Question

Two planets have the same average density but their radii are ${R_1}$ and ${R_2}$ ​. If acceleration due to gravity on these planets be ${g_1}$ and ${g_2}$. respectively, then
A. $\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_1}}}{{{R_2}}}$
B. $\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_2}}}{{{R_1}}}$
C. $\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_1}^2}}{{{R_2}^2}}$
D. $\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_1}^3}}{{{R_2}^3}}$

Answer 1

Hint: Since the problem is based on gravitational acceleration hence, consider the effect of gravitational force on acceleration due to the gravity of two planets at infinite distance. Also, as we all know that the parameters vary with each other hence, analyze every aspect of the solution needed for the question and then present the answer with a proper explanation.

Complete answer:
We know that, the magnitude of gravitational force and gravitational acceleration, according to Newton, is given as:
$F = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}$ … (1)
And, $g = \dfrac{4}{3}\pi Gr\rho $ … (2)
where,
F = Gravitational force between two planets
${m_1}$ and ${m_2}$ are the masses of two planets
r = distance between the centres of ${m_1}$ and ${m_2}$
G = Universal Gravitational Constant
$\rho $ = density of the planet

Since,the average density of the two planets is same. Therefore, $\rho $ will be same for both planets.
As, ${g_1}$ and ${g_2}$ are acceleration due to gravity of two planets & ${R_1}$ and ${R_2}$ are radii of two planets (given)

Now, applying these values in eq. (2), we get
${g_1} = \dfrac{4}{3}\pi G{R_1}\rho $ … (3)
${g_2} = \dfrac{4}{3}\pi G{R_2}\rho $ … (4)

Divide eq. (3) by eq. (4), we get
$\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{\dfrac{4}{3}\pi G{R_1}\rho }}{{\dfrac{4}{3}\pi G{R_2}\rho }}$
$ \Rightarrow \dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_1}}}{{{R_2}}}$

Thus, the relation between radii and acceleration due to gravity of two planets is $\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_1}}}{{{R_2}}}$.

Hence, the correct option is (A) $\dfrac{{{g_1}}}{{{g_2}}} = \dfrac{{{R_1}}}{{{R_2}}}$.

Note:This is a conceptual-based problem hence, it is essential that the given question is to be analyzed very carefully to give a precise explanation. While writing a solution, support your explanation by providing proper reasons with the help of formulas and scientific relations and correlate the terms used with each other that might help in the solution.