Two discs of moment of inertia $I_1 = 4 \ kg \ m^2$ and $I_2 = 2 \ kg \ m^2$ about their central axes and normal to their planes,rotating with angular speeds $10 \ rad/s$ and $4 \ rad/s$ respectively,are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is . . . . . . $J$.

$A$ block of mass $m$ is attached to a pulley disc of equal mass $m$ and radius $r$ by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of $5\, m/s$. Its velocity when the string becomes taut will be

$A$ sphere of mass $M$ and radius $R$ is attached by a light rod of length $l$ to a point $P$. The sphere rolls without slipping on a circular track as shown. It is released from the horizontal position. The angular momentum of the system about $P$ when the rod becomes vertical is:

$A$ thin hollow cylinder open at both ends is moving with the same speed $v$ in two different cases:
$(i)$ It slides without rotating.
$(ii)$ It rolls without slipping.
Find the ratio of the kinetic energies in the two cases.

In the following problems,indicate the correct direction of the friction force acting on the cylinder,which is pulled on a rough surface by a constant force $F$. $A$ cylinder is pulled horizontally by a force $F$ acting at a point above the centre of mass of the cylinder,as shown in the figure. The friction force can be given by which of the following diagrams?

$A$ cord of negligible mass is wound round the rim of a flywheel of mass $20 \; kg$ and radius $20 \; cm$. $A$ steady pull of $25 \; N$ is applied on the cord as shown in the figure. The flywheel is mounted on a horizontal axle with frictionless bearings.
$(a)$ Compute the angular acceleration of the wheel.
$(b)$ Find the work done by the pull,when $2 \; m$ of the cord is unwound.
$(c)$ Find also the kinetic energy of the wheel at this point. Assume that the wheel starts from rest.
$(d)$ Compare answers to parts $(b)$ and $(c)$.