$A$ stick of length $l$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any way. $A$ ball of mass $m$ moving with speed $v$ collides elastically with the stick at one of its ends as shown in the figure. If after the collision the ball comes to rest,what should be the mass of the ball?

$A$ plank of mass $M$ is placed over a smooth inclined plane and a sphere of mass $m$ is also placed over the plank. Friction is sufficient between the sphere and the plank. If the plank and the sphere are released from rest,the frictional force on the sphere is:

$A$ rod $AB$ is free to rotate in a vertical plane about a horizontal axis through $A$ as shown in the figure. It is slightly disturbed from rest in its position of unstable equilibrium and when it is next vertical,the end $B$ collides with a fixed peg and rebounds. If the rod comes to instantaneous rest when $AB$ is horizontal (as shown in the figure),then:

Consider regular polygons with number of sides $n=3, 4, 5, \ldots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

$A$ motor of an engine rotates at an angular speed of $100 \ rpm$ about its axis. When the switch is turned off,it comes to rest in $15 \ s$. How many rotations does it complete before coming to rest?

$A$ uniform rod of length $l$ and density $\rho$ is revolving about a vertical axis passing through its one end. If $\omega$ is the angular velocity of the rod,then the centrifugal force per unit area of the rod is