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OSCILLATIONS, , 7.1 Introduction, , Dear students, while studying the circular motion and the, projectile motion you have learnt that how the forces acting in, specific manners affect the trajectory of the particle. Even in, Std. IX, you have learnt about the wave motion, concept of, periodic motion (harmonic motion) and oscillatory motions, their, characteristics such as frequency, periodic time, amplitude etc., , The study of periodic motion is very much important in, physics. This motion plays fundamental role in understanding, generation and propagation of sound waves and electromagnetic, waves. You know that constituent particles like atoms, molecules, and ions too have oscillatory motion., , In this chapter first we refresh our concept of periodic, (harmonic) and oscillatory motion, and study such motion under, a position dependent forces. The mathematical formulations for, potential energy, kinetic energy and total mechanical energy will, also be seen. We shall also study the damping of oscillation,, forced oscillation and phenomenon of resonance., , 7.2 Periodic Motion and Oscillatory Motion, , If a body repeats its motion along a certain path, about, a fixed point, at a definite interval of time, it is said to, have a periodic motion,, , The motion of hands of a clock, motion of moon around the, earth, and the revolution of earth around the sun are the best, examples of periodic motion., , If a body repeatedly moves to and fro, back and forth,, or up and down about a fixed point in a definite interval, of time, such a motion is called an oscillatory motion. The, body performing such motion is called an oscillator., , The motion of the bob of the pendulum and the motion of a, | loaded spring are the known examples for oscillatory motion.

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OSCILLATIONS, , 157, , , , All oscillatory motions are periodic, motions but all periodic motions may not be, oscillatory. Like the motion of hands of a clock,, motion of the earth around the sun are periodic, but not an oscillatory. The concept of to and fro,, ‘back and forth or up and down about some fixed, point is not present in these cases., , ‘We will see that the oscillatory motion can, be expressed in terms of sine and cosine, functions. The sine and cosine trigonometric, functions are periodic functions having period of, 2m rad. In mathematics, these functions are, known as harmonic functions. Hence, oscillatory, motion is also called as harmonic motion., , 7.3 Simple Harmonic Motion (SHM), , Simple harmonic motion is a simplest type, of periodic motion,, , The periodic motion of a body about a, fixed point, om a linear path, under the, influence of the force acting towards the, fixed point and proportional to displacement, of the body from the fixed point is called a, simple harmonic motion (SHM)., , @ @®) ©, , , , Sr Ss, , eS, ltt ttt, , it, , 1, 14, ', , 1, , 4, , , , , , , , , , , A body performing simple harmonic motion, is known as simple harmonic oscillator (SHO)., , Let us consider a massless spring obeying, Hook’s Law. It is suspended vertically from a, rigid support as shown in Figure 7.1 and an object, of mass m is tied at its lower end. When we, pull an object downward and release, it will, perform (almost) simple harmonic motion., , Now use Figure 7.1 to understand some, basic terms associated with simple harmonic, motion., , Equilibrium position (Mean position), , The point about which the simple harmonic, oscillator performs simple harmonic motion is, known as equilibrium position or mean, position,, , In the Figure 7.1 (a), (e) and (i), the object, is at mean position., , Displacement, , The distance of the oscillator at any instant, from the equilibrium position is known as the, displacement of the oscillator at that instant., , Oo @ md, , , , position, , , , , , , , ‘Simple harmonic motion of a massive object attached to a spring and displacement—time graph, Figure 7.1

