$A$ rod of length $1\,m$ is standing vertically. When its upper end is released and it falls such that the lower end touches the ground without slipping,the speed of the upper end when it hits the ground is:

$A$ spool is pulled vertically by a constant force $F (< Mg)$ as shown in the figure. Which of the following diagrams correctly represents the direction of the friction force acting on the spool?

$A$ 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 ....... $J$.

$A$ roller is made by joining together two cones at their vertices $O$. It is kept on two rails $AB$ and $CD$,which are placed asymmetrically (see figure),with its axis perpendicular to $CD$ and its centre $O$ at the centre of the line joining $AB$ and $CD$ (see figure). It is given a light push so that it starts rolling with its centre $O$ moving parallel to $CD$ in the direction shown. As it moves,the roller will tend to

$A$ 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
$(A) \mu g \sqrt{\frac{M}{k}}$ $(B) \mu g \sqrt{\frac{M}{2k}}$ $(C) \mu g \sqrt{\frac{3M}{k}}$ $(D) \mu g \sqrt{\frac{5M}{2k}}$

$A$ thin hollow cylinder slides with velocity $v$ without rotating. It then rolls without slipping with the same velocity $v$. Find the ratio of the kinetic energies in the two cases.