The kinetic energy and potential energy of a particle executing S.H.M. are equal when displacement in terms of amplitude 'A' is:

(1) $\frac{A}{2}$

(2) $\frac{A}{\sqrt{2}}$

(3) $\frac{A\sqrt{2}}{3}$

(4) $A\sqrt{2}$

Two identical pendulums oscillate with a constant phase difference $\frac{\mathrm{\pi }}{4}$ and the same amplitude. If the maximum velocity of one is v, the maximum velocity of the other will be:

(1) v

(2) $\sqrt{2}v$

(3) 2v

(4) $\frac{v}{\sqrt{2}}$

A simple pendulum suspended from the ceiling of a stationary lift has period T₀. When the lift descends at a steady speed, the period is T₁, and when it descends with constant downward acceleration, the period is T₂. Which one of the following is true?

(1) T₀ = T₁ = T₂

(2) T₀ = T₁ < T₂

(3) T₀ = T₁ > T₂

(4) T₀ < T₁ < T₂

If a Second's pendulum is moved to a planet where the acceleration due to gravity is 4 times the acceleration due to gravity on earth, to keep the time period of the second's pendulum same, the length of the second's pendulum on the planet should be made:

(1) 2 times

(2) 4 times

(3) 8 times

(4) 15 times

Two pendulums of lengths 1.21 m and 1.0 m start vibrating. At some instant, the two are at the mean position in the same phase. After how many vibrations of the longer pendulum, the two will be in the phase again?

(1) 10

(2) 11

(3) 20

(4) 21

The time period of oscillations of a simple pendulum is 1 minute. If its length is increased by 44%, then its new time period of oscillations will be:

(1) 96 s

(2) 58 s

(3) 82 s

(4) 72 s

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) $2\pi \sqrt{\frac{l}{3g}}$