OSCILLATIONS CHAPTER FOURTEEN 14.1 INTRODUCTION

Transcription

1 CHAPTER FOURTEEN OSCILLATIONS 14.1 INTRODUCTION 14.1 Introduction 14. Periodic and oscilatory otions 14.3 Siple haronic otion 14.4 Siple haronic otion and unifor circular otion 14.5 Velocity and acceleration in siple haronic otion 14.6 Force law for siple haronic otion 14.7 Energy in siple haronic otion 14.8 Soe systes executing SHM 14.9 Daped siple haronic otion Forced oscillations and resonance Suary Points to ponder Exercises Additional Exercises Appendix In our daily life we coe across various kinds of otions. You have already learnt about soe of the, e.g. rectilinear otion and otion of a projectile. Both these otions are non-repetitive. We have also learnt about unifor circular otion and orbital otion of planets in the solar syste. In these cases, the otion is repeated after a certain interval of tie, that is, it is periodic. In your childhood you ust have enjoyed rocking in a cradle or swinging on a swing. Both these otions are repetitive in nature but different fro the periodic otion of a planet. Here, the object oves to and fro about a ean position. The pendulu of a wall clock executes a siilar otion. There are leaves and branches of a tree oscillating in breeze, boats bobbing at anchor and the surging pistons in the engines of cars. All these objects execute a periodic to and fro otion. Such a otion is tered as oscillatory otion. In this chapter we study this otion. The study of oscillatory otion is basic to physics; its concepts are required for the understanding of any physical phenoena. In usical instruents like the sitar, the guitar or the violin, we coe across vibrating strings that produce pleasing sounds. The ebranes in drus and diaphrags in telephone and speaker systes vibrate to and fro about their ean positions. The vibrations of air olecules ake the propagation of sound possible. Siilarly, the atos in a solid oscillate about their ean positions and convey the sensation of teperature. The oscillations of electrons in the antennas of radio, TV and satellite transitters convey inforation. The description of a periodic otion in general, and oscillatory otion in particular, requires soe fundaental concepts like period, frequency, displaceent, aplitude and phase. These concepts are developed in the next section.

2 OSCILLATIONS PERIODIC AND OSCILLATORY MOTIONS Fig 14.1 shows soe periodic otions. Suppose an insect clibs up a rap and falls down it coes back to the initial point and repeats the process identically. If you draw a graph of its height above the ground versus tie, it would look soething like Fig (a). If a child clibs up a step, coes down, and repeats the process, its height above the ground would look like that in Fig 14.1 (b). When you play the gae of bouncing a ball off the ground, between your pal and the ground, its height versus tie graph would look like the one in Fig 14.1 (c). Note that both the curved parts in Fig 14.1 (c) are sections of a parabola given by the Newton s equation of otion (see section 3.6), 1 h=ut + gt for downward otion, and 1 h=ut gt for upward otion, with different values of u in each case. These are exaples of periodic otion. Thus, a otion that repeats itself at regular intervals of tie is called periodic otion. Fig 14.1 Exaples of periodic otion. The period T is shown in each case. Very often the body undergoing periodic otion has an equilibriu position soewhere inside its path. When the body is at this position no net external force acts on it. Therefore, if it is left there at rest, it reains there forever. If the body is given a sall displaceent fro the position, a force coes into play which tries to bring the body back to the equilibriu point, giving rise to oscillations or vibrations. For exaple, a ball placed in a bowl will be in equilibriu at the botto. If displaced a little fro the point, it will perfor oscillations in the bowl. Every oscillatory otion is periodic, but every periodic otion need not be oscillatory. Circular otion is a periodic otion, but it is not oscillatory. There is no significant difference between oscillations and vibrations. It sees that when the frequency is sall, we call it oscillation (like the oscillation of a branch of a tree), while when the frequency is high, we call it vibration (like the vibration of a string of a usical instruent). Siple haronic otion is the siplest for of oscillatory otion. This otion arises when the force on the oscillating body is directly proportional to its displaceent fro the ean position, which is also the equilibriu position. Further, at any point in its oscillation, this force is directed towards the ean position. In practice, oscillating bodies eventually coe to rest at their equilibriu positions, because of the daping due to friction and other dissipative causes. However, they can be forced to reain oscillating by eans of soe external periodic agency. We discuss the phenoena of daped and forced oscillations later in the chapter. Any aterial ediu can be pictured as a collection of a large nuber of coupled oscillators. The collective oscillations of the constituents of a ediu anifest theselves as waves. Exaples of waves include water waves, seisic waves, electroagnetic waves. We shall study the wave phenoenon in the next chapter Period and frequency We have seen that any otion that repeats itself at regular intervals of tie is called periodic otion. The sallest interval of tie after which the otion is repeated is called its period. Let us denote the period by the sybol T. Its SI unit is second. For periodic otions,

