The gravitational potential energy of a body of mass $m$ at a distance $r$ from the center of the Earth is $U$. Express the weight of the body at this point in terms of $U$.

If three particles each of mass $M$ are placed at the corners of an equilateral triangle of side $a$,the potential energy of the system and the work done if the side of the triangle is changed from $a$ to $2a$,are

If the potential at the surface of the Earth is assigned a zero value,then the potential at the center of the Earth will be (Mass $= M$,Radius $= R$).

$A$ thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$. The work required to take a unit mass from point $P$ on its axis to infinity is

Gravitational potential difference between the surface of a planet and a point situated at a height of $20\,m$ above its surface is $2\,J/kg$. If the gravitational field is uniform,then the work done in taking a $5\,kg$ body to a height of $4\,m$ above the surface will be ........ $J$.

$A$ tunnel is dug across the diameter of the Earth. $A$ ball is released from the surface of the Earth into the tunnel. The velocity of the ball when it is at a distance $\frac{R}{2}$ from the center of the Earth is (where $R$ = radius of the Earth and $M$ = mass of the Earth):