Gravitation

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 centers of the two masses.

\(\vec{F}(r)=f(r)\hat{r}\)

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

Gravitation:

Gravitation is the force of attraction between any two bodies in this universe, yes everything attracts everything else in the universe with a certain amount of force, but this force is too weak to be observed in most of the cases due to very large distance of separation. This was first observed by Sir Isaac Newton. He proposed the universal law of gravitation in the year 1680.

Gravitational Force Formula

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

\(F=G \frac{m_{1}m_{2}}{r^{2}}\)
Where,

  • F is the Gravitational force between two objects measured using N.
  • G is the universal gravity Constant with a value of 6.674 × 10-11 Nm2kg2.
  • m1 is the mass of one massive body measured using Kilogram.
  • m2 is the mass of another massive body measured using Kilogram.
  • r is the separation between them measured using kilometre.

Calculation of force of gravitation:

GravitationConsider two bodies A and B with mass m1 and m2 respectively and r is the separation between them. Then, according to Newton’s law of gravitation body:

  • A and B will attract each other with the same force (F1 = F2 = F) which is proportional to the product of their masses
  • Inversely proportional to the square of the distance between them.

It is mathematically expressed as-

\(F\propto m_{1}m_{2}\)And\(F\propto \frac{1}{r^{2}}\)
Thus, combining both we can write-

\(F\propto \frac{m_{1}m_{2}}{r^{2}}\)
They can be equated by removing the proportionality symbol and introducing proportionality constant in this expression.

\(\Rightarrow F=G \frac{m_{1}m_{2}}{r^{2}}\)
Where G is the universal gravitational constant. Which has the value of approximately 6.674 × 10-11 Nm2kg2. Its values remain same at any part of the universe. do not confuse this with ‘g’ which stands for acceleration due to gravity, which varies from one massive body to another. Refer to the link given below to learn the relation between G and g. now we know why gravity is so weak between distant bodies and strong on the surface of the earth, it is because the force of gravity reduces exponentially with increase in the distance between the bodies.

Newton’s Law of Gravitation

According to the Newton’s laws 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 \(F\propto ({M}_{1}{M}_{2})\ldots \ldots ..\left( 5 \right)\) and inversely proportional to the square of distance between their centre

\(\left( F\propto \frac{1}{{r}^{2}} \right)\ldots \ldots ..\left( 6 \right)\).

Combining equations 5 and 6 we get,

\(F\propto \frac{{M}_{1}{M}_{2}}{{r}^{2}}\).

\(F=G\frac{{M}_{1}{M}_{2}}{{r}^{2}}\ldots \ldots \ldots \left( 7 \right)\)

Comparing equations (7) and (1)

\(f\left( r \right)=~G\frac{{M}_{1}{M}_{2}}{{r}^{2}}\ldots \ldots \ldots \ldots \ldots \ldots .\left( 8 \right)\) \(f\left( r \right)is~a~variable,~conservative~and~Non~contact~force.\) As \(f\left( r \right)\) varies inversely as a square of ‘r’ it is also known as inverse square law force. The proportionality constant \(\left( G \right)\) in the above equation is known as gravitational constant. The value of gravitational constant is \(6.67\times {10}^{-11}\frac{N{m}^{2}}{k{g}^{2}}\) in SI units, \(6.67\times {10}^{-8}\frac{dynec{m}^{2}}{{g}^{2}}\) in CGS units. Dimension formula of G is\(\left[ {M}^{-1}{L}^{3}{T}^{-2} \right]\).

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.

In 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, force on due to .

\({{F}_{12}}=~-\frac{G{{m}_{1}}{{m}_{2}}}{{{\left| {{r}_{12}} \right|}^{2}}}{{\hat{r}}_{12}}\) \({{F}_{12}}=~-\frac{G{{m}_{1}}{{m}_{2}}}{{{\left| {{r}_{1}}-{{r}_{2}} \right|}^{3}}}\left( {{r}_{1}}-{{r}_{2}} \right)\ldots \ldots \ldots \ldots \ldots ..\left( 9 \right)\)

Where, \({{\hat{r}}_{12}}\) is a unit vector pointing from \({{m}_{2}}\) to \({{m}_{1}}\) .

The negative sign in Eq.6 indicates the direction of force \({{F}_{12}}\) is opposite to that of \({{\hat{r}}_{12}}\).

Similarly, force on \({{m}_{2}}\) due to \({{m}_{1}}\) \({{F}_{21}}=~-\frac{G{{m}_{1}}{{m}_{2}}}{{{\left| {{r}_{21}} \right|}^{2}}}{{\hat{r}}_{12}}\ldots \ldots \ldots \ldots \left( 10 \right)\)

From equations (9) and (10) we get

\({{F}_{12}}=~-{{F}_{21}}\)

As \({{F}_{12}}\) and \({{F}_{21}}\) are directed towards the centres of the two particles, so gravitational force is conservative in nature.

Principle of Superposition of Gravitational Forces

Newton’s law of gravitation answers only the interaction between two particles, if system contains n particles there are \(\frac{n\left( n-1 \right)}{2}\) such interactions, according to principle of superposition if each of these interactions acts independently and uninfluenced by the other bodies, the resultant can be expressed as the vector summation of these interactions

\(F={{F}_{12}}+{{F}_{13}}+{{F}_{14}}\ldots ..+{{F}_{1n}}\ldots \ldots \ldots \ldots \ldots ..\left( 10 \right)\)

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\(\omega\). The centripetal force acting on the test mass for its circular motion is

\(F=mr{{\omega }^{2}}=mr{{\left( \frac{2\pi }{T} \right)}^{2}}\)

We know from kepler’s 3rd law

\({{T}^{2}}\propto {{r}^{3}}\)

Using this in force equation we get,

\(F=\frac{4{{\pi }^{2}}mr}{K{{r}^{3}}}\) \(\left( K=\frac{4{{\pi }^{2}}}{GM} \right)\)

⇒ \(F=\frac{GMm}{{{r}^{2}}}\) Which is Newton’s law of gravitation.

Also Read:


Practise This Question

Three particles each of mass m, are situated at the vertices of an equilateral triangle of side length a. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original separation a. Find the initial velocity that should be given to each particle and also the time required for the circular motion