$A$ wheel of moment of inertia $2 \ kg \ m^2$ is rotating about an axis passing through its centre and perpendicular to its plane at a speed of $60 \ rad \ s^{-1}$. Due to friction, it comes to rest in $5$ minutes. The angular momentum of the wheel three minutes before it stops rotating is:

$A$ solid metallic sphere of radius '$R$' having moment of inertia '$I$' about its diameter is melted and recast into a solid disc of radius '$r$' of uniform thickness. The moment of inertia of the disc about an axis passing through its edge and perpendicular to its plane is also equal to '$I$'. The ratio $\frac{r}{R}$ is

$A$ thin uniform bar lies on a frictionless horizontal surface and is free to move in any way on the surface. Its mass is $0.300 \, kg$ and length is $2 \, m$. Two particles,each of mass $0.100 \, kg$,are moving on the same surface towards the two ends of the bar in a direction perpendicular to the bar. One particle moves with a velocity of $10 \, m/s$ towards one end,and the other moves with a velocity of $5 \, m/s$ towards the other end. If the collision between the particles and the bar is completely elastic and both particles strike the bar simultaneously,find the velocity of the centre of mass of the bar after the collision in $m/s$.

Fill in the blanks:
$(1)$ In rotational motion,the role played by ............ is analogous to the role played by mass in linear motion.
$(2)$ For a rigid body in rotational motion,if a particle at a distance of $10 \ cm$ from the fixed axis of rotation has an angular velocity of $10 \ rad/s$,then the linear velocity of a particle at a distance of $5 \ cm$ from the axis of rotation is ............
$(3)$ The $SI$ unit $J \cdot s^{-2}$ is the unit of the physical quantity ............
$(4)$ The condition for a body to roll without slipping down an inclined plane with friction is ............

$A$ wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. $A$ constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point $Q$. Which of the following options is/are correct?

State whether the following statements are True or False:
$(1)$ The angular acceleration of an object rotating with a constant angular velocity is always zero.
$(2)$ An object can have a moment of inertia without energy.
$(3)$ The radius of gyration of an object is a constant quantity.
$(4)$ $A$ figure skater spins faster when they pull their arms in because their moment of inertia decreases.