Gravitational Potential Energy

A hollow sphere (mass $\mathrm{m}$, radius $\mathrm{R}$ ) is put inside a hollow cone (mass $\mathrm{m}$, radius $3\mathrm{R}$ and semi-angle $30°$ ) as shown in the figure. The whole arrangement is rigidly fixed to the ground at the apex of the cone. Both the sphere and the cone are made of the same material $\left(\alpha ={10}^{-4}/{}^{\circ }\mathrm{C}\right)$. Considering that major changes in dimensions could happen only due to heating effects, if the increase in potential energy (in Joule) of the system if it's temperature is increased by $100°\mathrm{C}$ is $∆U$, what is the value of $10∆U$?

$\mathrm{m}=0.5 \mathrm{kg}$  

$\mathrm{R}=\frac{\sqrt{3}-1}{2} \mathrm{m}$

A planet of mass '$m$' moves in an elliptical orbit around an unknown star of mass '$M$' such that its maximum and minimum distances from the star are equal to $r_1$ and $r_2$ respectively. The angular momentum of the planet relative to the centre of the star is

A body of mass $m \mathrm{kg}$ starts falling from a point $2R$ above the Earth's surface. Its kinetic energy when it has fallen to a point $R$ above the Earth's surface is [$R-$Radius of Earth, $M—$Mass of Earth,$G—$Gravitational Constant]

A satellite of mass $\mathrm{m}$ is orbiting the earth in a circular orbit at a height $2\mathrm{R}$ above earth surface where $\mathrm{R}$ is the radius of the earth. If it starts losing energy at a constant rate $\mathrm{\beta },$ then it will fall on the earth surface after the time $\mathrm{t}=\frac{\mathrm{GMm}}{\mathrm{n\beta R}}.$ Assuming that the satellite is approximately in a circular orbit at all times, find the value of $\mathrm{n}.$