$A$ uniform rod of mass $M$ is hinged at its upper end. $A$ particle of mass $m$ moving horizontally strikes the rod at its mid-point elastically. If the particle comes to rest after the collision,find the value of $M/m$.

Four particles each of mass $m$ are lying symmetrically on the rim of a disc of mass $M$ and radius $R$. The moment of inertia of the system about an axis passing through one of the particles and perpendicular to the plane of the disc is:

$A$ spherical rigid ball is released from rest and starts rolling down an inclined plane from height $h=7 \, m$,as shown in the figure. It hits a block at rest on the horizontal plane (assume elastic collision). If the mass of both the ball and the block is $m$ and the ball is rolling without sliding,then the speed of the block after collision is close to ............. $m/s$.

$A$ uniform disc is rolling on a horizontal surface. At a certain instant,$B$ is the point of contact and $A$ is the topmost point of the disc,where $R$ is the radius of the disc.

$Assertion$ : $A$ helicopter must necessarily have two propellers.
$Reason$ : Two propellers are provided in a helicopter in order to conserve linear momentum.

$A$ thin ring of mass $2 \ kg$ has a radius of $0.5 \ m$. It is rolling without slipping on a horizontal plane with a velocity of $1 \ m/s$. $A$ small ball of mass $0.1 \ kg$ moving in the opposite direction with a velocity of $20 \ m/s$ hits the ring at a height of $0.75 \ m$ and moves vertically upward with a velocity of $10 \ m/s$ after the collision. Immediately after the collision: