The displacement y of a particle executing periodic motion is given by $y=4{\cos }^2\left(t/2\right)\sin \left(1000t\right).$ This expression may be considered to be a result of the superposition of  ........... independent harmonic motions

1. Two         

2. Three

3. Four         

4. Five

A particle of mass m is attached to three identical springs A, B and C each of force constant k a shown in figure. If the particle of mass m is pushed slightly against the spring A and released then the time period of oscillations is -

(a) $2\mathrm{\pi }\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}}$          (b) $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}$

(c) $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$            (d) $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{3\mathrm{k}}}$

The graph shows the variation of displacement of a particle executing S.H.M. with time. We infer from this graph that -

(1) The force is zero at time T/8

(2) The velocity is maximum at time T/4

(3) The acceleration is maximum at time T

(4) The P.E. is equal to total energy at time T/4

For a particle executing S.H.M. the displacement x is given by $A<mspace\; width="1em"></mspace>\cos <mspace\; width="1em"></mspace>\omega t$. Identify the graph which represents the variation of potential energy (P.E.) as a function of time t and displacement x.

(a) I, III          (b) II, IV

(c) II, III         (d) I, IV

1. \(25~\text{Hz}\)
2. \(50~\text{Hz}\)
3. \(12.25~\text{Hz}\)
4. \(33.3~\text{Hz}\)

A body performs S.H.M. . Its kinetic energy K varies with time t as indicated by graph

(a)    (b) 

(c)      (d) 

The amplitude of a damped oscillator decreases to 0.9 times its original magnitude in 5 s. In another 10 s, it will decrease to $\mathrm{\alpha }$ times its original magnitude, where $\mathrm{\alpha }$ equals

1. 0.7

2. 0.81

3. 0.729

4. 0.6

A particle performs SHM on x-axis with amplitude A and time period T. The time taken by the particle to travel a distance $\frac{\mathrm{A}}{5}$ starting from rest is

1. $\frac{\mathrm{T}}{20}$

2. $\frac{\mathrm{T}}{2\mathrm{\pi }}{\cos }^{-1}\left(\frac{4}{5}\right)$

3. $\frac{\mathrm{T}}{2\mathrm{\pi }}{\cos }^{-1}\left(\frac{1}{5}\right)$

4. $\frac{\mathrm{T}}{2\mathrm{\pi }}{\sin }^{-1}\left(\frac{1}{5}\right)$

Two simple pendulums of length 5 m and 20 m respectively are given small linear displacement in one direction at the same time. They will again be in the phase when the pendulum of shorter length has  completed how many oscillations?

1. 5

2. 1

3. 2

4. 3