A simple pendulum is suspended in a car. The car starts moving on a horizontal road according to equation  $x\, = \,\frac{g}{2}\,\sqrt 3 {t^2}$. Find the time period of oscillation of the pendulum.

A pendulum bob has a speed of $3\, m/s$ at its lowest position. The pendulum is $0.5\, m$ long. The speed of the bob, when the length makes an angle of ${60^o}$ to the vertical, will be ..... $m/s$ (If $g = 10\,m/{s^2}$)

A pendulum clock that keeps correct time on the earth is taken to the moon it will run (it is given that $g_{Moon} = g_{Earth}/6$ )

Two pendulums begin to swing simultaneously. If the ratio of the frequency of oscillations of the two is $7 : 8$, then the ratio of lengths of the two pendulums will be

A simple pendulum of length $L$ and mass (bob) $M$ is oscillating in a plane about a vertical line between angular limits $ - \varphi $ and $ + \varphi $. For an angular displacement $\theta (|\theta | < \varphi )$, the tension in the string and the velocity of the bob are $T$ and $ v$ respectively. The following relations hold good under the above conditions

Two simple pendulums of length $1\, m$ and $4\, m$ respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to