3 338 which are either too fast or too slow on the scale of seconds, other convenient units of tie are used. The period of vibrations of a quartz crystal is expressed in units of icroseconds (10 6 s) abbreviated as μs. On the other hand, the orbital period of the planet Mercury is 88 earth days. The Halley s coet appears after every 76 years. The reciprocal of T gives the nuber of repetitions that occur per unit tie. This quantity is called the frequency of the periodic otion. It is represented by the sybol ν. The relation between v and T is v = 1/T (14.1) The unit of ν is thus s 1. After the discoverer of radio waves, Heinrich Rudolph Hertz ( ), a special nae has been given to the unit of frequency. It is called hertz (abbreviated as Hz). Thus, 1 hertz = 1 Hz =1 oscillation per second =1s 1 (14.) Note, that the frequency, ν, is not necessarily an integer. Exaple 14.1 On an average a huan heart is found to beat 75 ties in a inute. Calculate its frequency and period. Answer The beat frequency of heart = 75/(1 in) = 75/(60 s) = 1.5 s 1 = 1.5 Hz The tie period T = 1/(1.5 s 1 ) = 0.8 s 14.. Displaceent In section 4., we defined displaceent of a particle as the change in its position vector. In Fig. 14.(a) A block attached to a spring, the other end of which is fixed to a rigid wall. The block oves on a frictionless surface. The otion of the block can be described in ters of its distance or displaceent x fro the wall. PHYSICS this chapter, we use the ter displaceent in a ore general sense. It refers to change with tie of any physical property under consideration. For exaple, in case of rectilinear otion of a steel ball on a surface, the distance fro the starting point as a function of tie is its position displaceent. The choice of origin is a atter of convenience. Consider a block attached to a spring, the other end of which is fixed to a rigid wall [see Fig.14.(a)]. Generally it is convenient to easure displaceent of the body fro its equilibriu position. For an oscillating siple pendulu, the angle fro the vertical as a function of tie ay be regarded as a displaceent variable [see Fig.14.(b)]. The ter displaceent is not always to be referred Fig.14.(b) An oscillating siple pendulu; its otion can be described in ters of angular displaceent θ fro the vertical. in the context of position only. There can be any other kinds of displaceent variables. The voltage across a capacitor, changing with tie in an a.c. circuit, is also a displaceent variable. In the sae way, pressure variations in tie in the propagation of sound wave, the changing electric and agnetic fields in a light wave are exaples of displaceent in different contexts. The displaceent variable ay take both positive and negative values. In experients on oscillations, the displaceent is easured for different ties. The displaceent can be represented by a atheatical function of tie. In case of periodic otion, this function is periodic in tie. One of the siplest periodic functions is given by f (t) = A cos ωt (14.3a) If the arguent of this function, ωt, is increased by an integral ultiple of π radians,

4 OSCILLATIONS 339 the value of the function reains the sae. The function f (t ) is then periodic and its period, T, is given by π T = (14.3b) ω Thus, the function f (t) is periodic with period T, f (t) = f (t+t ) The sae result is obviously correct if we consider a sine function, f (t ) = A sin ωt. Further, a linear cobination of sine and cosine functions like, f (t) = A sin ωt + B cos ωt (14.3c) is also a periodic function with the sae period T. Taking, A = D cos φ and B = D sin φ Eq. (14.3c) can be written as, f (t) = D sin (ωt + φ ), Here D and φ are constant given by D = A + B 1 and tan B A (14.3d) The great iportance of periodic sine and cosine functions is due to a rearkable result proved by the French atheatician, Jean Baptiste Joseph Fourier ( ): Any periodic function can be expressed as a superposition of sine and cosine functions of different tie periods with suitable coefficients. Exaple 14. Which of the following functions of tie represent (a) periodic and (b) non-periodic otion? Give the period for each case of periodic otion [ω is any positive constant]. (i) sin ωt + cos ωt (ii) sin ωt + cos ωt + sin 4 ωt (iii) e ωt (iv) log (ωt) Answer (i) sin ωt + cos ωt is a periodic function, it can also be written as sin (ωt + π/4). Now sin (ωt + π/4)= sin (ωt + π/4+π) (ii) (iii) (iv) = sin [ω (t + π/ω) + π/4] The periodic tie of the function is π/ω. This is an exaple of a periodic otion. It can be noted that each ter represents a periodic function with a different angular frequency. Since period is the least interval of tie after which a function repeats its value, sin ωt has a period T 0 = π/ω; cos ωt has a period π/ω =T 0 /; and sin 4 ωt has a period π/4ω = T 0 /4. The period of the first ter is a ultiple of the periods of the last two ters. Therefore, the sallest interval of tie after which the su of the three ters repeats is T 0, and thus the su is a periodic function with a period π/ω. The function e ωt is not periodic, it decreases onotonically with increasing tie and tends to zero as t and thus, never repeats its value. The function log(ωt) increases onotonically with tie t. It, therefore, never repeats its value and is a non-periodic function. It ay be noted that as t, log(ωt) diverges to. It, therefore, cannot represent any kind of physical displaceent SIMPLE HARMONIC MOTION Let us consider a particle vibrating back and forth about the origin of an x-axis between the liits +A and A as shown in Fig In between these extree positions the particle Fig A particle vibrating back and forth about the origin of x-axis, between the liits +A and A. oves in such a anner that its speed is axiu when it is at the origin and zero when it is at ± A. The tie t is chosen to be zero when the particle is at +A and it returns to +A at t = T. In this section we will describe this otion. Later, we shall discuss how to achieve it. To study the otion of this particle, we record its positions as a function of tie

5 340 by taking snapshots at regular intervals of tie. A set of such snapshots is shown in Fig The position of the particle with reference to the origin gives its displaceent at any instant of tie. For such a otion the displaceent x(t) of the particle fro a certain chosen origin is found to vary with tie as, PHYSICS Phase x(t) = A cos (ω t + φ ) Displaceent Aplitude Angular Phase frequency constant { Fig A reference of the quantities in Eq. (14.4). The otion represented by Eq. (14.4) is called siple haronic otion (SHM); a ter that eans the periodic otion is a sinusoidal function of tie. Equation (14.4), in which the sinusoidal function is a cosine function, is plotted in Fig The quantities that deterine Fig Fig A sequence of snapshots (taken at equal intervals of tie) showing the position of a particle as it oscillates back and forth about the origin along an x-axis, between the liits +A and A. The length of the vector arrows is scaled to indicate the speed of the particle. The speed is axiu when the particle is at the origin and zero when it is at ± A. If the tie t is chosen to be zero when the particle is at +A, then the particle returns to +A at t = T, where T is the period of the otion. The otion is then repeated. It is represented by Eq. (14.4) for φ = 0. A graph of x as a function of tie for the otion represented by Eq. (14.4). x (t) = A cos (ωt + φ) (14.4) in which A, ω, and φ are constants. Fig (a) A plot of displaceent as a function of tie as obtained fro Eq. (14.4) with φ = 0. The curves 1 and are for two different aplitudes A and B. the shape of the graph are displayed in Fig along with their naes. We shall now define these quantities. The quantity A is called the aplitude of the otion. It is a positive constant which represents the agnitude of the axiu displaceent of the particle in either direction. The cosine function in Eq. (14.4) varies between the liits ±1, so the displaceent x(t) varies between the liits ± A. In Fig (a), the curves 1 and are plots of Eq. (14.4) for two different aplitudes A and B. The difference between these curves illustrates the significance of aplitude. The tie varying quantity, (ωt + φ ), in Eq. (14.4) is called the phase of the otion. It describes the state of otion at a given tie. The constant φ is called the phase constant (or phase angle). The value of φ depends on the displaceent and velocity of the particle at t = 0. This can be understood better by considering Fig. 14.7(b). In this figure, the curves 3 and 4 represent plots of Eq. (14.4) for two values of the phase constant φ. It can be seen that the phase constant signifies the initial conditions. The constant ω, called the angular frequency of the otion, is related to the period T. To get

