The period of a simple pendulum,whose bob is a hollow metallic sphere,is $T$. The period is $T_1$ when the bob is filled with sand,$T_2$ when it is filled with mercury and $T_3$ when it is half filled with mercury. Which of the following is true?

$A$ 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. Which of the following relations hold good under the above conditions?

The sum of kinetic energy and potential energy of a simple pendulum bob is $0.02 \text{ J}$. The speed of the simple pendulum bob at the equilibrium position is approximately: (Consider mass of the bob = $20 \text{ g}$) (in $\text{ m/s}$)

$A$ 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 $Q$?

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 a number of oscillations equal to

The length of a simple pendulum is $l$ and its maximum angular displacement is $\theta$. What is its maximum kinetic energy $(K.E.)$?