A ballet dancer is rotating at angular velocity $\omega$ on smooth horizontal floor. The ballet dancer folds his body close to his axis of rotation by which his radius of gyration decreases by $1/4^{th}$ of his initial radius of gyration, his final angular velocity is

A uniform rod of length $L$ lies on a smooth horizontal table. The rod has a mass $M.$ A particle of mass $m$ moving with speed $v$ strikes the rod perpendicularly at one of the ends of the rod sticks to it after collision.
Find the velocity of the centre of mass $C$ and the angular, velocity of the system about the centre of mass after the collision.

From a circular disc of radius R and mass 9 M , a small disc of radius R/3 is removed from the disc. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is

A cubical block of side a is moving with velocity V on a horizontal smooth plane as shown in Figure. It hits a ridge at point O. The angular speed of the block after it hits O is

A slender uniform rod of mass $M$ and length $l$ is pivoted at one ens so that it can rotate in a vertical plane, Fig. There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is

A balloons of mass M is descending at a constant acceleration $\alpha .$ When a mass m is released from the balloon, it starts rising. With the same acceleration $\alpha .$ Assuming that its volumes does not change, what is the valule of m?

In the shown planar frame made of thin uniform rods, the length of section $AB$ and $EF$ is $l_1$ and its thermal linear coefficient of expansion is ${\alpha }_1.$ The length of section $CD$ is $l_2$ and its thermal linear coefficient of expansion is ${\alpha }_2.CB\text{and}DE$ are of same length having thermal linear coefficient of expansion ${\alpha }_2.$ point $A,B,E\text{and}F$ reside on same line, that is, sections $AB\text{and}EF$ overlap. then the ratio of $\frac{l_1}{l_2}$ for which the distance between end $A$ and end $F$ remains the same at all temperatures, is

If the system is released, then the acceleration of the centre of mass of the system is :
.

The capacitance of the capacitors of plate areas $A_1$ and $A_2\left(A_1<A_2\right)$ at a distance d is

Two block $A$ and $B$ of masses $3m$ and $m$ respectively are connected by a massless and nextensible string. The whole system is suspended by a massless spring as shown in figure. The magnitudes of acceleration of $A$ and $B$ immediately after the string is cut, are resectively