6 OSCILLATIONS 341 We have had an introduction to siple haronic otion. In the next section we will discuss the siplest exaple of siple haronic otion. It will be shown that the projection of unifor circular otion on a diaeter of the circle executes siple haronic otion. Fig (b) A plot obtained fro Eq The curves 3 and 4 are for φ = 0 and -π/4 respectively. The aplitude A is sae for both the plots. their relationship, let us consider Eq. (14.4) with φ = 0; it then reduces to, x(t ) = A cos ωt (14.5) Now since the otion is periodic with a period T, the displaceent x (t) ust return to its initial value after one period of the otion; that is, x (t) ust be equal to x (t + T ) for all t. Applying this condition to Eq. (14.5) leads to, A cos ωt = A cos ω(t + T ) (14.6) As the cosine function first repeats itself when its arguent (the phase) has increased by π, Eq. (14.6) gives, ω(t + T ) = ωt + π or ωt = π Thus, the angular frequency is, ω = π/ T (14.7) The SI unit of angular frequency is radians per second. To illustrate the significance of period T, sinusoidal functions with two different periods are plotted in Fig Fig Plots of Eq. (14.4) for φ = 0 for two different periods. In this plot the SHM represented by curve a, has a period T and that represented by curve b, has a period T / =T/. Exaple 14.3 Which of the following functions of tie represent (a) siple haronic otion and (b) periodic but not siple haronic? Give the period for each case. (1) sin ωt cos ωt () sin ωt Answer (a) sin ωt cos ωt = sin ωt sin (π/ ωt) = cos (π/4) sin (ωt π/4) = sin (ωt π/4) This function represents a siple haronic otion having a period T = π/ω and a phase angle ( π/4) or (7π/4) (b) sin ωt = ½ ½ cos ωt The function is periodic having a period T = π/ω. It also represents a haronic otion with the point of equilibriu occurring at ½ instead of zero SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION In 1610, Galileo discovered four principal oons of the planet Jupiter. To hi, each oon seeed to ove back and forth relative to the planet in a siple haronic otion; the disc of the planet foring the id point of the otion. The record of his observations, written in his own hand, is still available. Based on his data, the position of the oon Callisto relative to Jupiter is plotted in Fig In this figure, the circles represent Galileo s data points and the curve drawn is a best fit to the data. The curve obeys Eq. (14.4), which is the displaceent function for SHM. It gives a period of about 16.8 days. It is now well known that Callisto oves with essentially a constant speed in an alost circular orbit around Jupiter. Its true otion is unifor circular otion. What Galileo saw and what we can also see, with a good pair of binoculars, is the projection of this unifor circular otion on a line in the plane of otion. This can easily be visualised by perforing a

7 34 Fig The angle between Jupiter and its oon Callisto as seen fro earth. The circles are based on Galileo s easureents of The curve is a best fit suggesting a siple haronic otion.at Jupiter s ean distance,10 inutes of arc corresponds to about 10 6 k. siple experient. Tie a ball to the end of a string and ake it ove in a horizontal plane about a fixed point with a constant angular speed. The ball would then perfor a unifor circular otion in the horizontal plane. Observe the ball sideways or fro the front, fixing your attention in the plane of otion. The ball will appear to execute to and fro otion along a horizontal line with the point of rotation as the idpoint. You could alternatively observe the shadow of the ball on a wall which is perpendicular to the plane of the circle. In this process what we are observing is the otion of the ball on a diaeter of the circle noral to the direction of viewing. This experient provides an analogy to Galileo s observation. In Fig , we show the otion of a reference particle P executing a unifor circular otion with (constant) angular speed ω in a reference circle. The radius A of the circle is the agnitude of the particle s position vector. At any tie t, the angular position of the particle Fig The otion of a reference particle P executing a unifor circular otion with (constant) angular speed ω in a reference circle of radius A. PHYSICS is ωt + φ, where φ is its angular position at t = 0. The projection of particle P on the x-axis is a point P, which we can take as a second particle. The projection of the position vector of particle P on the x-axis gives the location x(t) of P. Thus we have, x(t) = A cos (ωt + φ ) which is the sae as Eq. (14.4). This shows that if the reference particle P oves in a unifor circular otion, its projection particle P executes a siple haronic otion along a diaeter of the circle. Fro Galileo s observation and the above considerations, we are led to the conclusion that circular otion viewed edge-on is siple haronic otion. In a ore foral language we can say that : Siple haronic otion is the projection of unifor circular otion on a diaeter of the circle in which the latter otion takes place. Exaple 14.4 Fig depicts two circular otions. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on the figures. Obtain the siple haronic otions of the x-projection of the radius vector of the rotating particle P in each case. Fig Answer (a) At t = 0, OP akes an angle of 45 o = π/4 rad with the (positive direction of) x-axis. After tie t, it covers an angle π t in the T anticlockwise sense, and akes an angle of π π t + with the x-axis. T 4 The projection of OP on the x-axis at tie t is given by,

