Question

The earth revolves around the sun due to gravitational attraction. Suppose that the sun and the earth are point particles with their existing masses and that Bohr's quantization rule for angular momentum is valid in the case of gravitation. (a) Calculate the minimum radius the earth can have for its orbit. (b) What is the value of the principal quantum number n for the present?

Answer 1

We have,

Mass of the Earth $=Me=6.0$ X ${10}^{24}kg$

Mass of the Sun $=Ms=2.0$ X ${10}^{30}kg$

Distance between Earth and Sun $=1.2$ X ${10}^{11}m$

Also we have the equations,

$mvr=\frac{nh}{2\pi }$ or $m^2{v}^2{r}^2=\frac{n^2{h}^2}{4{\pi }^2}$ .....(1)

$\frac{GMeMs}{r^2}=\frac{Me{v}^2}{r}$ or $v^2=\frac{GMs}{r}$ .....(2)

Dividing (1) and (2), we get

${Me}^{2}r=\frac{n^2{h}^2}{4{\pi }^{2}GMs}$

For $n=1$

Radius of the Earth, $r=\sqrt{\frac{h^2}{4{\pi }^{2}GMsM{e}^2}}=2.29$ X ${10}^{-138}m$