
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
164.7k+ views
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.
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.
Recently Updated Pages
Uniform Acceleration - Definition, Equation, Examples, and FAQs

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line

Motion in a Straight Line Class 11 Notes: CBSE Physics Chapter 2

Important Questions for CBSE Class 11 Physics Chapter 1 - Units and Measurement