8 OSCILLATIONS 343 (b) x (t) = A cos ( t + T 4 For T = 4 s, x(t) = A cos t which is a SHM of aplitude A, period 4 s, and an initial phase* =. 4 In this case at t = 0, OP akes an angle of 90 o = ) with the x-axis. After a tie t, it covers an angle of t T in the clockwise sense and akes an angle of t T with the x-axis. The projection of OP on the x-axis at tie t is given by 14.5 VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION It can be seen easily that the agnitude of velocity, v, with which the reference particle P (Fig ) is oving in a circle is related to its angular speed, ω, as v = ω A (14.8) where A is the radius of the circle described by the particle P. The agnitude of the velocity vector v of the projection particle is ωa ; its projection on the x-axis at any tie t, as shown in Fig. 14.1, is v(t) = ωa sin (ωt + φ ) (14.9) x(t) = B cos T t = B sin ( t T For T = 30 s, x(t) = B sin 15 t ) Writing this as x (t) = B cos t, and 15 coparing with Eq. (14.4). We find that this represents a SHM of aplitude B, period 30 s, Fig The velocity, v (t), of the particle P is the projection of the velocity v of the reference particle, P. The negative sign appears because the velocity coponent of P is directed towards the left, in the negative direction of x. Equation (14.9) expresses the instantaneous velocity of the particle P (projection of P). Therefore, it expresses the instantaneous velocity of a particle executing SHM. Equation (14.9) can also be obtained by differentiating Eq. (14.4) with respect to tie as, and an initial phase of. d v(t) = x () t dt (14.10) * The natural unit of angle is radian, defined through the ratio of arc to radius. Angle is a diensionless quantity. Therefore it is not always necessary to ention the unit radian when we use π, its ultiples or subultiples. The conversion between radian and degree is not siilar to that between etre and centietre or ile. If the arguent of a trigonoetric function is stated without units, it is understood that the unit is radian. On the other hand, if degree is to be used as the unit of angle, then it ust be shown explicitly. For exaple, sin(15 0 ) eans sine of 15 degree, but sin(15) eans sine of 15 radians. Hereafter, we will often drop rad as the unit, and it should be understood that whenever angle is entioned as a nuerical value, without units, it is to be taken as radians.

9 344 PHYSICS Fig The acceleration, a(t), of the particle P is the projection of the acceleration a of the reference particle P. We have seen that a particle executing a unifor circular otion is subjected to a radial acceleration a directed towards the centre. Figure shows such a radial acceleration, a, of the reference particle P executing unifor circular otion. The agnitude of the radial acceleration of P is ω A. Its projection on the x-axis at any tie t is, a (t) = ω A cos (ωt + φ) = ω x (t) (14.11) which is the acceleration of the particle P (the projection of particle P). Equation (14.11), therefore, represents the instantaneous acceleration of the particle P, which is executing SHM. Thus Eq. (14.11) expresses the acceleration of a particle executing SHM. It is an iportant result for SHM. It shows that in SHM, the acceleration is proportional to the displaceent and is always directed towards the ean position. Eq. (14.11) can also be obtained by differentiating Eq. (14.9) with respect to tie as, d at () = vt () (14.1) dt The inter-relationship between the displaceent of a particle executing siple haronic otion, its velocity and acceleration can be seen in Fig In this figure (a) is a plot of Eq. (14.4) with φ = 0 and (b) depicts Eq. (14.9) also with φ = 0. Siilar to the aplitude A in Eq. (14.4), the positive quantity ω A in Eq. (14.9) is called the velocity aplitude v. In Fig (b), it can be seen that the velocity of the Fig The particle displaceent, velocity and acceleration in a siple haronic otion. (a) The displaceent x (t) of a particle executing SHM with phase angle φ equal to zero. (b) The velocity v (t) of the particle. (c) The acceleration a (t) of the particle. oscillating particle varies between the liits ± v = ± ωa. Note that the curve of v(t) is shifted (to the left) fro the curve of x(t) by one quarter period and thus the particle velocity lags behind the displaceent by a phase angle of π/; when the agnitude of displaceent is the greatest, the agnitude of the velocity is the least. When the agnitude of displaceent is the least, the velocity is the greatest. Figure14.14(c) depicts the variation of the particle acceleration a(t). It is seen that when the displaceent has its greatest positive value, the acceleration has its greatest negative value and vice versa. When the displaceent is zero, the acceleration is also zero. Exaple 14.5 A body oscillates with SHM according to the equation (in SI units), x = 5 cos [π t + π/4]. At t = 1.5 s, calculate the (a) displaceent, (b) speed and (c) acceleration of the body. Answer The angular frequency ω of the body = π s 1 and its tie period T = 1 s. At t = 1.5 s (a) displaceent = (5.0 ) cos [(π s 1 ) 1.5 s + π/4]

