A ring of radius R is hung by a nail on its periphery such that it can freely rotate in its vertical plane. The time period of the ring for small oscillations is:

1.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{R}}{\mathrm{g}}}$

2.  $\mathrm{T} = \mathrm{\pi }\sqrt{\frac{\mathrm{R}}{\mathrm{g}}}$

3.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{2\mathrm{R}}{\mathrm{g}}}$

4.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{3\mathrm{R}}{5\mathrm{g}}}$

The equation of S.H.M. is given as x = Asin(0.02$\mathrm{\pi t}$), where t is in seconds. With what time period the potential energy oscillates? 

1.  200 s

2.  100 s

3.  50 s

4.  10 s

In a stationary lift, a spring-block system oscillates with a frequency \(f.\) When the lift accelerates, the frequency becomes \(f'\) . Then:

                     
1. \(\frac{\pi}{2}~\text{s}\)
2. \(\frac{1}{2}~\text{s}\)
3. \(\pi~\text{s}\)
4. \(1~\text{s}\)

The equation of a SHM is given as $\mathrm{x} = 5\sin \left(4\mathrm{\pi t} + \frac{\mathrm{\pi }}{3}\right)$, where \(\mathrm t\) is in seconds and \(\mathrm x\) in meters. During a complete cycle, the average speed of the oscillator is:
1. zero
2. \(10\) m/s 
3. \(20\) m/s
4. \(40\) m/s

The equation of a simple harmonic oscillator is given as $\mathrm{y} = \mathrm{Asin}\left(20\mathrm{\pi t} + \frac{\mathrm{\pi }}{3}\right)$, where t is in seconds. The frequency with which kinetic energy oscillates is

1.  5 Hz

2.  10 Hz 

3.  20 Hz

4.  40 Hz

What is the period of oscillation of the block shown in the figure?

If a simple pendulum is brought deep inside a mine from the earth's surface, its time period of oscillation will: