$A$ particle falls freely near the surface of the earth. Consider a fixed point $O$ (not vertically below the particle) on the ground.

$A$ disc having velocity $v_0$ and angular speed $\omega_0$ in the anticlockwise direction is placed on a rough plank. Initially,the plank is at rest. (Assume the length of the plank is very large). Choose the $INCORRECT$ option.

$A$ solid sphere of mass $m$,radius $R$,having moment of inertia about an axis passing through its center of mass as $I$ is recast into a disc of thickness $t$ whose moment of inertia about an axis passing through the rim (edge) and perpendicular to its plane remains $I$. Then the radius of the disc is:

$ABC$ is an equilateral triangle with $O$ as its centre. $\vec F_1, \vec F_2$ and $\vec F_3$ represent three forces acting along the sides $AB, BC$ and $AC$ respectively. If the total torque about $O$ is zero,then the magnitude of $\vec F_3$ is

$A$ rod of mass $m$ and length $l$ is hinged at one end to a horizontal floor and stands vertically. If it is allowed to fall,the velocity with which its upper end strikes the floor is:

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 $2$.