$A$ thin uniform straight rod of mass $2 \, kg$ and length $1 \, m$ is free to rotate about its upper end when at rest. It receives an impulsive blow of $10 \, Ns$ at its lowest point,normal to its length as shown in the figure. The kinetic energy of the rod just after impact is ........ $J$.

The general motion of a rigid body can be considered to be a combination of $(i)$ a motion of the centre of mass about an axis,and $(ii)$ its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider,for example,a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless stick,as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed $\omega$,the motion at any instant can be taken as a combination of $(i)$ a rotation of the centre of mass of the disc about the $z$-axis,and $(ii)$ a rotation of the disc about an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points $P$ and $Q$). Both the motions have the same angular speed $\omega$ in this case. Now consider two similar systems as shown in the figure: Case $(a)$ the disc with its face vertical and parallel to the $x-z$ plane; Case $(b)$ the disc with its face making an angle of $45^{\circ}$ with the $x-y$ plane,its horizontal diameter parallel to the $x$-axis. In both the cases,the disc is welded at point $P$,and systems are rotated with constant angular speed $\omega$ about the $z$-axis.
$1.$ Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?
$(A)$ It is $\sqrt{2} \omega$ for both the cases.
$(B)$ It is $\omega$ for case $(a)$; and $\frac{\omega}{\sqrt{2}}$ for case $(b)$.
$(C)$ It is $\omega$ for case $(a)$; and $\sqrt{2} \omega$ for case $(b)$.
$(D)$ It is $\omega$ for both the cases.
$2.$ Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?
$(A)$ It is vertical for both the cases $(a)$ and $(b)$.
$(B)$ It is vertical for case $(a)$; and is at $45^{\circ}$ to the $x-z$ plane and lies in the plane of the disc for case $(b)$.
$(C)$ It is horizontal for case $(a)$; and is at $45^{\circ}$ to the $x-z$ plane and is normal to the plane of the disc for case $(b)$.
$(D)$ It is vertical for case $(a)$; and is at $45^{\circ}$ to the $x-z$ plane and is normal to the plane of the disc for case $(b)$.
Give the answer for question $1$ and $2$.

$A$ cubical block of side $a = 30\,cm$ is moving with velocity $v = 2\,m/s$ on a smooth horizontal surface. The surface has a small bump at a point $O$ as shown in the figure. The angular velocity (in $rad/s$) of the block immediately after it hits the bump is:

$A$ rotating body has angular momentum $L$. If its frequency is doubled and its kinetic energy is halved,what will be its new angular momentum?

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 ............

$A$ thin ring of mass $2 \ kg$ and radius $1 \ m$ is rolling without slipping on a horizontal plane with velocity $1 \ m/s$. $A$ small ball of mass $1 \ kg$,moving with velocity $2 \ m/s$ in the opposite direction,hits the ring at a height of $1.8 \ m$ and goes vertically up with velocity $1 \ m/s$. Immediately after the collision:
$(A)$ the ring has pure rotation about its stationary $CM$.
$(B)$ the ring comes to a complete stop.
$(C)$ friction between the ring and the ground is to the left.
$(D)$ there is no friction between the ring and the ground.