10 OSCILLATIONS 345 (b) (c) = (5.0 ) cos [(3π + π/4)] = = Using Eq. (14.9), the speed of the body = (5.0 )(π s 1 ) sin [(π s 1 ) 1.5 s + π/4] = (5.0 )(π s 1 ) sin [(3π + π/4)] = 10π s 1 = s 1 Using Eq.(14.10), the acceleration of the body = (π s 1 ) displaceent = (π s 1 ) ( ) = 140 s 14.6 FORCE LAW FOR SIMPLE HARMONIC MOTION In Section14.3, we described the siple haronic otion. Now we discuss how it can be generated. Newton s second law of otion relates the force acting on a syste and the corresponding acceleration produced. Therefore, if we know how the acceleration of a particle varies with tie, this law can be used to learn about the force, which ust act on the particle to give it that acceleration. If we cobine Newton s second law and Eq. (14.11), we find that for siple haronic otion, F (t ) = a = ω x (t) or F (t) = k x (t ) (14.13) where k = ω (14.14a) or k ω = (14.14b) Equation (14.13) gives the force acting on the particle. It is proportional to the displaceent and directed in an opposite direction. Therefore, it is a restoring force. Note that unlike the centripetal force for unifor circular otion that is constant in agnitude, the restoring force for SHM is tie dependent. The force law expressed by Eq. (14.13) can be taken as an alternative definition of siple haronic otion. It states : Siple haronic otion is the otion executed by a particle subject to a force, which is proportional to the displaceent of the particle and is directed towards the ean position. Since the force F is proportional to x rather than to soe other power of x, such a syste is also referred to as a linear haronic oscillator. Systes in which the restoring force is a nonlinear function of x are tered as non-linear haronic or anharonic oscillators. Exaple 14.6 Two identical springs of spring constant k are attached to a block of ass and to fixed supports as shown in Fig Show that when the ass is displaced fro its equilibriu position on either side, it executes a siple haronic otion. Find the period of oscillations. Fig Answer Let the ass be displaced by a sall distance x to the right side of the equilibriu position, as shown in Fig Under this situation the spring on the left side gets Fig elongated by a length equal to x and that on the right side gets copressed by the sae length. The forces acting on the ass are then, F 1 = k x (force exerted by the spring on the left side, trying to pull the ass towards the ean position) F = k x (force exerted by the spring on the right side, trying to push the ass towards the ean position)

11 346 PHYSICS The net force, F, acting on the ass is then given by, F = kx Hence the force acting on the ass is proportional to the displaceent and is directed towards the ean position; therefore, the otion executed by the ass is siple haronic. The tie period of oscillations is, T= π k 14.7 ENERGY IN SIMPLE HARMONIC MOTION A particle executing siple haronic otion has kinetic and potential energies, both varying between the liits, zero and axiu. In section14.5 we have seen that the velocity of a particle executing SHM, is a periodic function of tie. It is zero at the extree positions of displaceent. Therefore, the kinetic energy (K) of such a particle, which is defined as 1 K = v 1 ω sin ( ω + φ) = A t 1 = k A sin ( ωt + φ ) (14.15) is also a periodic function of tie, being zero when the displaceent is axiu and axiu when the particle is at the ean position. Note, since the sign of v is iaterial in K, the period of K is T/. What is the potential energy (PE) of a particle executing siple haronic otion? In Chapter 6, we have seen that the concept of potential energy is possible only for conservative forces. The spring force F = kx is a conservative force, with associated potential energy 1 U = k x 14.16) Hence the potential energy of a particle executing siple haronic otion is, U(x) = 1 1 k x = k A t cos ( ω + φ ) (14.17) Thus the potential energy of a particle executing siple haronic otion is also periodic, with period T/, being zero at the ean position and axiu at the extree displaceents. It follows fro Eqs. (14.15) and (14.17) that the total energy, E, of the syste is, E = U + K 1 1 cos ( ω + φ) + sin ( ω + φ) = k A t k A t 1 cos ( ω + φ) + sin ( ω + φ) = k A t t The quantity within the square brackets above is unity and we have, 1 E = k A (14.18) The total echanical energy of a haronic oscillator is thus independent of tie as expected for otion under any conservative force. The tie and displaceent dependence of the potential and kinetic energies of a linear siple haronic oscillator are shown in Fig It is observed that in a linear haronic oscillator, all energies are positive and peak twice during every period. For x = 0, the energy is all kinetic and for x = ± A it is all potential. In between these extree positions, the potential energy increases at the expense of kinetic energy. This behaviour of a linear haronic oscillator suggests that it possesses an eleent of springiness and an eleent of inertia. The forer stores its potential energy and the latter stores its kinetic energy.

12 OSCILLATIONS 347 Its displaceent at any tie t is then given by, x(t) = 0.1 cos (7.07t) Therefore, when the particle is 5 c away fro the ean position, we have 0.05 = 0.1 cos (7.07t) Or cos (7.07t) = 0.5 and hence sin (7.07t) = 3 = 0.866, Fig (a) Potential energy U(t), kinetic energy K(t) and the total energy E as functions of tie t for a linear haronic oscillator. All energies are positive and the potential and kinetic energies peak twice in every period of the oscillator. (b) Potential energy U(x), kinetic energy K(x) and the total energy E as functions of position x for a linear haronic oscillator with aplitude A. For x = 0, the energy is all kinetic and for x = ± A it is all potential. Exaple 14.7 A block whose ass is 1 kg is fastened to a spring. The spring has a spring constant of 50 N 1. The block is pulled to a distance x = 10 c fro its equilibriu position at x = 0 on a frictionless surface fro rest at t = 0. Calculate the kinetic, potential and total energies of the block when it is 5 c away fro the ean position. Answer The block executes SHM, its angular frequency, as given by Eq. (14.14b), is ω = k 50 N 1 = 1kg = 7.07 rad s 1 Then the velocity of the block at x = 5 c is = s 1 = 0.61 s 1 Hence the K.E. of the block, 1 = v = ½[1kg (0.613 s 1 ) ] = 0.19 J The P.E. of the block, 1 = k x = ½(50 N ) = J The total energy of the block at x = 5 c, = K.E. + P.E. = 0.5 J we also know that at axiu displaceent, K.E. is zero and hence the total energy of the syste is equal to the P.E. Therefore, the total energy of the syste, = ½(50 N ) = 0.5 J which is sae as the su of the two energies at a displaceent of 5 c. This is in confority with the principle of conservation of energy SOME SYSTEMS EXECUTING SIMPLE HARMONIC MOTION There are no physical exaples of absolutely pure siple haronic otion. In practice we coe across systes that execute siple haronic otion approxiately under certain conditions. In the subsequent part of this section, we discuss the otion executed by soe such systes.

