$A$ small sphere of radius $r$ is placed on a concave surface of radius of curvature $R$ a little away from the center. When the sphere is released, it oscillates. Assuming the oscillation to be simple harmonic motion and $r << R$, then the time period is

An object of mass $3 \,kg$ is tied by a string of negligible mass to a ceiling and held such that the string is taut. The object is released suddenly such that the string remains taut. Its acceleration when released is (acceleration due to gravity $= 10 \,ms^{-2}$) (in $\,ms^{-2}$)

To show that a simple pendulum executes simple harmonic motion,it is necessary to assume that

$A$ 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 $Q$?

For a simple pendulum, the graph of time period $(T)$ versus length $(l)$ is depicted by:

$A$ simple pendulum oscillates with an angular amplitude $\theta$. If the maximum tension in the string is $4$ times the minimum tension,then the value of $\theta$ is