An ideal spring with spring-constant K is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially un-stretched. Then the maximum extension in the spring will be:
1. 4 Mg/K 
2. 2 Mg/K
3. Mg/K 
4. Mg/2K

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 $A \cos \omega t$. 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. $\frac{\mathrm{T}}{2\mathrm{\pi }}{\sin }^{-1}\left(\frac{1}{5}\right)$