Gravitation

Gravitation is an inherent property of all matter. Two bodies attract each other by virtue of their mass. This force between two bodies of any size (an atom or a galaxy) signifies existence of matter and is known as gravitational force.

Gravitational force is weakest of four fundamental forces. It is, therefore, experienced only when at least one of two bodies has considerable mass. This presents difficulties in setting up illustrations with terrestrial objects. On the other hand, gravitation is the force that sets up our universe and governs motions of all celestial bodies. Orbital motions of satellites – both natural and artificial – are governed by gravitational force.

Newton derived a law to quantify gravitational force between two “particles”. The famous incidence of an apple falling from a tree stimulated Newton’s mind to analyze observations and carry out series of calculations that finally led him to propose universal law of gravitation. A possible sequence of reasoning, leading to the postulation is given here :

1: The same force of attraction works between “Earth (E) and an apple (A)” and between “Earth (E) and Moon (M)”.

2: From the analysis of data available at that time, he observed that the ratio of forces of attraction for the above two pairs is equal to the ratio of square of distance involved as :

$$\frac{F_{ME}}{F_{AE}}=\frac{r_{{AE}^2}}{r_{{ME}^2}}$$

3: From above relation, Newton concluded that the force of attraction between any pair of two bodies is inversely proportional to the square of linear distance between them.

$$F\propto \frac{1}{r^2}$$

4: From second law of motion, force is proportional to mass of the body being subjected to gravitational force. From third law of motion, forces exist in equal and opposite pair. Hence, gravitational force is also proportional to the mass of other body. Newton concluded that force of gravitation is proportional to the product of mass of two bodies.

$$F\propto m_1{m}_2$$

5: Combining two “proportional” equations and introducing a constant of proportionality,"G", Newton proposed the gravitational law as :

$$\Rightarrow F=\frac{G{m}_1{m}_2}{r^2}$$

In order to emphasize the universal character of gravitational force, the constant “G” is known as “Universal gravitational constant”. Its value is :

$$G=6.67X{10}^{-11}N-m^2/k{g}^2$$

This law was formulated based on the observations on real bodies - small (apple) and big (Earth and moon). However, Newton’s law of gravitation is stated strictly for two particles. The force pair acts between two particles, along the line joining their positions as shown in the figure :

Extension of this law to real bodies like Earth and Moon pair can be understood as bodies are separated by large distance (about $0.4X{10}^6$ km), compared to dimension of bodies (in thousands km). Two bodies, therefore, can be treated as particles.

On the other hand, Earth can not be treated as particle for “Earth and Apple” pair, based on the reasoning of large distance. Apple is right on the surface of Earth. Newton proved a theorem that every spherical shell (hollow sphere) behaves like a particle for a particle external to it. Extending this theorem, Earth, being continuous composition of infinite numbers of shells of different radii, behaves - as a whole - like a particle for external object like an apple.