$A$ 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 sphere of radius $R$ is placed on a rough horizontal surface and given a linear velocity $v_0$ and angular velocity $\omega_0$ as shown. The sphere comes to rest after moving some distance to the right. It follows that:

$A$ uniform body of mass $M$ and radius $R$ has a small mass $m$ attached at its edge as shown in the figure. The system is placed on a perfectly rough horizontal surface such that mass $m$ is at the same horizontal level as the centre of the body. It is assumed that there is no slipping at point $A$. If $I_A$ is the moment of inertia of the combined system about the point of contact $A$,then the normal reaction at point $A$ just after the system is released from rest is ........ $N$. ($M = 6 \ kg$,$m = 2 \ kg$,$I_A = 4 \ kg \ m^2$,$R = 1 \ m$,$g = 10 \ m/s^2$)

On a solid sphere lying on a horizontal surface,a force $F$ is applied at a height of $R/2$ from the centre of mass. The initial acceleration of a point at the top of the sphere is (there is no slipping at any point).

The moment of inertia of a ring about an axis passing through its centre and perpendicular to its plane is $I$. It is rotating with angular velocity $\omega$. Another identical ring is gently placed on it so that their centres coincide. If both rings are rotating about the same axis, then the loss in kinetic energy is

$ABC$ is an equilateral triangle with $O$ as its centre. $\vec F_1, \vec F_2$ and $\vec F_3$ represent three forces acting along the sides $AB, BC$ and $AC$ respectively. If the total torque about $O$ is zero,then the magnitude of $\vec F_3$ is