Disc $A$ and disc $B$ (with a hole) have equal mass and radius,and a string is wrapped around them as shown. An equal force $F$ is applied to the string of each body. Friction is sufficient for rolling. After time $t$,the velocity of $A$ is $v_A$ and that of $B$ is $v_B$,and their kinetic energies are $k_A$ and $k_B$ respectively. Then:

$A$ uniform rod $AB$ of mass $2 \ kg$ and length $30 \ cm$ is at rest on a smooth horizontal surface. An impulse of force $0.2 \ Ns$ is applied to end $B$. The time taken by the rod to turn through a right angle will be $\frac{\pi}{X} \ s$,where $X = \text{ . . . . . . }$.

Which one of the following four graphs best depicts the variation with $x$ of the moment of inertia $I$ of a uniform triangular lamina about an axis parallel to its base at a distance $x$ from it?

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 $2$.

Three particles,each of mass $m$,are placed at the corners of an equilateral triangle of side $l$. Which of the following is/are correct?

Which of the following statements are correct?
$(a)$ Centre of mass of a body always coincides with the centre of gravity of the body.
$(b)$ Centre of mass of a body is the point at which the total gravitational torque on the body is zero.
$(c)$ $A$ couple on a body produces both translational and rotational motion in a body.
$(d)$ Mechanical advantage greater than $1$ means that small effort can be used to lift a large load.