Find the minimum height $h$ of the obstacle so that the sphere of radius $R$ can stay in equilibrium on an inclined plane of angle $\theta$.

The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. Which graph represents the variation of $I$ with respect to $R$,keeping $M$ constant?

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.

$A$ disc of radius $R$ is rolling purely on a flat horizontal surface with a constant angular velocity $\omega$. The angle between the velocity and acceleration vectors of point $P$ (which is at the same horizontal level as the center $C$) is

$A$ uniform disc is rolling on a horizontal surface. At a certain instant,$B$ is the point of contact and $A$ is the topmost point of the disc,where $R$ is the radius of the disc.

$A$ wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. $A$ constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point $Q$. Which of the following options is/are correct?