# Gravitation - Newton's Law of Gravitation

Gravitation is the force of attraction between any two bodies. All the objects in the universe attract each other with a certain amount of force, but in most of the cases, the force is too weak to be observed due to the very large distance of separation. This force of attraction was first observed by Sir Isaac Newton and was presented as Newton’s law of gravitation in the year 1680.

## What is Gravitational Force?

Each body in this universe attracts other bodies towards itself with a force known as Gravitational Force, thus gravitation is a study of the interaction between two masses. Out of the two masses, the heavier one is called source mass and the lighter one is called test mass.

Gravitational force is a central force which depends only on the position of test mass from the source mass and always acts along the line joining the centres of the two masses.

$\vec{F}$ (r) = f(r) r^

The core problem of gravitation has always been in understanding the interaction between the two masses and the relativistic effects associated with it.

⇒ Check:

### History of Gravitational Theory:

Ptolemy has proposed geocentric model which failed in understanding planetary motions led to the development of heliocentric model by Nicholas Copernicus whose idea is based on rotation of a test mass around the source mass in circular orbits, although the model correctly predicts the position of planets and their motions but has failed in explaining many aspects like the occurrence of seasons which led the construction of a model based on Kepler’s laws of planetary motion.

## Newton’s Law of Gravitation

According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is,

• Directly proportional to the product of their masses i.e. F ∝ (M1M2) . . . . (1)
• Inversely proportional to the square of the distance between their centre i.e. (F ∝ 1/r2) . . . . (2)

On combining equations (1) and (2) we get,

F ∝ M1M2/r2

F = G × [M1M2]/r2 . . . . (7)

Or, f(r) = GM1M2/r[f(r)is a variable, Non-contact, and conservative force]

As f(r) varies inversely as a square of ‘r’ it is also known as inverse square law force. The proportionality constant (G) in the above equation is known as gravitational constant.

The dimension formula of G is[M-1L3T-2]. Also, the value of the gravitational constant,

• In SI units: 6.67 × 10-11 Nmkg-2,
• In CGS units: 6.67×10-8 dyne cmg-2

### Vector Form of Newton’s Law of Gravitation

The vector form of Newton’s law of gravitation signifies that the gravitational forces acting between the two particles form action-reaction pair.

Newton’s Law of Gravitation in Vector Form

From the above figure, it can be seen that the two particles of masses and are placed at a distance, therefore according to Newton’s law of gravitation, the force on m1 due to mi.e. F12 is given by,

F12 = [-G m1m2]/|r12|2 r^12

F12 = [-Gm1m2/|r1 – r2|3] (r1 – r2) . . . . . . . . . (9)

Where r^12 is a unit vector pointing from m2 to m1.

The negative sign in Equation (9) indicates that the direction of force F12 is opposite to that of r^12.

Similarly, the force on m2 due to m1 i.e. F21 is given by,

F21 = -Gm1m2/|r21|2 r^12 . . . . . . . . . . (10)

From equations (9) and (10) we get,

F12 = – F21

As F12 and F21 are directed towards the centres of the two particles, so the gravitational force is conservative in nature.

## Gravitational Force Formula

Gravitational force is explained using Newton’s law of gravitation. Gravitational force decides how much we weigh and how far a ball travels when thrown before it lands on the ground.

According to Newton’s law of gravitation, every particle 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 them.

Mathematically it can be represented as,

F = Gm1m2/r2
Where,

• F is the Gravitational force between two objects measured in Newton (N).
• G is the Universal Gravitational Constant with a value of 6.674 × 10-11 Nm2kg-2.
• mis the mass of one massive body measured in kg.
• m2 is the mass of another massive body measured in kg.
• r is the separation between them measured in kilometre (Km).

### Principle of Superposition of Gravitational Forces

Newton’s law of gravitation answers only the interaction between two particles if the system contains ‘n’ particles there are n(n – 1)/2 such interactions.

According to the principle of superposition, if each of these interactions acts independently and uninfluenced by the other bodies, the results can be expressed as the vector summation of these interactions;

F = F12 + F13 + F14 . . . . . . + F1n.

It states that:

“The resultant gravitational force F acting on a particle due to the number of point masses is equal to the vector sum of forces exerted by the individual masses on the given particle”.

## Derivation of Newton’s law of Gravitation from Kepler’s law

Suppose a test mass is revolving around a source mass in a nearly circular orbit of radius ‘r’, with a constant angular speed (ω). The centripetal force acting on the test mass for its circular motion is,

F = mrω2 = mr × (2π/T)2

According to Kepler’s 3rd law, T2 ∝ r3

Using this in force equation we get,

F = 4π2mr/Kr3 [Where, K = 4π2/GM]

⇒ F = GMm/r2, which is the equation of Newton’s law of gravitation.