13 348 PHYSICS and the period, T, of the oscillator is given by, T = k (14.1) Equations (14.0) and (14.1) tell us that a large angular frequency and hence a sall period is associated with a stiff spring (high k) and a light block (sall ). Fig A linear siple haronic oscillator consisting of a block of ass attached to a spring. The block oves over a frictionless surface. Once pulled to the side and released, it executes siple haronic otion Oscillations due to a Spring The siplest observable exaple of siple haronic otion is the sall oscillations of a block of ass fixed to a spring, which in turn is fixed to a rigid wall as shown in Fig The block is placed on a frictionless horizontal surface. If the block is pulled on one side and is released, it then executes a to and fro otion about a ean position. Let x = 0, indicate the position of the centre of the block when the spring is in equilibriu. The positions arked as A and +A indicate the axiu displaceents to the left and the right of the ean position. We have already learnt that springs have special properties, which were first discovered by the English physicist Robert Hooke. He had shown that such a syste when defored, is subject to a restoring force, the agnitude of which is proportional to the deforation or the displaceent and acts in opposite direction. This is known as Hooke s law (Chapter 9). It holds good for displaceents sall in coparison to the length of the spring. At any tie t, if the displaceent of the block fro its ean position is x, the restoring force F acting on the block is, F (x) = k x (14.19) The constant of proportionality, k, is called the spring constant, its value is governed by the elastic properties of the spring. A stiff spring has large k and a soft spring has sall k. Equation (14.19) is sae as the force law for SHM and therefore the syste executes a siple haronic otion. Fro Eq. (14.14) we have, ω = k (14.0) Exaple 14.8 A 5 kg collar is attached to a spring of spring constant 500 N 1. It slides without friction over a horizontal rod. The collar is displaced fro its equilibriu position by 10.0 c and released. Calculate (a) the period of oscillation, (b) the axiu speed and (c) axiu acceleration of the collar. Answer (a) The period of oscillation as given by Eq. (14.1) is, (b) (c) T = k 5.0 kg = π N = (π/10) s = 0.63 s The velocity of the collar executing SHM is given by, v(t) = Aω sin (ωt + φ) The axiu speed is given by, v = Aω = 0.1 = 0.1 k 500 N 1 5 kg = 1 s 1 and it occurs at x = 0 The acceleration of the collar at the displaceent x (t) fro the equilibriu is given by, a (t) = ω x(t) = k x(t) Therefore the axiu acceleration is, a ax = ω A

14 OSCILLATIONS N 1 = x kg = 10 s and it occurs at the extreities The Siple Pendulu It is said that Galileo easured the periods of a swinging chandelier in a church by his pulse beats. He observed that the otion of the chandelier was periodic. The syste is a kind of pendulu. You can also ake your own pendulu by tying a piece of stone to a long unstretchable thread, approxiately 100 c long. Suspend your pendulu fro a suitable support so that it is free to oscillate. Displace the stone to one side by a sall distance and let it go. The stone executes a to and fro otion, it is periodic with a period of about two seconds. Is this otion siple haronic? To answer this question, we consider a siple pendulu, which consists of a particle of ass (called the bob of the pendulu) suspended fro one end of an unstretchable, assless string of length L fixed at the other end as shown in Fig (a). The bob is free to swing to and fro in the plane of the page, to the left and right of a vertical line through the pivot point. The forces acting on the bob are the force T, tension in the string and the gravitational force F g (= g), as shown in Fig (b). The string akes an angle θ with the vertical. We resolve the force F g into a radial coponent F g cos θ and a tangential coponent F g sin θ. The radial coponent is cancelled by the tension, since there is no otion along the length of the string. The tangential coponent produces a restoring torque about the pendulu s pivot point. This Fig (a) A siple pendulu. (b) The forces acting on the bob are the force due to gravity, F g (= g), and the tension T in the string. (b) The tangential coponent F g of the gravitational force is a restoring force that tends to bring the pendulu back to the central position. torque always acts opposite to the displaceent of the bob so as to bring it back towards its central location. The central location is called the equilibriu position (θ = 0), because at this position the pendulu would be at rest if it were not swinging. The restoring torque τ is given by, τ = L (F g sinθ ) (14.) where the negative sign indicates that the torque acts to reduce θ, and L is the length of the oent ar of the force F g sin θ about the pivot point. For rotational otion we have, τ = I α (14.3) where I is the pendulu s rotational inertia about the pivot point and α is its angular acceleration about that point. Fro Eqs. (14.) and (14.3) we have, L (F g sin θ ) = I α (14.4) Substituting the agnitude of F g, i.e. g, we have, L g sin θ = I α Or, (b) gl α = sin (14.5) I We can siplify Eq. (14.5) if we assue that the displaceent θ is sall. We know that sin θ can be expressed as, (a) sin ±... 3! 5! where θ is in radians. (14.6)

15 350 PHYSICS Now if θ is sall, sin θ can be approxiated by θ and Eq. (14.5) can then be written as, gl α = θ (14.7) I In Table 14.1, we have listed the angle θ in degrees, its equivalent in radians, and the value of the function sin θ. Fro this table it can be seen that for θ as large as 0 degrees, sin θ is nearly the sae as θ expressed in radians. Table 14.1 sin θ as a function of angle θ SHM - how sall should the aplitude be? When you perfor the experient to deterine the tie period of a siple pendulu, your teacher tells you to keep the aplitude sall. But have you ever asked how sall is sall? Should the aplitude to 5 0, 0, 1 0, or 0.5 0? Or could it be 10 0, 0 0, or 30 0? To appreciate this, it would be better to easure the tie period for different aplitudes, up to large aplitudes. Of course, for large oscillations, you will have to take care that the pendulu oscillates in a vertical plane. Let us denote the tie period for sall-aplitude oscillations as T (0) and write the tie period for aplitude θ 0 as T(θ 0 ) = ct (0), where c is the ultiplying factor. If you plot a graph of c versus θ 0, you will get values soewhat like this: θ 0 : Equation (14.7) is the angular analogue of Eq. (14.11) and tells us that the angular acceleration of the pendulu is proportional to the angular displaceent θ but opposite in sign. Thus as the pendulu oves to the right, its pull to the left increases until it stops and begins to return to the left. Siilarly, when it oves towards left, its acceleration to the right tends to return it to the right and so on, as it swings to and fro in SHM. Thus the otion of a siple pendulu swinging through sall angles is approxiately SHM. Coparing Eq. (14.7) with Eq. (14.11), we see that the angular frequency of the pendulu is, ω = gl I and the period of the pendulu, T, is given by, T I gl (14.8) All the ass of a siple pendulu is centred in the ass of the bob, which is at a radius of L fro the pivot point. Therefore, for this syste, we can write I = L and substituting this in Eq. (14.8) we get, c : This eans that the error in the tie period is about % at an aplitude of 0 0, 5% at an aplitude of 50 0, and 10% at an aplitude of 70 0 and 18% at an aplitude of In the experient, you will never be able to easure T (0) because this eans there are no oscillations. Even theoretically, sin θ is exactly equal to θ only for θ = 0. There will be soe inaccuracy for all other values of θ. The difference increases with increasing θ. Therefore we have to decide how uch error we can tolerate. No easureent is ever perfectly accurate. You ust also consider questions like these: What is the accuracy of the stopwatch? What is your own accuracy in starting and stopping the stopwatch? You will realise that the accuracy in your easureents at this level is never better than 5% or 10%. Since the above table shows that the tie period of the pendulu increases hardly by 5% at an aplitude of 50 0 over its low aplitude value, you could very well keep the aplitude to be 50 in your experients.

16 OSCILLATIONS 351 L T (14.9) g Equation (14.9) represents a siple expression for the tie period of a siple pendulu. Exaple 14.9 What is the length of a siple pendulu, which ticks seconds? Answer Fro Eq. (14.9), the tie period of a siple pendulu is given by, turn, exerts an inhibiting drag force (viscous drag) on it and thus on the entire oscillating syste. With tie, the echanical energy of the blockspring syste decreases, as energy is transferred to the theral energy of the liquid and vane. Let the daping force exerted by the liquid on the syste be* F d. Its agnitude is proportional to the velocity v of the vane or the T L g Fro this relation one gets, gt L 4 The tie period of a siple pendulu, which ticks seconds, is s. Therefore, for g = 9.8 s and T = s, L is 9.8( s ) 4(s ) 4 = DAMPED SIMPLE HARMONIC MOTION We know that the otion of a siple pendulu, swinging in air, dies out eventually. Why does it happen? This is because the air drag and the friction at the support oppose the otion of the pendulu and dissipate its energy gradually. The pendulu is said to execute daped oscillations. In daped oscillations, although the energy of the syste is continuously dissipated, the oscillations reain apparently periodic. The dissipating forces are generally the frictional forces. To understand the effect of such external forces on the otion of an oscillator, let us consider a syste as shown in Fig Here a block of ass oscillates vertically on a spring with spring constant k. The block is connected to a vane through a rod (the vane and the rod are considered to be assless). The vane is suberged in a liquid. As the block oscillates up and down, the vane also oves along with it in the liquid. The up and down otion of the vane displaces the liquid, which in Fig A daped siple haronic oscillator. The vane iersed in a liquid exerts a daping force on the block as it oscillates up and down. block. The force acts in a direction opposite to the direction of v. This assuption is valid only when the vane oves slowly. Then for the otion along the x-axis (vertical direction as shown in Fig. 14.0), we have F d = b v (14.30) where b is a daping constant that depends on the characteristics of the liquid and the vane. The negative sign akes it clear that the force is opposite to the velocity at every oent. When the ass is attached to the spring and released, the spring will elongate a little and the ass will settle at soe height. This position, shown by O in Fig 14.0, is the equilibriu position of the ass. If the ass is pulled down or pushed up a little, the restoring force on the block due to the spring is F S = kx, where x is the displaceent of the ass fro its equilibriu position. Thus the total force acting * Under gravity, the block will be at a certain equilibriu position O on the spring; x here represents the displaceent fro that position.

17 35 on the ass at any tie t is F = k x b v. If a(t) is the acceleration of the ass at tie t, then by Newton s second law of otion for force coponents along the x-axis, we have a(t) = k x(t) b v(t) (14.31) Here we have dropped the vector notation because we are discussing one-diensional otion. Substituting dx/dt for v(t) and d x/dt for the acceleration a(t) and rearranging gives us the differential equation, dx b k x 0 dt d x dt (14.3) The solution of Eq. (14.3) describes the otion of the block under the influence of a daping force which is proportional to velocity. The solution is found to be of the for x(t) = A e b t/ cos (ω t + φ ) (14.33) where a is the aplitude and ω is the angular frequency of the daped oscillator given by, k b ω ' = (14.34) 4 In this function, the cosine function has a period π/ω but the function x(t) is not strictly periodic because of the factor e b t/ which decreases continuously with tie. However, if the decrease is sall in one tie period T, the otion represented by Eq. (14.33) is approxiately periodic. The solution, Eq. (14.33), can be graphically represented as shown in Fig We can PHYSICS regard it as a cosine function whose aplitude, which is Ae b t/, gradually decreases with tie. If b = 0 (there is no daping), then Eqs. (14.33) and (14.34) reduce to Eqs. (14.4) and (14.14b), expressions for the displaceent and angular frequency of an undaped oscillator. We have seen that the echanical energy of an undaped oscillator is constant and is given by Eq. (14.18) (E =1/ k A ). If the oscillator is daped, the echanical energy is not constant but decreases with tie. If the daping is sall, we can find E (t) by replacing A in Eq. (14.18) by Ae bt/, the aplitude of the daped oscillations. Thus we find, 1 b t/ Et () = k A e (14.35) Equation (14.35) shows that the total energy of the syste decreases exponentially with tie. Note that sall daping eans that the diensionless ratio b is uch less than 1. k Exaple For the daped oscillator shown in Fig. 14.0, the ass of the block is 00 g, k = 90 N 1 and the daping constant b is 40 g s 1. Calculate (a) the period of oscillation, (b) tie taken for its aplitude of vibrations to drop to half of its initial value and (c) the tie taken for its echanical energy to drop to half its initial value. Answer (a) We see that k = = 18 kg N 1 = kg s ; therefore k = 4.43 kg s 1, and b = 0.04 kg s 1. Therefore b is uch less than k. Hence the tie period T fro Eq. (14.34) is given by T k Fig Displaceent as a function of tie in daped haronic oscillations. Daping goes on increasing successively fro curve a to d. 0. kg 90 N ±1 = 0.3 s (b) Now, fro Eq. (14.33), the tie, T 1/, for the aplitude to drop to half of its initial value is given by,

