A ring of radius $4a$ is rigidly fixed in vertical position on a table. A small disc of mass $m$ and radius $a$ is released as shown in the fig. When the disc rolls down, without slipping, to the lowest point of the ring, then its speed will be

In a gravity free space, a man of mass $M$ standing at a height $h$ above the floor, throws a ball of mass $m$ straight down with a speed $u$ . When the ball reaches the floor, the distance of the man above the floor will be

The moment of inertia of a uniform thin rod of length $L$ and mass $M$ about an axis passing through the rod from a point at a distance of $L/3$ from one of its ends perpendicular to the rod is

Four particles of masses $m_1 = 2m,\, m_2 = 4m,\, m_3 = m$ and $m_4$ are placed at four corners of a square. What should be the value of $m_4$ so that the centres of mass of all the four particles are exactly at the centre of the square?

A massless string is wrapped round a disc of mass $M$ and radius $R$. Another end is tied to a mass $m$ which is initially at height $h$ from ground level as shown in the fig. If the mass is released then its velocity while touching the ground level will be

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined  plane making an angle $\theta $ with the horizontal. then its acceleration is