$A$ horizontal platform is rotating with a uniform angular velocity about a vertical axis passing through its center. At some instant, a viscous liquid of mass $m$ is dropped at its center, which is free to spread and eventually falls off the edge. During this time interval, the angular velocity

$A$ solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown below. The cube is free to rotate about the fixed axis $AB$. $A$ bullet of mass $m (< M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed $\omega_{c}$,so that the cube just topples over. Then,$\omega_{c}$ is (Note: the moment of inertia of the cube about an axis passing through the centre of mass and parallel to the edge is $2Ma^{2}/3$)

State whether the following statements are True or False:
$(1)$ Angular position $\theta$ is a scalar,while angular displacement $\Delta \theta$ is a vector.
$(2)$ The relation between linear velocity $\vec{v}$ and angular velocity $\vec{\omega}$ for a particle in rotational motion is given by $\vec{v} = \vec{r} \times \vec{\omega}$.
$(3)$ The moment of inertia of a rigid body is constant.
$(4)$ The moment of momentum is called angular momentum.

$A$ bullet of mass $m$ is fired horizontally into a large sphere of mass $M$ and radius $R$ resting on a smooth horizontal table. The bullet hits the sphere at a height $h$ from the table and sticks to its surface. If the sphere starts rolling without slipping immediately on impact,then

On a solid sphere lying on a horizontal surface,a force $F$ is applied at a height of $R/2$ from the centre of mass. The initial acceleration of a point at the top of the sphere is (there is no slipping at any point).

$A$ 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 $^{\circ}$)