Question

The moon revolves around the earth 13 times per year. If the ratio of the distance of the earth from the sun to the distance of the moon from the earth is 392, find the ratio of mass of the sun to the mass of the earth.
A. 365
B. 356
C. 3.56 $\times$ 10$^5$
D. 1

Answer 1

Hint: The gravitational force imparted the earth on the moon gives the necessary centripetal force for the moon through which it is orbiting the earth. The moon appears to be orbiting around the earth but is orbiting the centre of gravity that is inside the earth.

Complete step by step solution:
Given:
(a) The ratio of time taken by earth around sun ${T_e}$ and the time taken by moon around earth ${T_m}$ is
$\dfrac{{{T_e}}}{{{T_m}}} = 13$
(b) Ratio of the distance of the earth from the sun ${R_e}$ to the distance of the moon from the earth ${R_m}$ is Re/Rm=392.
Express the relation for time period Te of earth around sun of mass Ms.
${T_e} \propto \dfrac{{R_e^3}}{{{M_s}}}$ ……. (I)
Here ${R_e}$ is the sun-earth distance.
Express the relation for the time period Tm of moon around earth of mass Me.
${T_m} \propto \dfrac{{R_m^3}}{{{M_e}}}$ …… (II)
Here ${R_m}$ is the moon earth distance.
Divide eq (I) by eq (II), it comes out to be,
\[\begin{array}{l}
{\left( {\dfrac{{{T_e}}}{{{T_m}}}} \right)^2} = \left( {\dfrac{{{M_e}}}{{{M_s}}}} \right)\dfrac{{R_e^3}}{{R_m^3}}\\
\left( {\dfrac{{{M_s}}}{{{M_e}}}} \right) = \dfrac{{R_e^3}}{{R_m^3}} \times {\left( {\dfrac{{{T_m}}}{{{T_e}}}} \right)^2}
\end{array}\]
Substitute $\dfrac{{{T_e}}}{{{T_m}}} = 13$,$\dfrac{{{R_e}}}{{{R_m}}} = 392$ , it comes out to be,
\[\begin{array}{l}
\left( {\dfrac{{{M_s}}}{{{M_e}}}} \right) = {392^3} \times {13^2}\\
\left( {\dfrac{{{M_s}}}{{{M_e}}}} \right) = 3.56 \times {10^5}
\end{array}\]

Therefore, the correct option is (C).

Note: There is no quantity or friction to stop the Moon from orbiting, so it keeps on going round-round. The Moon is travelling in a vacuum so there is no drag to make it slow it down. It moves around the Earth because it is in orbit around the Earth. The Moon is keen to fall to the Earth, but since the Moon is also moving side-along, it keeps missing the Earth.