Correct Answer - Option 1 :
\(\frac{g_1}{g_2} =\frac{R_1}{R_2}\)
The correct answer is option 1) i.e. \(\frac{g_1}{g_2} =\frac{R_1}{R_2}\)
CONCEPT:
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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 \(g =\frac{GM}{R^2}\),
For planet 1: \(g_1 =\frac{GM_1}{R_1^2}\)
For planet 2: \(g_2 =\frac{GM_2}{R_2^2}\)
Given that both the planets have the same average density. Let the average density of both the planets be ρ.
⇒ Mass = ρ × volume
\(⇒ M_1 = ρ × \frac{4}{3}\pi R_1^3\)
\(⇒ M_2 = ρ × \frac{4}{3}\pi R_2^3\)
Ratio = \(\frac{g_1}{g_2} =\frac{\frac{GM_1}{R_1^2}}{\frac{GM_2}{R_2^2}}\)
\(\Rightarrow \frac{g_1}{g_2} ={\frac{M_1}{R_1^2}}\times {\frac{R_2^2}{M_2}}\)
\(\Rightarrow \frac{g_1}{g_2} ={\frac{(ρ × \frac{4}{3}\pi R_1^3)}{R_1^2}}\times {\frac{R_2^2}{(ρ × \frac{4}{3}\pi R_2^3)}}\)
\(\Rightarrow \frac{g_1}{g_2} =\frac{R_1}{R_2}\)
