s B = 960 N causes the springs to copress 3.0 * 0. Therefore (Eq. ), the spring constant is k = F = 960 N 3.0 * 0 = 6.5 * 04 N. If the car is loaded with 300 kg, Hooke s law gives or 4.5 c. NOTE We could have obtained without solving for k: since is proportional to F, if 00 kg copresses the spring 3.0 c, then.5 ties the force will copress the spring.5 ties as uch, or 4.5 c. Any vibrating syste for which the restoring force is directly proportional to the negative of the displaceent (as in Eq., F = k) is said to ehibit siple haronic otion (SHM). Such a syste is often called a siple haronic oscillator (SHO). Most solid aterials stretch or copress according to Eq. as long as the displaceent is not too great. Because of this, any natural vibrations are siple haronic, or sufficiently close to it that they can be treated using this SHM odel. CONCEPTUAL EXAMPLE Is the otion siple haronic? Which of the following represent a siple haronic oscillator: F = 0.5, F =.3y, (c) F = 8.6, (d) F = 4u? RESPONSE Both and (d) represent siple haronic oscillators because they give the force as inus a constant ties a displaceent. The displaceent need not be, but the inus sign is required to restore the syste to equilibriu, which is why (c) is not a SHO. = F k = (300 kg)a9.8 s B A6.5 * 0 4 N B Energy in the Siple Haronic Oscillator With forces that are not constant, such as here with siple haronic otion, it is often convenient and useful to use the energy approach. To stretch or copress a spring, work has to be done. Hence potential energy is stored in a stretched or copressed spring. Indeed, elastic potential energy is given by pe = k. The total echanical energy E of a ass spring syste is the su of the kinetic and potential energies, E = v + k, = 4.5 * 0, where v is the velocity of the ass when it is a distance fro the equilibriu position. As long as there is no friction, the total echanical energy E (3) FIGURE 4 Photo of a car s spring. (Also visible is the shock absorber, in red see Section 5.) SHM SHO Total energy of SHO Robert Reiff/Getty Iages, Inc. Tai The word haronic refers to the otion being sinusoidal, which we discuss in Section 3. It is siple when there is sinusoidal otion of a single frequency. 333
PE E = ka = A (v = 0) = A reains constant. As the ass oscillates back and forth, the energy continuously changes fro potential energy to kinetic energy, and back again (Fig. 5). At the etree points, = A and = A (Fig. 5a, c), all the energy is stored in the spring as potential energy (and is the sae whether the spring is copressed or stretched to the full aplitude). At these etree points, the ass stops oentarily as it changes direction, so v = 0 and E = va E = (0) + ka = ka. (4a) KE = A = A E = ka Thus, the total echanical energy of a siple haronic oscillator is proportional to the square of the aplitude. At the equilibriu point, (Fig. 5b), all the energy is kinetic: E = v a + k(0) = v a, (4b) PE (c) = A = A (v = 0) v a where represents the aiu velocity during the otion (which occurs at ). At interediate points (Fig. 5d), the energy is part kinetic and part potential; because energy is conserved (we use Eqs. 3 and 4a), v + k = ka. (4c) PE KE (d) = A E = v + k = A FIGURE 5 Energy changes fro potential energy to kinetic energy and back again as the spring oscillates. Fro this conservation of energy equation, we can obtain the velocity as a function of position. Solving for v, we have v = k AA - B = k A a - A b. Fro Eqs. 4a and 4b, we have v a = ka, so v a = (k )A. Inserting this into the equation above and taking the square root, we have v = &v a B - A. (5) This gives the velocity of the object at any position. The object oves back and forth, so its velocity can be either in the ± or direction, but its agnitude depends only on the agnitude of. CONCEPTUAL EXAMPLE 3 Doubling the aplitude. Suppose the spring in Fig. 5 is stretched twice as far (to = A). What happens to the energy of the syste, the aiu velocity of the oscillating ass, (c) the aiu acceleration of the ass? RESPONSE Fro Eq. 4a, the total energy is proportional to the square of the aplitude A, so stretching it twice as far quadruples the energy ( = 4). You ay protest, I did work stretching the spring fro to = A. Don t I do the sae work stretching it fro A to A? No. The force you eert is proportional to the displaceent, so for the second displaceent, fro = A to A, you do ore work than for the first displaceent ( to A). Fro Eq. 4b, we can see that since the energy is quadrupled, the aiu velocity ust be doubled. Cv a r E r A.D (c) Since the force is twice as great when we stretch the spring twice as far, the acceleration is also twice as great: a r F r. EXERCISE B Suppose the spring in Fig. 5 is copressed to = A, but is given a push to the right so that the initial speed of the ass is v 0. What effect does this push have on the energy of the syste, the aiu velocity, (c) the aiu acceleration? 334