From a solid sphere of mass $M$ and radius $R\text{,}$ a spherical portion of radius $\left(\frac{R}{2}\right)$ is removed as shown in the figure. Taking gravitational potential $V=0$ at $r=\infty ,$ the potential at the centre of the cavity thus formed is ($G=$gravitational constant)

A solid body of constant heat capacity $1 \mathrm{J} {}^{\circ }\mathrm{C}^{-1}$ is being heated by keeping it in contact with reservoirs in two ways
(i) Sequentially keeping in contact with $2$ reservoirs such that each reservoir supplies the same amount of heat.
(ii) Sequentially keeping in contact with $8$ reservoirs such that each reservoir supplies the same amount of heat.

In both the cases, the body is brought from initial temperature $100 °\mathrm{C}$ to final temperature $200 °\mathrm{C}$. Entropy change of the body in the two cases respectively, is

A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O.$ These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant, the angular speed of the system is $\frac{8}{9}\omega$ and one of the masses is at a distance of $\frac{3}{5}R$ from $O.$ At this instant, the distance of the other mass from $O$ is,