$A$ 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:

$A$ thin rod of length '$L$' lies along the $x$-axis with its ends at $x = 0$ and $x = L$. Its linear mass density $\lambda$ varies with $x$ as $\lambda = k{\left( {\frac{x}{L}} \right)^n}$,where $n$ is a non-negative constant. If the position $x_{CM}$ of the center of mass of the rod is plotted against '$n$',which of the following graphs best approximates the dependence of $x_{CM}$ on $n$?

Two point-like objects of masses $20 \text{ g}$ and $30 \text{ g}$ are fixed at the two ends of a rigid massless rod of length $10 \text{ cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass,as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{-8} \text{ N m rad}^{-1}$. The angular frequency of the oscillations is $n \times 10^{-3} \text{ rad s}^{-1}$. The value of $n$ is

$A$ rod of mass $M$ and length $L$ is lying on a horizontal frictionless surface. $A$ particle of mass $m$ travelling along the surface hits at one end of the rod with velocity $u$ in a direction perpendicular to the rod. The collision is completely elastic. After the collision, the particle comes to rest. The ratio of masses $\left(\frac{m}{M}\right)$ is $\frac{1}{x}$. The value of $x$ will be ..... .

$A$ body of mass $1 \,kg$ is suspended by a weightless string which passes over a pulley of mass $2 \,kg$ as shown in the figure. The mass is released from a height of $1.6 \,m$ from the ground. With what velocity does it strike the ground?

Two flywheels are connected by a non-slipping belt as shown in the figure. $I_1 = 4 \ kg \ m^2$,$r_1 = 20 \ cm$,$I_2 = 20 \ kg \ m^2$,and $r_2 = 30 \ cm$. $A$ torque of $10 \ Nm$ is applied on the smaller wheel. Match the entries of column $I$ with appropriate entries of column $II$.

Quantities	Their numerical values (in $SI$ units)
a. Angular acceleration of smaller wheel	$1$. $5/3$
b. Torque on the larger wheel	$2$. $100/3$
c. Angular acceleration of larger wheel	$3$. $5/2$