The variation of the potential energy of the harmonic oscillator is shown in the figure. The spring constant will be:
           
1. \(1\times10^{2}~\text{N/m}\)
2. \(150~\text{N/m}\)
3. \(0.667\times10^{2}~\text{N/m}\)
4. \(3\times10^{2}~\text{N/m}\)

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

The displacement of a particle is represented by the equation $\mathrm{y}=3 \cos \left(\frac{\mathrm{\pi }}{4}-2\mathrm{\omega t}\right)$. The motion of the particle is.

1. simple harmonic with period $2\pi /\mathrm{\omega }$

2. simple harmonic with period $\mathrm{\pi }/\mathrm{\omega }$

3. periodic but not simple harmonic

4. non-periodic

The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is the maximum speed?

1. 100 m/min

2. 200 m/min

3. 300 m/min

4. 50 m/min

Two points are located at a distance of 10 m and 15 m from the source of oscillation. The period of oscillation is 0.05s and the velocity of the wave is 300 m/s. What is the phase difference between the oscillations of two points?

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{6}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{2\mathrm{\pi }}{3}$

A body has a time period ${\mathrm{T}}_1$ under the action of one force and ${\mathrm{T}}_2$ under the action of another force, the square of the time period when both the forces are acting in the same direction is

1. ${\mathrm{T}}_1^2{\mathrm{T}}_2^2$

2. ${\mathrm{T}}_1^2{\mathrm{T}}_2^2$

3. ${\mathrm{T}}_1^2+{\mathrm{T}}_2^2$

4. ${\mathrm{T}}_1^2{\mathrm{T}}_2^2/({\mathrm{T}}_1^2+{\mathrm{T}}_2^2)$

Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution and the initial position are indicated on the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P is:

(1) x(t) = B sin $\left(\frac{2\mathrm{\pi t}}{30}\right)$

(2) x(t) = B cos $\left(\frac{\mathrm{\pi t}}{15}\right)$

(3) x(t) = B sin $\left(\frac{\mathrm{\pi t}}{15}+\frac{\mathrm{\pi }}{2}\right)$

(4) x(t) = B $\left(\frac{\mathrm{\pi t}}{15}+\frac{\mathrm{\pi }}{2}\right)$