¶Newton's Law of Universal Gravitation

The law of universal gravitation, formulated by Sir Isaac Newton in 1687, states that every point mass in the universe 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 them.

This relationship is expressed mathematically as: $F = G \frac{m_1 m_2}{r^2}$

¶Key Components

• $F$ (Gravitational Force): The magnitude of the attractive force between the two bodies, measured in Newtons (N). This force is mutual; body 1 attracts body 2 with the exact same force that body 2 attracts body 1 (Newton's Third Law).
• $G$ (Gravitational Constant): An empirical physical constant. Its value is approximately $6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$. This extremely small value explains why gravitational attraction is only noticeable when at least one mass is extraordinarily large (like a planet).
• $m_1, m_2$ (Masses): The masses of the two interacting objects, measured in kilograms (kg).
• $r$ (Distance): The straight-line distance between the centres of mass of the two objects, measured in metres (m).

¶The Inverse-Square Law

The most defining characteristic of gravity is its $1/r^2$ decay. If you double the distance between two objects ($2r$), the gravitational force decreases to $(1/2)^2$, or one-quarter ($1/4$) of its original strength. If you triple the distance, the force becomes one-ninth ($1/9$).

¶Vector Representation

While the formula above calculates the magnitude of the force, force is fundamentally a vector quantity. The complete vector form of Newton's law is: $\vec{F}_{12} = -G \frac{m_1 m_2}{|\vec{r}_{12}|^2} \hat{r}_{12}$

Where $\vec{F}_{12}$ is the force exerted on mass 1 by mass 2, $\vec{r}_{12}$ is the distance vector from mass 1 to mass 2, and $\hat{r}_{12}$ is the unit vector pointing from mass 1 to mass 2. The negative sign explicitly indicates that the force is attractive (pulling mass 1 towards mass 2).

¶Limitations

Newton's law is an excellent approximation for most engineering and orbital mechanics applications. However, it is fundamentally superseded by Albert Einstein's Theory of General Relativity (1915), which describes gravity not as a force, but as the curvature of spacetime caused by mass and energy. Newton's formulation breaks down under extreme conditions, such as near black holes or when describing the precise orbit of Mercury.

¶References

• The Feynman Lectures on Physics, Vol. I, Ch. 7: The Theory of Gravitation (Richard P. Feynman)
• Philosophiæ Naturalis Principia Mathematica (Isaac Newton, 1687)
• NASA JPL: Basics of Space Flight - Gravity & Mechanics