A uniform disc of mass M and radius R is suspended in a vertical plane from a point on its periphery. Its time period of oscillation is:

(1) $2\pi \sqrt{\frac{3R}{g}}$

(2) $2\pi \sqrt{\frac{R}{3g}}$

(3) $2\pi \sqrt{\frac{2R}{3g}}$

(4) $2\pi \sqrt{\frac{3R}{2g}}$

A block of mass m hangs from three springs having the same spring constant k. If the mass slightly displaced downwards, the time period oscillations will be:

(1) $2\pi \sqrt{\frac{m}{3k}}$

(2) $2\pi \sqrt{\frac{3m}{2k}}$

(3) $2\pi \sqrt{\frac{2m}{3k}}$

(4) $2\pi \sqrt{\frac{3k}{m}}$

A spring block system in horizontal oscillation has a time-period T. Now the spring is cut into four equal parts and the block is re-connected with one of the parts. The new time period of the vertical oscillations will be:

(1) $\frac{T}{\sqrt{2}}$

(2) 2T

(3) $\frac{T}{2}$

(4) $\frac{T}{2\sqrt{2}}$

A block of mass m is suspended separately by two different springs having time periods t₁ and t₂. If the same mass is connected to the parallel combination of both springs, then its time period is given by:

(1) $\frac{t_1{t}_2}{t_1+t_2}$

(2) $\frac{t_1{t}_2}{\sqrt{t_1^2+t_2^2}}$

(3) $\sqrt{\frac{t_1{t}_2}{t_1+t_2}}$

(4) $t_1+t_2$

In damped oscillations, the damping force is directly proportional to the speed of the oscillator. If amplitude becomes half of its maximum value in 1 sec, then after 2 sec, the amplitude of the damped oscillation for which data is given, will be: (Initial amplitude = $A_0$)

1. $\frac{1}{4}{A}_0$

2. $\frac{1}{2}{A}_0$

3. $A_0$

4. $\frac{\sqrt{3}{A}_0}{2}$

For forced oscillations, a particle oscillates in a simple harmonic fashion with a frequency equal to:

1. the frequency of driving force.

2. the mean of frequency of driving force and natural frequency of the body.

3. the difference of frequency of driving force and natural frequency of the body.

4. the natural frequency of the body.

Resonance is a special case of:
1. forced oscillations
2. damped oscillations
3. undamped oscillations
4. coupled oscillations

The SHM of a particle is given by the equation $x=2 \sin \omega t + 4 cos\omega t$. Its amplitude of oscillations is:

(1) 4 units

(2) 2 units

(3) 6 units

(4) 2$\sqrt{5}$ units

Which of the diagrams represents variation of the total mechanical energy of a pendulum oscillating in air as a function of time?

1. 

2. 

3. 

4. 

The rotation of the earth about its axis is: