$A$ rod of length $1\,m$ is standing vertically. When its upper end is released and it falls such that the lower end touches the ground without slipping,the speed of the upper end when it hits the ground is:

Two point-like objects of masses $20 \text{ g}$ and $30 \text{ g}$ are fixed at the two ends of a rigid massless rod of length $10 \text{ cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass,as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{-8} \text{ N m rad}^{-1}$. The angular frequency of the oscillations is $n \times 10^{-3} \text{ rad s}^{-1}$. The value of $n$ is

$A$ uniform rod of mass $m$ and length $l$ hinged at its end is released from rest when it is in the horizontal position. The normal reaction at the hinge when the rod becomes vertical is:

Four spheres,each of mass $M$ and diameter $2a$,are placed at the corners of a square of side $b$. The moment of inertia of this system about an axis along one of the sides of the square is:

Four point masses are fastened to the corners of a frame of negligible mass lying in the $xy$ plane as shown in the figure. Let $\omega$ be the angular speed of rotation. Then:

$A$ bullet of mass $m$ is fired horizontally into a large sphere of mass $M$ and radius $R$ resting on a smooth horizontal table. The bullet hits the sphere at a height $h$ from the table and sticks to its surface. If the sphere starts rolling without slipping immediately on impact,then