Test: Kepler’s laws & Universal law of gravitation (6 August) - JEE MCQ

We know, the gravitational force 
,
This is valid when R is the centre to centre distance between the two masses M and m.
At a distance r < R, the mass of Earth cannot be taken as M.

The new mass (M'):
The mass of earth will be contained for a radius r. Let us assume that the mean density of the earth is ρ and that earth is a sphere with a radius equal to the distance r. We know, mass = density × volume

Therefore the gravitational force, F = 

Hence the gravitational force, F ∝ r

We know that Gravitational force = 

and,

The SI unit of the universal gravitational constant is N m²/kg²
The SI unit of the acceleration due to gravity is the same as that of acceleration.
Hence, the SI unit of the acceleration due to gravity is m/s².
So, the SI unit of 

Explanation for the Gravitational Force Between the Earth and the Moon

• According to Newton's Third Law of Motion, every action has an equal and opposite reaction. This law applies to the gravitational forces between two bodies as well.
• The gravitational force with which the Earth attracts the Moon is equal to the gravitational force with which the Moon attracts the Earth. This is because the forces are action and reaction pairs, as described by Newton's Third Law.
• Therefore, if the Earth attracts the Moon with a gravitational force of 1020 N, then the Moon attracts the Earth with the same gravitational force of 1020 N.

Hence, the correct answer is b. 10²⁰ N.

Gravitational force between two objects is mathematically represented as:
F = G * (m1*m2) / r²
Applying the Law of Gravitation to the Given Problem
Given that the initial force F is due to two objects each of mass m separated by a distance r.
Substituting these into the formula gives:
F = G * (m*m) / r²
The problem then states that the mass of each of the objects is doubled (to 2m) and the distance between them is also doubled (to 2r).
The new force F' using these values in the formula would be: F' = G * (2m*2m) / (2r)²

Simplifying this gives:
F' = 4G * (m*m) / 4*r²
This simplifies further to:
F' = G * (m*m) / r²
Comparing this with the original equation for F, we find that:
F' = F

Thus, the force remains the same (F) even when the mass of each of the objects is doubled and the distance between them is also doubled. Hence, the correct answer is A: F.

Kepler's third law relates the period (T) of an orbiting body to its semi-major axis (a). The potential energy is inversely proportional to T^2/3, as per Kepler's third law.

We know that the force of gravitation is inversely proportional to square of the distance between the two bodies,

i.e. F∝ r^-2

Hence, when the distance between them will be doubled, the force will be reduced by 4 times

So, the ratio will be 4:1