Question

A particle of mass m is moving in a circular orbit under the influence of the central force\( f(r) = −kr\), corresponding to the potential energy \(v(r) = \frac{kr^2}{2}\), where \(k\) is a positive force constant and \( r\) is the radial distance from the origin. According to the Bohr’s quantization rule, the angular momentum of the particle is given by \(L= nℏ\), where\( ℏ = ℎ/(2pi), ℎ\) is the Planck’s constant, and \(n\) a positive integer. If \(v\) and \(E\) are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

• A. \(r^2\)=\(nh\sqrt{\frac{1}{mk}}\)
• B. \(v^2\)=\(nh\sqrt{\frac{k}{m^3}}\)
• C. \(\frac{L}{mr^2}\)=\(\sqrt{\frac{k}{m}}\)
• D. \(E\)=\(\frac{nh}{2}\sqrt{\frac{k}{m}}\)

Answer 1

The Correct Option is A, B, C

The correct option is (A): \(r^2\)=\(nh\sqrt{\frac{1}{mk}}\), (B): \(v^2\)=\(nh\sqrt{\frac{k}{m^3}}\) and (C): \(\frac{L}{mr^2}\)=\(\sqrt{\frac{k}{m}}\)