$A$ tunnel is dug in the Earth across one of its diameters. Two masses $m$ and $2m$ are dropped from the ends of the tunnel. The masses collide, stick to each other, and perform $S.H.M.$ The amplitude of the $S.H.M.$ will be:

Three point masses,each of mass $m$,are kept at the corners of an equilateral triangle of side $L$. The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to

State whether the following statements are true or false:
$(a)$ The value of acceleration due to gravity increases as we go to a depth $d$ below the Earth's surface.
$(b)$ If the escape velocity for an object of mass $m$ on the Earth's surface is $11.2 \, km/s$,then the escape velocity for an object of mass $2m$ is $22.4 \, km/s$.
$(c)$ If the Earth's gravitational attraction suddenly disappears,the mass of an object becomes zero,but its weight remains the same.

Out of the following,the only incorrect statement about satellites is

Answer the following:
$(a)$ You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?
$(b)$ An astronaut inside a small space ship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size,can he hope to detect gravity?
$(c)$ If you compare the gravitational force on the earth due to the sun to that due to the moon,you would find that the Sun's pull is greater than the moon's pull. However,the tidal effect of the moon's pull is greater than the tidal effect of the sun. Why?

Two bodies of masses $m_1$ and $m_2$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is