Consider a sphere of mass $m$ and radius $R$ performing pure rolling motion on a rough surface with velocity $v_0$ as shown in the figure. It makes an elastic impact with a smooth wall,moves back,and eventually starts pure rolling again.

$A$ rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical,its lower half separates without exerting any reaction at the breaking point. Then the maximum angle '$\theta$' made by the hinged upper half with the vertical is ......... $^o$.

If the Earth suddenly stops rotating about its own axis,the increase in its temperature will be

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 $2$.

$A$ uniform thin wooden plank $AB$ of length $L$ and mass $M$ is kept on a table with its $B$ end slightly outside the edge of the table. When an impulse $J$ is given to the end $B$,the plank moves up with the centre of mass rising a distance $h$ from the surface of the table. Then,

$A$ uniform rod $AB$ of mass $m$ and length $l$ is at rest on a smooth horizontal surface. An impulse $J$ is applied to the end $B$,perpendicular to the rod in the horizontal direction. Find the speed of particle $P$ at a distance $\frac{l}{6}$ from the centre towards $A$ of the rod after time $t = \frac{\pi m l}{12 J}$.