Due to gravitational attraction, the earth revolves round the sun. Assume that the sun and the earth are point particles with their existing masses, and that Bohr's quantization rule for angular momentum holds true for gravitation. Determine the value of the principal quantum number n for the present radius. (Given: mass of the earth = 6.0 10²⁴ kg , mass of the sun = 2.0 10³⁰ kg , earth–sun distance = 1.5 10¹¹ m ).

Step-by-step solution

According to Bohr's quantization rule, the angular momentum of the earth is given by: L = nh 2 The angular momentum of the earth revolving in a circular orbit is: L = mvr The orbital speed v of the earth is obtained by equating the gravitational force to the centripetal force: GMm r^2 = mv^2 r v = GM r Substituting v into the expression for angular momentum: L = m GM r r = m GMr Equating the two expressions for angular momentum: nh 2 = m GMr n = 2 m h GMr Substituting the given values ( G = 6.67 10⁻¹¹ N m ^2 kg ⁻²