Test: Universal Law of Gravitation - JEE MCQ

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 

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

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.

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

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.

The value of G is universally constant = 6.67 × 10^-8 dyne.cm²/g²

We know:
1 dyne = 10^-5 N
1 cm = 10^-2 m
1 g = 10^-3 kg

⇒ 6.67 × 10^-8 × ( 10^-5) .(10^-2)²/(10^-3)² Nm²/kg² = 6.67 × 10^-8 { 10^-5× 10^-4 /10^-6} Nm²/kg²
= 6.67 × 10^-8 × 10^-3 N.m²/kg²= 6.67× 10^-11 Nm²/kg²

• Gravitational force strictly follows Newton’s Third Law of motion i.e. for every action force there is an equal and opposite reaction force.
• Gravitational forces add up like vectors i.e. they obey the principle of superposition.

Acceleration (a) = 9.8 m/s²

Weight of the body = 9.8 N

Since, Weight of the body = mg

⇒ Mass = Weight/g = 9.8/9.8 = 1 kg

We know that,

F = m × a

⇒ F = 1 kg × 9.8 m/s²

⇒ F = 9.8 Newton

Acceleration due to gravity at a height “h” is given by

g’ = g (R/R+h)²

Here,

g is the acceleration due to gravity on the surface

R is the radius of the earth

As g’ is given as g/9, we get

g/9 = g(R/R+h)²

⅓ = R/(R+h)

h=2R

The gravitational force of the earth acts on every object having mass. When a ball is thrown upwards the gravitational force of the earth acts against the velocity of the object and pulls the ball downwards, i.e, towards itself.

Thus, gravity pulls a ball back to earth.

The universal law of gravitation:

• Sir Isaac Newton put forward the universal law of gravitation in 1687 and used it to explain the observed motions of the planets and moons.
• The law states that every particle attracts every other particle in the universe with force directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
  
• Formula, Gravitational force,  where, F is the gravitational force between bodies, m1, and m2 are the masses of the bodies, d is the distance between the centers of two bodies, and G is the universal gravitational constant.

• Here, universal gravitational constant, G = 6.67 × 10^-11 Nm/kg²

• The Universal Gravitational Law can explain almost anything, right from how an apple falls from a tree to why the moon revolves around the Earth.

Newton's third law of motion

• It states that for every action, there is an equal and opposite reaction.
• Action and reaction always act on two different bodies.

Explanation:

By Newton's Third Law and Newton's Law of Universal gravitation, the gravitational force the Earth exerts on the Moon is exactly the same as the force the Moon exerts on the Earth. 
F_(Earth) = F_(Moon)
Therefore, the moon attracts the earth with a gravitational force of 10²⁰N.

Concept:

Newton's law of gravitation: The force of attraction between any objects in the universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

• The force acts along the line joining the two bodies.
• The gravitational force is a central force that is It acts along the line joining the centers of two bodies.
• It is a conservative force. This means that the work done by the gravitational force in displacing a body from one point to another is only dependent on the initial and final positions of the body and is independent of the path followed.

• The moon revolves around the earth in a circular orbit (not perfectly circular). Sun exerts a gravitational force on both, the earth and moon. The major force acting on the moon is due to the gravitational force of attraction of the sun and earth and the moon is not always on the line joining the sun and earth. 
• Two forces acting on the moon have different lines of action or the forces are not central, so it's motion will not be strictly elliptical.

Hence, option (2) is the correct answer.

Concept 

• : Newton's law of gravitation states 
• "Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres."

Explanation 

Newton's law of gravitation can be represented mathematically as: F = G x (m₁ x m₂/ r²)

Where:

• F is the force of attraction between the two bodies.
• m1 and m2 are the masses of the two bodies.
• r is the distance between the centres of the two bodies.
• G is the gravitational constant, a proportional constant that in the SI system is approximately 6.674 x10^-11N(m/kg)².
• So, if you have two bodies with masses m1 and m2, and they are separated by a distance r, then the force of gravity between them is directly proportional to the product of their masses and inversely proportional to the square of the distance between them,
• Real-world applications of this law include everything from explaining the motion of planets in our solar system to the rolling of a ball down a hill. It's one of the cornerstones of classical physics and is essential in understanding the larger structure of the universe.

Concept:

Newton's law of gravitation: The force of attraction between any objects in the universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

• The force acts along the line joining the two bodies.
• The gravitational force is a central force that is It acts along the line joining the centers of two bodies.
• It is a conservative force. This means that the work done by the gravitational force in displacing a body from one point to another is only dependent on the initial and final positions of the body and is independent of the path followed.

Explanation:

• The gravitational force on mercury due to the sun is very large than that due to the earth.
• As observed from the earth, the sun appears to move around the earth in a circular orbit though, in reality, the earth moves around the sun due to gravitational force between them
• All planets move around the sun due to the huge gravitational force of the sun acting on them.
• Since the gravitational force on mercury due to the earth is smaller as compared to that on it due to the sun therefore it revolves around the sun and not around the earth.

Hence, option (3) is the correct answer.

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:

 

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 (F) acting on an object of mass m placed on the surface of Earth is 
 

Where R is the radius of the earth. 

CALCULATION:

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, 

Hence the gravitational force, F ∝ r