The potential energy between two atoms is given by the function $U(r) = a/r^{12} - b/r^{6}$. Find the equilibrium distance between them.

Given in the figures are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case,specify the regions,if any,in which the particle cannot be found for the given energy. Also,indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

The force acting on a particle is given by the formula $F(x) = -kx + ax^3$. Find the graph of its potential energy $U(x)$ for $x \ge 0$.

If the potential energy of a particle is given by $U = A - Bx^2$,then the force is proportional to which of the following?

Calculate the work done in rotating a vertical meter scale of mass $400 \, g$ by an angle of ${60^o}$. (in $, J$)

If the potential energy of a particle of mass $0.1 \ kg$ moving along $x$-axis is $U(x) = 5x(x-4) \ J$,then the speed of the particle is maximum at a position of