Question

If the mass of the earth is doubled and the distance of the moon revolving around the earth is also doubled, then, find the new time period of revolution of moon.
(Take the present time of revolution as 28 days)
A. 6
B. 36
C. 56
D. 112

Answer 1

In this question, we need to use the concept of the centripetal force on the moon, which is provided by the gravitational force of the earth acting on the moon. If we equate both the formulas and incorporate the fact that the total distance covered is equal to the time taken multiplied by the tangential velocity of the moon, then we get that the expression for the time period of revolution is, $T = 2\pi \sqrt {\dfrac{{{r^3}}}{{G \cdot {m_e}}}}$ .

Complete step by step answer:
According to the question;
Given:
If the initial mass of the earth is ${M_1}$, then the new mass of the earth is, ${M_2} = 2{M_1}$.
The present time period of revolution is, ${T_1} = 28{\rm{ days}}$.
Again, if the initial distance between the moon and the earth is ${R_1}$, then the new distance from the earth is, ${R_2} = 2{R_1}$.
Now, the relation for the time period of revolution for the moon is given as, $T = 2\pi \sqrt {\dfrac{{{r^3}}}{{G \cdot {m_e}}}}$.
So, the time period, $T\alpha \sqrt {\dfrac{{{r^3}}}{{{m_e}}}}$
Taking the ratio of the time period in the old and in the new case we have;
$\dfrac{{{T_1}}}{{{T_2}}} = \sqrt {\dfrac{{{R_1}^3 \times {M_2}}}{{{R_2}^3 \times {M_1}}}}$
Substituting the values of mass and the distance in the above expressions we get;

\;\;\;\;\dfrac{{{T_1}}}{{{T_2}}} = \sqrt {\dfrac{{{R_1}^3 \times {M_2}}}{{{R_2}^3 \times {M_1}}}} \\\ \Rightarrow \dfrac{{{T_1}}}{{{T_2}}} = \sqrt {{{\left( {\dfrac{1}{2}} \right)}^3} \times 2} \\\ \Rightarrow \dfrac{{{T_1}}}{{{T_2}}} = \sqrt {\dfrac{1}{4}} \end{array}$$ $$\begin{array}{l} \Rightarrow \dfrac{{{T_1}}}{{{T_2}}} = \dfrac{1}{2}\\\ \Rightarrow {T_2} = 2{T_1} = 2 \times 28 = 56 \end{array}$$ Therefore, the new time period of revolution of the moon is 56 days **So, the correct answer is “Option C”.** **Note:** In the current question, we considered that even after the change in the mass of the earth and the distance in between, the moon is revolving under the effect of the gravitational force of earth only. In other words, we can calculate the time of revolution of a body using this formula only if the condition that the centripetal force is due to the gravitational force on the body is satisfied.