potential energy Questions

If $g$ is the acceleration due to gravity on the earth’s surface, the gain in the potential energy of object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth is:

A charged particle $q$ is shot towards another charged particle $Q$ which is fixed. With a speed $v$, it approaches $Q$ up to a closest distance $r$ and then returns. If $q$ were given a speed $2v$, the closest distance of approach would be:

A uniform chain of length $2\text{m}$ is kept on a table such that a length of $60\text{cm}$ hangs freely from the edge of the table. The total mass of the chain is $4\text{kg}$. What is the work done in pulling the entire chain on the table?

A particle of mass $10\text{ g}$ is kept on the surface of a uniform sphere of mass $100\text{ kg}$ and radius $10\text{ cm}$. Find the work to be done against the gravitational force between them to take the particle far away from the sphere [$G=6.67\times 10^{-11}{\text{ Nm}}^2/{\text{kg}}^2$ ]:

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U(x)=\frac{a}{x^{12}}-\frac{b}{x^6}$, where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is $D=[U(x=\infty )-U_{atequilibrium}]$, $D$ is: