Question

a particle of mass m is moving on a horizontal circle of radius r under a centripetal force equal to (k/r^2 ) where k is a constant.what is its potential energy?

Answer 1

Let us assume that the force here is conservative as only conservative forces account to the potential energy. The potential energy is given by

U=−∫∞rF.drU=−∫r∞F.dr

The potential energy is

U=−krU=−kr

Centripetal force makes a particle move in circular motion, therefore

mv2r=kr2mv2r=kr2

mv2=krmv2=kr

Since the kinetic energy of the particle is

K=mv22K=mv22

K=k2rK=k2r

And finally by the conservation of energy

E=U+KE=U+K

E=−kr+k2rE=−kr+k2r

Therefore the total energy of the system is given by

E=−k2r