$A$ block of mass $m = 1 \, kg$ slides with velocity $v = 6 \, m/s$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about $O$ and swings as a result of the collision,making an angle $\theta$ before momentarily coming to rest. If the rod has mass $M = 2 \, kg$ and length $l = 1 \, m$,the value of $\theta$ is approximately (Take $g = 10 \, m/s^2$) (in $^{\circ}$)

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

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 $2$.

$STATEMENT-1$ If there is no external torque on a body about its center of mass,then the velocity of the center of mass remains constant. because
$STATEMENT-2$ The linear momentum of an isolated system remains constant.

$A$ thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement$(s)$ is/are correct,when the rod makes an angle $60^{\circ}$ with vertical? [$g$ is the acceleration due to gravity]
$(1)$ The radial acceleration of the rod's center of mass will be $\frac{3g}{4}$
$(2)$ The angular acceleration of the rod will be $\frac{3\sqrt{3}g}{4L}$
$(3)$ The angular speed of the rod will be $\sqrt{\frac{3g}{2L}}$
$(4)$ The normal reaction force from the floor on the rod will be $\frac{Mg}{16}$

$A$ projectile of mass $3m$ explodes at the highest point of its path. It breaks into three equal parts. One part retraces its path,the second one comes to rest. The distance of the third part from the point of projection when it finally lands on the ground is ........$m.$ (The range of the projectile was $100\,m$ if no explosion would have taken place.)