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

Answer: (A), (B), (C)

$r^2$=$nh\sqrt{\frac{1}{mk}}$

Answer 2

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