A ring of radius $3a$ is fixed rigidly on a table. A small ring whose mass is $m$ and radius $a$ rolls without slipping inside it as shown in the figure. The small ring is released from position $A$. When it reaches the lowest point, the speed of the centre of the ring at that time would be,

The moment of inertia of semicircular plate of radius $R$ and mass $M$ about axis $AA\mathit{\text{'}}$ in its plane passing through its centre is

The figure shows a uniform rod lying along the $x$-axis. The locus of all the points lying on the $xy$-plane, about which the moment of inertia of the rod is same as that about $O$ is:

A man can move on a horizontal plank supported symmetrically as shown. The variation of normal reaction on support A with distance $x$ of the man from the end of the plank is best represented by

Find minimum height of obstacle so that the sphere can stay in equilibrium

A disc of radius $R$ is rolling purely on a flat horizontal surface, with a constant angular velocity. The angle between the velocity and acceleration vectors of point $P$ is

The portion $AB$ of the wedge shown in figure is rough and $BC$ is smooth. A solid cylinder rolls without slipping from $A$ to $B$. The ratio of translational kinetic energy to rotational kinetic energy when the cylinder reaches point $C$ is,

A ring of mass $m$ and radius $R$ has three particles attached to the ring as shown in the figure. The centre of the ring has a speed $v_0$. The kinetic energy of the system is : (slipping is absent)