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14%20-%20Oscillations

# 14%20-%20Oscillations - 14 OSCILLATIONS Page 1 14.1...

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14 - OSCILLATIONS Page 1 14.1 Periodic and Oscillatory motion Motion of a system at regular interval of time on a definite path about a definite point is known as a periodic motion, e.g., uniform circular motion of a particle. To and fro motion of a system on a linear path is called an oscillatory motion, e.g., motion of the bob of a simple pendulum. 14.2 Simple harmonic motion This is the simplest type of periodic motion which can be understood by considering the following example. Suppose a body of mass m is suspended at the lower end of a massless elastic spring obeying Hooke’s law which is fixed to a rigid support in the vertical position. The spring elongates by length Δ l and attains equilibrium as shown in Fig. ( b ) Here two forces act on the body. ( 1 ) Its weight, mg, downwards and ( 2 ) the restoring force developed in the spring, k Δ l , upwards, where k = force constant of the spring. For equilibrium, mg = k Δ l ( 1 ) The spring is constrained to move in the vertical direction only. Now, suppose the body is given some energy in its equilibrium condition and it undergoes displacement y in the upward direction as shown in Fig. ( c ). Two forces act on the body in this displaced condition also. ( 1 ) Its weight, mg, downwards and ( 2 ) the restoring force developed in the spring, k ( Δ l - y ), upwards. The resultant force acting on the body in this condition is given by F = - mg + k ( Δ l - y ) ( 2 ) From equations ( 1 ) and ( 2 ), F = - ky

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14 - OSCILLATIONS Page 2 Displacement: The distance of the body at any instant from the equilibrium point is known as its displacement at that instant. The displacements along the positive Y-axis are taken as positive and those on the negative Y-axis are taken as negative. In the equation, F = - ky, F is negative when y is positive and vice versa. Thus, the resultant force acting on the body is proportional to the displacement and is directed opposite to the displacement, i.e., towards the equilibrium point. Differential equation of simple harmonic motion ( SHM ) According to Newton’s second law of motion, F = ma = m dt dv = m 2 2 dt y d = - ky ( for spring-type oscillator as above ) 2 2 dt y d = - m k y = - ω 0 2 y ( taking m k = ω 0 2 ) 2 2 dt y d + ω 0 2 y = 0 This is the differential equation of SHM. To obtain the solution of the above differential equation is to obtain y as a function of t such that on twice differentiating y w.r.t. t, we get back the same function y with a negative sign. Both the sine and the cosine functions possess such a property. Hence, taking y = A 1 sin ω 0 t + A 2 cos ω 0 t as a possible solution and differentiating twice w.r.t. t, dt dy = A 1 ω 0 cos ω 0 t - A 2 ω 0 sin ω 0 t and 2 2 dt y d = - A 1 ω 0 2 sin ω 0 t - A 2 ω 0 2 cos ω 0 t = - ω 0 2 ( A 1 sin ω 0 t + A 2 cos ω 0 t ) = - ω 0 2 y Thus, y t = A 1 sin ω 0 t + A 2 cos ω 0 t is the solution of the differential equation and is known as its general solution, where y t is the displacement of simple harmonic oscillator ( SHO ) at time t.
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