$A$ bar of length $L$ carrying a small mass $m$ at one of its ends rotates with a uniform angular speed $\omega$ in a vertical plane about the midpoint of the bar. During the rotation, at some instant of time when the bar is horizontal, the mass is detached from the bar but the bar continues to rotate with the same $\omega$. The mass moves vertically up, comes back, and reaches the bar at the same point. At that place, the acceleration due to gravity is $g$.

$A$ uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(CM)$ at a distance $L$ from the wall. The disk rolls without slipping with velocity $\vec{V}_0 = V_0 \hat{i}$. The coefficient of friction is $\mu$.
$1.$ The net external force acting on the disk when its centre of mass is at displacement $x$ with respect to its equilibrium position is
$(A) -kx$ $(B) -2kx$ $(C) -\frac{2kx}{3}$ $(D) -\frac{4kx}{3}$
$2.$ The centre of mass of the disk undergoes simple harmonic motion with angular frequency $\omega$ equal to
$(A) \sqrt{\frac{k}{M}}$ $(B) \sqrt{\frac{2k}{M}}$ $(C) \sqrt{\frac{2k}{3M}}$ $(D) \sqrt{\frac{4k}{3M}}$
$3.$ The maximum value of $V_0$ for which the disk will roll without slipping is
$(A) \mu g \sqrt{\frac{M}{k}}$ $(B) \mu g \sqrt{\frac{M}{2k}}$ $(C) \mu g \sqrt{\frac{3M}{k}}$ $(D) \mu g \sqrt{\frac{5M}{2k}}$

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery and sticks to it. The loss in kinetic energy due to the collision is ....... $J$.

Find the minimum height $h$ of the obstacle so that the sphere of radius $R$ can stay in equilibrium on an inclined plane of angle $\theta$.

$A$ solid sphere of mass $M$,radius $R$ and having moment of inertia about an axis passing through the centre of mass as $I$,is recast into a disc of thickness $t$,whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains $I$. Then,the radius of the disc will be

$A$ solid sphere of radius $R$ has a moment of inertia $I$ about its geometric axis. It is melted and recast into a disc of radius $r$ and thickness $t$. If the moment of inertia of this disc about an axis tangent to its edge and perpendicular to its plane is also $I$,find the value of $r$.