A particle of mass m , and angular momentum is moving in a circular orbit of radius r₀ under the influence of an attractive force F (r) = - k r^2 r . Keeping its angular momentum unchanged, the particle is displaced radially by a small distance r r₀ , due to which its radial distance varies periodically. The corresponding time period is:

Step-by-step solution

The equation of motion for the radial distance r is given by: m d^2r dt^2 = F(r) + ^2 mr^3 = - k r^2 + ^2 mr^3 For a circular orbit of radius r₀ , the radial acceleration is zero: - k r₀^2 + ^2 mr₀^3 = 0 r₀ = ^2 mk Let the particle be displaced by a small distance x such that r = r₀ + x . The restoring force is: m d^2x dt^2 = - k (r₀+x)^2 + ^2 m(r₀+x)^3 Using the binomial expansion for x r₀ : m d^2x dt^2 - k r₀^2 (1 - 2x r₀ ) + ^2 mr₀^3 (1 - 3x r₀ ) Since k r₀^2 = ^2 mr₀^3 , the constant terms cancel out: m d^2x dt