A small particle of mass m moves in such a way that its potential energy U = 1 2 m ω 2 r 2 where ω is constant and r is the distance of the particle from origin. Assuming Bohr’s quantization of momentum and circular orbit, the radius of n th orbit will be proportional to

Step-by-step solution

The data given is U = 1 2 m ω 2 r 2 From Bohr's quantization, m v r = n h 2 π ⇒ v 2 = n h 2 π r m 2       . . . ( i ) In an orbit the value of the kinetic energy is half of the potential energy, K = m ω 2 r 2 4 Substituting the value of equation (i) in the kinetic energy, K = 1 2 m n 2 h 2 4 π 2 m 2 r 2 = m ω 2 r 2 4 ⇒ n 2 h 2 2 π 2 m 2 ω 2 = r 4 ⇒ r ∝ n