Question

A particle moves along a closed trajectory in a central field of force where the particle's potential energy U = kr² (k is a positive constant, r is the distance of the particle from the centre O of the field). Find the mass of the particle if its minimum distance from the point O equals r₁ and its velocity at the point farthest from O equals v₂.

Answer 1

If r = radial velocity of the particle then the total .energy of the particle at any instant is 

where the second term is the kinetic energy of angular motion about the centre O. Then the extreme values of r are determined by r = 0 and solving the resulting quadratic equation

where r₁ is the minimum distance from O and r is the maximum distance. Then

Note: Eq. (1) can be derived from the standard expression for kinetic energy and angular momentum in plane polar coordinates :