Rotational Motion

The moment of inertia of a uniform cylinder of length $l$ and radius R about its perpendicular bisector is $I$. What is the ratio ${l \over R}$ such that the moment of inertia is minimum?

Two coaxial discs, having moments of inertia I₁ and I₁/2, are rotating with respective angular velocities $\omega$_1 and $\omega$₁/2 , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If E_f and E_i are the final and initial total energies, then (E_f - E_i) is:

A thin disc of mass M and radius R has mass per unit area $\sigma$(r) = kr² where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta$, where $\theta$ is the angle by which it has rotated, is given as k$\theta$². If its moment of inertia is I then the angular acceleration of the disc is :

A thin circular plate of mass M and radius R has its density varying as $\rho$(r) = $\rho$₀r with $\rho$₀ as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = aMR². The value of the coefficient a is :

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 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 :

The linear mass density of a thin rod AB of length L varies from A to B as $\lambda \left( x \right) = {\lambda _0}\left( {1 + {x \over L}} \right)$, where x is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is :

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period T₁ and, (ii) back and forth in a direction perpendicular to its plane, with a period T₂. The ratio ${{{T_1}} \over {{T_2}}}$ will be :

Consider two uniform discs of the same thickness and different radii R₁ = R and R₂ = $\alpha$R made of the same material. If the ratio of their moments of inertia I₁ and I₂ , respectively, about their axes is I₁ : I₂ = 1 : 16 then the value of $\alpha$ is :

Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is I = $M\left( {{{{R^2}} \over 4} + {{{L^2}} \over {12}}} \right)$. If such a cylinder is to be made for a given mass of a material, the ratio ${L \over R}$ for it to have minimum possible I is