$A$ 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:

$A$ sphere of mass $M$ and radius $R$ is attached by a light rod of length $l$ to a point $P$. The sphere rolls without slipping on a circular track as shown. It is released from the horizontal position. The angular momentum of the system about $P$ when the rod becomes vertical is:

Two identical rings $A$ and $B$ of same mass $M$ and radius $R$ are revolving. Ring $A$ revolves around its own diameter, and ring $B$ revolves about a tangential axis in its own plane. Both rings $A$ and $B$ have the same rotational kinetic energy. The ratio of the angular velocity of ring $B$ $(\omega_B)$ to that of ring $A$ $(\omega_A)$ is:

$A$ metal rod of length $L$ and mass $m$ is pivoted at one end. $A$ thin disk of mass $M$ and radius $R$ $(R < L)$ is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case $A$) The disc is not free to rotate about its center and (case $B$) the disc is free to rotate about its center. The rod-disc system performs $SHM$ in a vertical plane after being released from the same displaced position. Which of the following statement$(s)$ is (are) true?

State whether the following statements are True or False:
$(1)$ The angular acceleration of an object rotating with a constant angular velocity is always zero.
$(2)$ An object can have a moment of inertia without energy.
$(3)$ The radius of gyration of an object is a constant quantity.
$(4)$ $A$ figure skater spins faster when they pull their arms in because their moment of inertia decreases.

In the following problem,indicate the correct direction of the friction force acting on a cylinder of mass $M$ and radius $R$,which is pulled on a rough surface by a constant horizontal force $F$ applied at its center. The friction force can be represented by which of the following diagrams?