When you prepare for any examination, it will be very useful to work out the questions which appeared in earlier examinations. Here are two questions on oscillations which appeared in AIEEE 2006 question paper:

**(1)** Starting from the origin a body oscillates simple harmonically with a period of 2 s. After what time will the kinetic energy be 75% of the total energy?

(1) 1/12 s

(2) 1/6 s

(3) 1/4 s

(4) 1/3 s

This simple harmonic motion can be represented by the equation,

*y = A *sin* **ωt* where *y *is the displacement at the instant *t*, *A* is the amplitude and *ω* is the angular frequency.

The instantaneous velocity *v* of the particle is given by

*v* = *dy/dt* = *A**ω *cos*ωt*

The maximum velocity *v _{max} *of the particle is evidently

*A*

*ω*and the maximum kinetic energy which is equal to the total energy is ½

*m*

*v*

_{max}^{2}where

*m*is the mass of the particle. We have

½ *m**v*^{2} = (¾)(½)*m**v _{max}*

^{2}

Therefore, ½ *m** *(*A**ω *cos*ωt*)^{2} = (¾)(½)*m*(*A**ω*)^{2}* *from which cos*ωt* = (√3)/2

Therefore, *ωt* = π/6 so that *t* = π/6*ω* = π/(6×2π/*T* ) = **1/6 s** since the period *T *is 2 s.

**(2) **The maximum velocity of a particle executing simple harmonic motion with amplitude 7 mm is 4.4 ms^{–1}. The period of oscillation is

(1) 100 s

(b) 0.01 s

(c) 10 s

(d) 0.1 s

Since the maximum velocity *v _{max} *=

*A*

*ω*and the period

*T =*2π/

*ω*we have

*T = *2π*A*/*v _{max}* = 2π×7×10

^{–3}/4.4 =

**0.01 s**