Two rings of radius $R$ and $n R$ made of same material have the ratio of moment of inertia about an axis passing through centre is $1: 64$. The value of $n$ is

A thin but rigid semicircular wire frame of radius $r$ is hinged at $O$ and can rotate in its own vertical plane. A smooth peg $P$ starts from $O$ and moves horizontally with constant speed $2 v_0$ lifting the frame upward as shown in figure. Find the angular velocity $\omega$ of the frame when its diameter makes an angle of $60^{\circ}$ with vertical.

Four holes of radius 5 cm are cut from a thin square plate of 20 cm and mass 1 kg. The moment of inertia of the remaining portion about Z-axis is

A solid sphere of 80 kg and radius 15 m moving in a space becomes a circular disc of radius 20 m in 1 h. The rate of change of moment of Inertia in this process is .......

The angular momentum of a body placed at origin of mass 1 kg and having position vector $r = 3t\widehat i + 4\widehat j$ is

Three particles each of mass m are kept at the vertices of an equilateral triangle of side b. Moment of inertia of the system about an axis passing through the centroid and perpendicular to its plane is

A T-shaped object with dimensions shown in figure, is lying on a smooth floor. A force F is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C.

The wheel of a car, accelerated uniformly from rest, rotates through 5 radians during the first second. The angle (in radians) rotated during the next second is