Question

Mass m is in a potential well given by U(r) = m (a/(r¹⁰)– b/(r⁸)) where a and b are known positive constants, and r is the distance from m to the force center, with r > 0. NOTE: Consider this problem as one-dimensional motion in the spherical coordinate r.

Suppose that, when the atom is at equilibrium, it undergoes a small displacement x (where x << r₀) from equilibrium and is released from rest; the displacement of the atom is x + r₀, where r₀ is a constant equal to the equilibrium distance. i. Write and simplify the differential equation in terms of x (and, of course, the given constants) Show that, for small values of x, the particle undergoes simple harmonic motion ii. Use your results to find, in terms of a and b, an expression for T, the period of oscillation of the atom about its equilibrium point.

Answer 1

Potential

Force on a mass is  

In equilibrium net force on the mass is zero.

Consider a small displacement .

In this position, net force on the object is

Using the binomial approximation, if

Using the equilibrium condition, first term is zero.

Substituting , and

Comparing with standard equation for simple harmonic motion, ,

Time period of simpe harmonic motion is