Question

Two planets A and B have the same surface gravity, but planet B has twice the mass as that of planet A. If the radius of planet A is R, then the radius of planet B is
1. 0.707R
2. 2R
3. 1.414R
4. 4R

Answer 1

Correct Answer - Option 3 : 1.414R

The correct answer is option 3) i.e. 1.414R

CONCEPT:

• Law of Universal Gravitation: It states that all objects attract each other with a force that is proportional to the masses of two objects and inversely proportional to the square of the distance that separates their centres.

It is given mathematically as follows:

\(F = \frac{Gm_1m_2}{R^2}\)

Where m1 and m2 are the mass of two objects, G is the gravitational constant and R is the distance between their centres.

• From the Law of Universal Gravitation, the gravitational force acting on an object of mass m placed on the surface of Earth is:

\(F = \frac{GMm}{R^2}\)

Where R is the radius of the earth. 

From Newton's second law, F = ma = mg

\(⇒ mg =\frac{GMm}{R^2}\)

Therefore, acceleration due to gravity, \(g =\frac{GM}{R^2}\)

CALCULATION:

Let m_A and m_B be the mass of planets and R_A and R_B be their radii respectively.

Given that: m_B = 2m_A and R_A = R

Acceleration due to gravity, \(g =\frac{GM}{R^2}\)

Since the surface gravities are the same, acceleration due to gravity is the same i.e. gA = gB

\(⇒ \frac{Gm_A}{R_A^2} = \frac{Gm_B}{R_B^2}\)

\(⇒ \frac{m_A}{R_A^2} = \frac{2m_A}{R_B^2} ⇒ R_B^2 =2R_A^2\)

⇒ R_B = √2R_A = √2R = 1.414R