$A$ $metre$ scale is suspended vertically from a horizontal axis passing through one end of it. Its time period would be ....... $\sec$

$A$ 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:

The graph between the time period $(T)$ and the length $(l)$ of a simple pendulum is

$A$ block of mass $m$ is sliding on a fixed frictionless concave surface of radius $R$. It is released from rest at point $P$ which is at a height of $H \ll R$ from the lowest point $Q$.
$(a)$ What is the potential energy as a function of $\theta$,taking the lowest point $Q$ as the reference level for potential energy?
$(b)$ What is the kinetic energy as a function of $\theta$?
$(c)$ What is the time taken for the particle to reach from point $P$ to the lowest point $Q$?
$(d)$ How much force is exerted by the block on the concave surface at the point $Q$?

From a fixed support,two small identical spheres are suspended by means of strings of length $1 \,m$ each. They are pulled aside as shown and then released. $B$ is the mean position. Then the two spheres collide

$A$ simple pendulum hangs from the ceiling of a car. If the car accelerates with a uniform acceleration,the frequency of the simple pendulum will