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158, , PHYSICS, , , , In the Figure 7.1 (b) the displacement of the, oscillator at t= ¢, is y,. The displacement of the, oscillator at t = f, is —y, Figure 7.1(/)., Amplitude :, , The maximum displacement of the oscillator, on either side of mean position is called, amplitude of the oscillator., , As shown in the Figure 7.1 (c, g), the, maximum displacement achieved by the oscillator, is y,. Hence for this, y, be the amplitude of the, oscillator., , Periodic Time (Time period or period) :, , The time required to complete one oscillation, is known as periodic time (T) of the oscillator., , In other words, the least interval of time after, which the periodic motion of an oscillator repeat, itself is called as periodic time of the oscillator., , SI unit of periodic time is second (s)., , For the oscillator of Figure 7.1, t, — f, is the, periodic time., , Ferquency :, , Frequency of simple harmonic oscillator is, defined as the number of oscillations completed, by the simple harmonic oscillator in one second., or hertz (H,), , It is denoted by f, and f =, Angular frequency :, , 2m times the frequency of an oscillator is, called the angular frequency of that oscillator,, , It is denoted by @ (= 2nf), , Its SI uint is rad s“'., , If we draw a graph of displacement of SHO, against time, it appears as shown in the lower part, of the Figure 7.1. Such motion can be represented, aimeennmnns ita |e, , Od sin GR or) (7.3.1), , Displacement Z male Angular eS aa, at time ft frequency, , Its SI unit is s™, , ean, We know that the span of the sine function, is [-1, 1]. Hence the displacement y(t) of the, SHM varies between +A (See Figure 7.2). If, , another SHM is represented by y(t) = B sin, (@t + >) with B < A than it will be of type -2, curve shown in Figure 7.2, And if B > A, the, curve will be of type -3., , , , , , y(t) = B sin (wt+ 9); B>A, Al--- y(t) = A sin (ot+ 6), , (0) = B sin (t+ 6); B<A, (for ¢ = 0), , Displacement of SHM as a function of time, Figure 7.2, , The quantity (wt + ¢) is known as phase of, the SHM at time f and represents the state of, motion at that time., , The phase at t = 0 is called initial phase,, phase constant or epoch (¢) of SHM., , For one complete oscillation, phase of SHM, increases by 27 rad and hence after n oscillations, its phase is increased by 2nm rad., , As the motion is periodic with time period T,, the displacement of the oscillator at (¢ + T) must, be the same for any ¢., , ie., , WO = yt + T), , A sin(ot + $) = A sin [w(t + T) + $], , sin(@t + > + 2m) = sin (wt + WT + ), , ot+ > +2n =ot+ oT +o, , @T = 2n, , ome Cie me ives ly, = 4p =f T= fF) (73.2), , Velocity ;, Now the velocity of an oscillator is, , ne ae, , v(t) = @Acos(wt + ) (7.3.3), , From equation (7.3.3), , v = +Ao Ji —sin? (or + 4)

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OSCILLATIONS, , 159, , , , v = +0,fA? — A?sin? (or + 6), v= 40 JA? -y?, , At y = 0, v = +A = +y,,, , This is the maximum yelocity or velocity, amplitude (v,,) of the SHM,, , At y = +A (end point of the SHM), » = 0, Acceleration, , The acceleration of SHO is, , (73.4), , _ dt) _ @y), a(t) = a = dt?, a(t) = -@*Asin(wt + 9), , al) = -0 yo (735), , At y=0, a(Q = 0 and, at y = +A, a(t) = FO7A, , The graphs of particle displacement y(t),, velocity v(t) and acceleration a(t) against time 1, of a SHM are shown in Figure 7.3., , A + i |, , , , The variation of y(f), v(t) and a(r) with time, are also summerized in Table 7.1., , ‘Table 7.1, Values of y(f), WO) and ai), , , , @ Ampl, (i) time period a, , ‘Here, y is in metre and t is in second., , Solution :, , Compare y = 0.40 sin(440r + 0.61), , with y = A sin(wt + 0), , (i) The amplitude is A = 0.40 m, , (ii) Angular frequency @ = 440 rad/s, , jodic time T = 22 =2 x 22 » —L., Gi) Periodic time T = 4"=2 x 7 * Fao, , = 0.0143 s, , (iv) Initial phase $= 0.61 rad, 7.4 The Force Law for Simple Harmonic, , Motion, , It is seen from equation (7.3.5) that the, acceleration of simple harmonic oscillator is a, function of time. Hence we can use Newton's, second law of motion to answer a question : How, much force is needed for this acceleration ?, , We know that, , F=ma, , 2 F= mo), , This is restoring force,, , According to Hook’s Law, the restoring force, is given by, , F =k), , with k as spring constant, , (7.4.1), , (7.42)

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160, , PHYSICS, , , , Comparing equations (7.4.1) and (7.4.2),, k= mo, , O= = (7.4.3), and frequency of oscillation, , fe dt (744), The period of oscillation, , T= ; = anf (7.4.5), , In many cases, the simple harmonic motion, can also occur even without spring. In that case, k is called the force constant of SHM and it is, , restoring force per unit displacement &=—-8),, , 7.5 Differential Equation of Simple Harmonic, Motion, According to Newton’s second law of, motion,, , a) _ 2), , F=ma=m—7° =m (75.1), Comparing this with F = —ky(t), a’ y(t), a —ky(), ay) __k, «ae mr, d?y(1) 7, et —07y(t) (Co 7.4.3), 2, as — + w(f) = 0 (7.5.2), , This is the second order differential equation, of the simple harmonic motion. The solution of, this equation is of the type,, , y() = A sin ot, , or, , yt) = B cos wt, , or any linear combination of sine and cosine, functions,, , y(t) = A sin wt + B cos wt, , , , Solution :, (1) Body is pulled down by 3 cm, Hence its amplitude is 3 cm, , As the motion begins from the lower end, at, t=0,y=-A, , “. y =A sin (@f + ) gives, , -A=A sind, , . sing =-1, , = 3E rad, , [ke, = fee ee Kf 8, ~ fxd fE, = fiogs®, 9, , = 2", Tey, aT = 728, ( =), 103, “2s, (3) We know the phase is given by, 0=at+o, 0= 23% rad