A particle of mass m moves in circular orbits with potential energy U r = F r , where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by R and its speed and energy are denoted by v and E respectively, then for the n th orbit (here h is the Planck's constant)

Step-by-step solution

U = F · r F c = - d U d r = - F m v 2 r = F m u r = n h 2 π ⇒ m u r 2 m u 4 r = n 2 h 2 4 π 2 · F ⇒ m r 3 = n 2 h 2 4 π 2 F ⇒ r = n 2 h 2 4 π 2 m F 1 / 3 ⇒ r ∝ n 2 / 3 E = U + K = F · r + 1 / 2   m v 2 = F · r + 1 / 2 F · r E = 3 / 2 F · r Put value of r , E = 3 2 F r n 2 h 2 4 π 2 m F 1 / 3