The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. If the mass is kept constant,what will be the nature of the graph between $I$ and $R$?

$A$ solid sphere and a disc of the same radius are released from the same height on an inclined plane and they reach the bottom of the plane at different times. This is due to their:

State whether the following statements are true or false:
$(1)$ In the law of conservation of momentum, the center of mass velocity remains constant.
$(2)$ "If the resultant of internal forces on a system is zero, then the total linear momentum of the system remains constant." This is the statement of the law of conservation of linear momentum.
$(3)$ The position of the center of mass of a rigid body must always be inside the rigid body.
$(4)$ All particles of a rigid body undergoing rotational motion have the same linear velocity.

$A$ uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly,what happens to its speed of rotation?

The figure shows a system consisting of $(i)$ a ring of outer radius $3R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and $(ii)$ an inner disc of radius $2R$ rotating anti-clockwise with angular speed $\omega/2$. The ring and disc are separated by frictionless ball bearings. The system is in the $x-z$ plane. The point $P$ on the inner disc is at distance $R$ from the origin,where $OP$ makes an angle of $30^{\circ}$ with the horizontal. Then with respect to the horizontal surface,
$(A)$ the point $O$ has linear velocity $3R\omega\hat{i}$.
$(B)$ the point $P$ has a linear velocity $\frac{11}{4}R\omega\hat{i} + \frac{\sqrt{3}}{4}R\omega\hat{k}$.
$(C)$ the point $P$ has linear velocity $\frac{13}{4}R\omega\hat{i} - \frac{\sqrt{3}}{4}R\omega\hat{k}$.
$(D)$ The point $P$ has a linear velocity $(3 - \frac{\sqrt{3}}{4})R\omega\hat{i} + \frac{1}{4}R\omega\hat{k}$.

Fill in the blanks:
$(1)$ If the velocity of the center of mass of a body is $v_{cm} = 0$ and the angular velocity is $\omega = 0$,the body is said to be in ............. equilibrium.
$(2)$ Angular momentum is generated in a body when a ............. acts on it.
$(3)$ If a barrel is filled half with water,its center of gravity will move ............. .
$(4)$ The point at which the entire mass of a body is assumed to be concentrated is called the ............. .