If the potential energy of a system of four particles placed at the vertices of a square of side $l$ is $\frac{-5.41{\mathrm{Gm}}^{\alpha }}{l}$ and the potential at the centre of the square is $\frac{-4\sqrt{\beta }\mathrm{Gm}}{l}$, then find $\alpha +\beta$.

A body moving radially away from a planet of mass$\mathrm{M}$, when at distance $\mathrm{r}$ from planet, explodes in such a way that two of its many fragments move in mutually perpendicular circular orbits around the planet. What will be

(i) Then velocity in circular orbits?
(ii) Maximum distance between the two fragments before collision and
(iii) Magnitude of their relative velocity just before they collide?

A satellite$P$ is revolving around the Earth at a height $h=$radius of Earth $\left(R\right)$ above equator. Another satellite $Q$, is at a height $2 \mathrm{h}$ revolving in opposite direction. At an instant the two are at same vertical line passing through centre of sphere. If the least time after which again they are in this situation is given by$\frac{2\pi R^{{\displaystyle \frac{3}{2}}}6\sqrt{\alpha }}{\sqrt{GM}(2\sqrt{2}+3\sqrt{\beta })}$, then find $\alpha +\beta$. 

The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the equator barely provides the centripetal force needed for the rotation. (Why?) 

(i) Show then that the corresponding shortest period of rotation is given by $\mathrm{T}=\sqrt{\frac{3\pi }{\mathrm{Gp}}}$ where $\rho$ is the density of the planet, assumed to be homogeneous.

(ii) Evaluate the rotation period assuming a density of $3.0\mathrm{gm} {\mathrm{cm}}^{-3}$ typical of many planets, satellites, and asteroids. No such object is found to be spinning with a period shorter than found by this analysis.