The bob of a simple pendulum is a spherical hollow ball filled with water. $A$ plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation,till water is coming out,the time period of oscillation would

$A$ pendulum oscillates simple harmonically if and only if:
$I$. The size of the bob of the pendulum is negligible in comparison with the length of the pendulum.
$II$. The angular amplitude is less than $10^{\circ}$.
Choose the correct option.

$A$ brass cube of side $a$ and density $\rho$ is floating in mercury of density $\sigma$. If the cube is displaced a bit vertically,it executes $S.H.M.$ Its time period will be

If a simple pendulum is released from the given position,find the velocity of the bob when it reaches the lowest position. (Take $g = 10 \ m/s^2$) (in $m/s$)

The period of oscillation of a simple pendulum of length $L$ suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $\alpha$,is given by

$A$ trolley of mass $m_1$ is placed on a horizontal rigid pair of rails at the same height. $A$ mass $m_2$ is suspended from the trolley vertically by means of an ideal massless rope. The rope hangs between the rails without touching them. The trolley can move along smooth rails but cannot move in any other direction. The suspended mass is given a small oscillation and performs $SHM$ after being displaced slightly from its stable equilibrium position in two ways: first,perpendicular to the rails,and second,parallel to the rails. The ratio of the time periods of these two $SHM$s (second case to first case) is: