$A$ string of negligible mass going over a clamped pulley of mass $m$ supports a block of mass $M$ as shown in the figure. The force on the pulley by the clamp is given by

$A$ ball of mass $m$ is released from the top of a smooth movable wedge of mass $m$. When the ball collides with the floor,the velocity of the wedge is $v$. Then the maximum height attained by the ball after an elastic collision with the floor is: (Neglect any edge at the lower end of the wedge).

In the figures $(i), (ii)$ and $(iii)$ shown, the objects $A, B$ and $C$ have the same mass $m$. The string, spring, and pulley are massless. Object $C$ strikes object $B$ with velocity $u$ in each case and sticks to it. Find the ratio of the velocity of $B$ in case $(i)$ to $(ii)$ to $(iii)$.

$A$ uniform cylinder rests on a cart as shown. The coefficient of static friction between the cylinder and the cart is $0.5$. If the cylinder is $4\, cm$ in diameter and $10\, cm$ in height,which of the following is the minimum acceleration of the cart needed to cause the cylinder to tip over? (Assume $g = 10\, m/s^2$)

Consider the following statements $A$ and $B$ and identify the correct answer:
$A$. When a person walks on a rough surface,the direction of the frictional force exerted by the surface on the person is in the same direction as his motion.
$B$. When a cycle is in motion,the force of friction exerted by the ground on the front wheel is in the backward direction.

$A$ sphere is placed rotating with its centre initially at rest in a corner as shown in figures $(a)$ and $(b)$. The coefficient of friction between all surfaces and the sphere is $\mu = \frac{1}{3}$. Find the ratio of the frictional force $\frac{f_a}{f_b}$ exerted by the ground in situations $(a)$ and $(b)$.