Deriving Newton's Universal Law of Gravitation: A Comprehensive Guide

Newton's universal law of gravitation, also known as Newton's law of gravitation, posits 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. This fundamental principle can be mathematically expressed as:

F Gm?m?/r2

where F is the gravitational force, m? and m? are the masses of the objects, r is the distance between them, and G is the gravitational constant, approximately equal to 6.673 × 10-11 N·(m/kg)2.

Historical Context

The derivation of Newton's law of gravitation, though rooted in empirical observations, involves a profound understanding of physics and mathematics. Let us explore how Newton built upon the work of Kepler to formulate his law.

Observations of Celestial Bodies

Johannes Kepler, a key influence, formulated the laws of planetary motion based on extensive observational data. Kepler's Third Law, which states that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (a) of its orbit, provides a crucial starting point:

( T^2 k a^3 )

Centripetal Force

Newton recognized that for an object moving in a circular orbit, there must be a centripetal force acting towards the center. This force F can be expressed as:

( F frac{mv^2}{r} )

where m is the mass of the orbiting body, v is its orbital velocity, and r is the radius of the orbit.

Relating Orbital Velocity and Period

The orbital velocity v can be related to the period T and radius r:

( v frac{2pi r}{T} )

Substituting this into the centripetal force equation gives:

( F frac{m left( frac{2pi r}{T} right)^2}{r} frac{4pi^2 m r}{T^2} )

Derivation of Gravitational Force

Using Kepler's Third Law

From Kepler's Third Law, we have:

( T^2 k a^3 )

For circular orbits, a r, so:

( T^2 k r^3 )

This implies:

( frac{T^2}{r^3} k )

Thus, we can express T2 in terms of r:

( T^2 k r^3 )

Equating Forces

By equating the centripetal force expression and the gravitational force expression, we assume that the force exerted by the mass M of the central body on mass m of the orbiting body is given by:

( F frac{GMm}{r^2} )

Setting the Forces Equal

Now, we set the centripetal force equal to the gravitational force:

( frac{4pi^2 m r}{T^2} frac{GMm}{r^2} )

Cancelling m from both sides (assuming m ≠ 0), we get:

( frac{4pi^2 r^3}{T^2} GM )

Rearranging gives:

( T^2 frac{4pi^2 r^3}{GM} )

Final Formulation of the Law

The above equation suggests that the gravitational force F between two masses M and m separated by a distance r can be expressed as:

( F frac{GMm}{r^2} )

This is Newton's law of universal gravitation, which states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Conclusion

Newton's law of universal gravitation was a significant breakthrough in understanding the forces that govern the motion of celestial bodies and laid the groundwork for classical mechanics. The law connects terrestrial and celestial mechanics through a single principle, demonstrating the universality of gravitational interaction.