A simple pendulum is oscillating in a trolley moving on a horizontal straight road with constant acceleration a. If the direction of motion of the trolley is taken as positive x-direction and the vertically upward direction as positive y-direction, then the mean position of the pendulum makes an angle:

(1) ${\tan }^{-1}\left(\frac{g}{a}\right)$ with y-axis in +x direction

(2) ${\tan }^{-1}\left(\frac{\mathrm{a}}{g}\right)$ with y-axis in -x direction

(3) ${\tan }^{-1}\left(\frac{\mathrm{a}}{g}\right)$ with y-axis in +x direction

(4) ${\tan }^{-1}\left(\frac{g}{a}\right)$ with y-axis in -x direction

The time period of oscillations of a second's pendulum on the surface of a planet having mass and radius doubled of those of earth is:

(1) 4 s

(2) 1 s

(3) $\sqrt{2}$ s

(4) 2$\sqrt{2}$ s

A hollow metal sphere is filled with water through a small hole in it. It is hung by a long thread and is made to oscillate. Water slowly flows out of the hole at the bottom. Select the correct variation of its time period.

(1) The period will go on increasing till the sphere is emptied.

(2) The period will go on decreasing till the sphere is emptied.

(3) The period will not be affected at all.

(4) The period will increase first, then decrease to the initial value after the sphere is emptied.

A uniform rod of mass m and length l is suspended about its end. The time period of small angular oscillations is:

(1) $2\pi \sqrt{\frac{l}{g}}$

(2) $2\pi \sqrt{\frac{2l}{g}}$

(3) $2\pi \sqrt{\frac{2l}{3g}}$

(4) $2\pi \sqrt{\frac{l}{3g}}$

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.