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The acceleration due to gravity on Earth and Saturn are ge and gs respectively. If the radius and mass of Saturn is twice as that of the Earth, the ratio of their acceleration due to gravity is-
1. 2 : 1
2. 1 : 3
3. 1 : 4
4. 2 : 5

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Correct Answer - Option 1 : 2 : 1

The correct answer is option 1) i.e. 2 : 1

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

\(⇒ mg =\frac{GMm}{R^2}\)

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

EXPLANATION:

Using \(g =\frac{GM}{R^2}\)

Acceleration due to gravity on Earth, \(g_e =\frac{GM_e}{R_e^2}\)

Acceleration due to gravity on Saturn, \(g_s =\frac{GM_s}{R_s^2}\)

Given that:

Rs = 2Re and Ms = 2Me

\(\Rightarrow g_s =\frac{G(2M_e)}{(2R_e)^2}\)

Ratio = \(\frac{g_e}{g_s} =\frac{\frac{GM_e}{R_e^2}}{\frac{G2M_e}{(2R_e)^2}}\)

\(\Rightarrow \frac{g_e}{g_s} = \frac{(2R_e)^2}{2} \times \frac{1}{R_e^2}\)

\(\Rightarrow \frac{g_e}{g_s} = \frac{4}{2}=\frac{2}{1}\)

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