18 OSCILLATIONS 353 T = 1/ ln(1/) b/ s = 6.93 s (c) For calculating the tie, t 1/, for its echanical energy to drop to half its initial value we ake use of Eq. (14.35). Fro this equation we have, E (t 1/ )/E (0) = exp ( bt 1/ /) Or ½ = exp ( bt 1/ /) ln (1/) = (bt 1/ /) Or t 1/ g s 00 g = 3.46 s This is just half of the decay period for aplitude. This is not suprising, because, according to Eqs. (14.33) and (14.35), energy depends on the square of the aplitude. Notice that there is a factor of in the exponents of the two exponentials FORCED OSCILLATIONS AND RESONANCE A person swinging in a swing without anyone pushing it or a siple pendulu, displaced and released, are exaples of free oscillations. In both the cases, the aplitude of swing will gradually decrease and the syste would, ultiately, coe to a halt. Because of the everpresent dissipative forces, the free oscillations cannot be sustained in practice. They get daped as seen in section However, while swinging in a swing if you apply a push periodically by pressing your feet against the ground, you find that not only the oscillations can now be aintained but the aplitude can also be increased. Under this condition the swing has forced, or driven, oscillations. In case of a syste executing driven oscillations under the action of a haronic force, two angular frequencies are iportant : (1) the natural angular frequency ω of the syste, which is the angular frequency at which it will oscillate if it were displaced fro equilibriu position and then left to oscillate freely, and () the angular frequency ω d of the external force causing the driven oscillations. Suppose an external force F(t) of aplitude F 0 that varies periodically with tie is applied to a daped oscillator. Such a force can be represented as, F(t) = F o cos ω d t (14.36) The otion of a particle under the cobined action of a linear restoring force, daping force and a tie dependent driving force represented by Eq. (14.36) is given by, a(t) = k x(t) bv(t) + F o cos ω d t (14.37a) Substituting d x/dt for acceleration in Eq. (14.37a) and rearranging it, we get d x b dx kx F dt dt o cos ω d t (14.37b) This is the equation of an oscillator of ass on which a periodic force of (angular) frequency ω d is applied. The oscillator initially oscillates with its natural frequency ω. When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force. Its displaceent, after the natural oscillations die out, is given by x(t) = A cos (ω d t + φ ) (14.38) where t is the tie easured fro the oent when we apply the periodic force. The aplitude A is a function of the forced frequency ω d and the natural frequency ω. Analysis shows that it is given by A F 1/ d db (14.39a) and tan φ = v ο (14.39b) ωd xο where is the ass of the particle and v 0 and x 0 are the velocity and the displaceent of the particle at tie t = 0, which is the oent when we apply the periodic force. Equation (14.39) shows that the aplitude of the forced oscillator depends on the (angular) frequency of the driving force. We can see a different behaviour of the oscillator when ω d is far fro ω and when it is close to ω. We consider these two cases.

19 354 (a) Sall Daping, Driving Frequency far fro Natural Frequency : In this case, ω d b will be uch saller than (ω ω ), and we can d neglect that ter. Then Eq. (14.39) reduces to A F d (14.40) Figure 14. shows the dependence of the displaceent aplitude of an oscillator on the angular frequency of the driving force for different aounts of daping present in the syste. It ay be noted that in all the cases the aplitude is greatest when ω d /ω = 1. The curves in this figure show that saller the daping, the taller and narrower is the resonance peak. If we go on changing the driving frequency, the aplitude tends to infinity when it equals the natural frequency. But this is the ideal case of zero daping, a case which never arises in a real syste as the daping is never perfectly zero. You ust have experienced in a swing that when the tiing of your push exactly atches with the tie period of the swing, your swing gets the axiu aplitude. This aplitude is large, but not infinity, because there is always soe daping in your swing. This will becoe clear in the (b). (b) Driving Frequency Close to Natural Frequency : If ω d is very close to ω, (ω d ) PHYSICS would be uch less than ω d b, for any reasonable value of b, then Eq. (14.39) reduces to Fο A = (14.41) ωdb This akes it clear that the axiu possible aplitude for a given driving frequency is governed by the driving frequency and the daping, and is never infinity. The phenoenon of increase in aplitude when the driving force is close to the natural frequency of the oscillator is called resonance. In our daily life we encounter phenoena which involve resonance. Your experience with swings is a good exaple of resonance. You ight have realised that the skill in swinging to greater heights lies in the synchronisation of the rhyth of pushing against the ground with the natural frequency of the swing. To illustrate this point further, let us consider a set of five siple pendulus of assorted lengths suspended fro a coon rope as shown in Fig The pendulus 1 and 4 have the sae lengths and the others have different lengths. Now let us set pendulu 1 into otion. The energy fro this pendulu gets transferred to other pendulus through the connecting rope and they start oscillating. The driving force is provided through the connecting rope. The frequency of this force is the frequency with which pendulu 1 oscillates. If we observe the response of pendulus, 3 and 5, they first start oscillating with their natural frequencies Fig. 14. The aplitude of a forced oscillator as a function of the angular frequency of the driving force. The aplitude is greatest near ω d /ω = 1. The five curves correspond to different extents of daping present in the syste. Curve a corresponds to the least daping, and daping goes on increasing successively in curves b, c, d, e. Notice that the peak shifts to the left with increasing b. Fig A syste of five siple pendulus suspended fro a coon rope.