$Assertion$ : If polar ice melts, days will be shorter.
$Reason$ : Moment of inertia decreases and thus angular velocity increases.

An annular disk of mass $M$,inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$,as shown in the figure. At some time,an impulse $J_0 \hat{x}$ is applied at a height $h$ above the center of the disk. If $h=h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement$(s)$ is(are) correct?
$(A)$ For $\mu \neq 0$ and $a \rightarrow 0, h_m=b / 2$
$(B)$ For $\mu \neq 0$ and $a \rightarrow b, h_m=b$
$(C)$ For $h=h_m$,the initial angular velocity does not depend on the inner radius $a$.
$(D)$ For $\mu=0$ and $h=0$,the wheel always slides without rolling.

$A$ spherical rigid ball is released from rest and starts rolling down an inclined plane from height $h=7 \, m$,as shown in the figure. It hits a block at rest on the horizontal plane (assume elastic collision). If the mass of both the ball and the block is $m$ and the ball is rolling without sliding,then the speed of the block after collision is close to ............. $m/s$.

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.

State whether the following statements are True or False:
$(1)$ Angular position $\theta$ is a scalar,while angular displacement $\Delta \theta$ is a vector.
$(2)$ The relation between linear velocity $\vec{v}$ and angular velocity $\vec{\omega}$ for a particle in rotational motion is given by $\vec{v} = \vec{r} \times \vec{\omega}$.
$(3)$ The moment of inertia of a rigid body is constant.
$(4)$ The moment of momentum is called angular momentum.

$A$ particle is moving in a uniform circular motion with angular momentum $L$. If its angular frequency is doubled and its kinetic energy is halved,then the new angular momentum of the particle will be .......