Let the moment of inertia of a hollow cylinder of length $30\mathrm{ }\mathrm{cm}$ (inner radius $10\mathrm{ }\mathrm{cm}$ and outer radius $20\mathrm{ }\mathrm{cm}$), about its axis be $I$. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also $I,$ is:

A uniform rod is hinged as shown in the figure and is released from a horizontal position. The angular velocity of the rod as it passes the vertical position is:

A thin uniform rod of length $4 l$, mass $4 \mathrm{m}$ is bent at the points as shown in the figure. What is the moment of inertia of the rod about the axis passing point $O \&$ perpendicular to the plane of the papers.

Two uniform rods of equal length, but different masses are rigidly joined to form an L-shaped body, which is then pivoted about $O$ as shown in the figure. If in equilibrium the body is in the shown configuration, ratio $\frac{M}{m}$ will be:

Four forces tangent to the circle of radius $R$ are acting on a wheel as shown in the figure. The resultant equivalent one force system will be

Two points $A$ and $B$ on a disc have velocities $v_1$ and $v_2$  at some moment. Their directions make angles $60°$ and $30°$, respectively with the line of separation as shown in figure. The angular velocity of disc is

An equilateral uniform prism of mass $m$ rests on a rough horizontal surface with a coefficient of friction $\mu .$ A horizontal force $F$ is applied on the prism as shown in the figure. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, then the minimum force required to topple the prism is:

A uniform sphere of mass $m$ is given some angular velocity about a horizontal axis through its centre and gently placed on a plank of mass $m$. The coefficient of friction between the two is $\mu$. The plank rests on a smooth horizontal surface. The initial acceleration of the centre of the sphere relative to the plank will be: