# Newton's Law of Universal Gravitation

## Contents

## Physical Law

Every particle in the universe attracts every other particle with a force $F$ which is inversely proportional to the square of the distance between them:

- $\mathbf F \propto \dfrac {m_1 m_2} {r^2}$

The direction of the force on either particle is the same as the direction of the displacement vector to the other particle.

Using vector notation:

- $\mathbf F \propto \dfrac {m_1 m_2} {r^2} \hat {\mathbf r}$

where $\hat {\mathbf r}$ is the unit vector from one particle to the other.

### Gravitational Constant

The constant of proportion $G$ in Newton's Law of Universal Gravitation is a physical constant.

Its value in SI units is referred to as $G$ and is approximately equal to $6.674 \times 10^{-11} \, \mathrm N \, \mathrm m^2 \, \mathrm{kg}^{-2}$.

Thus the equation becomes:

- $\mathbf F = \dfrac {G m_1 m_2 \mathbf r} {r^3}$

## Also known as

**Newton's Law of Universal Gravitation** is also known as just **Newton's Law of Gravitation**.

## Source of Name

This entry was named for Isaac Newton.

## Historical Note

The popular tale has it that Newton had the idea while lying in the garden at his home in Woolsthorpe between the years of $1665$ and $1667$ and watching an apple fall from a tree.

This tale has the feel of an off-the-cuff comment that Newton may have made during the course of an interview with a journalist of the time, or whatever the equivalent may have been.

However, the inverse square law was in fact conjectured by Edmund Halley in $1684$, but he was unable to do anything to prove his conjecture. He discussed this with Christopher Wren and Robert Hooke, who claimed he had a proof of it. However, Halley disbelieved him.

Some months later, Halley had the chance to ask Newton what law of attraction would cause the planets to move in an elliptical orbit. Newton answered immediately that it would be an inverse square law, and claimed to have already calculated it.

Some sources state that the question was posed the other way round: that Halley asked how the planets would behave under a central force obeying the inverse square law.

The mathematical work that Newton he had performed to prove had supposedly been deduced from Kepler's Laws of Planetary Motion.

Having been thus spurred on by Halley, Newton went ahead to write and publish his *Philosophiae Naturalis Principia Mathematica*.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1966: Isaac Asimov:
*Understanding Physics*: $\text{I}$: Chapter $4$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: $(11)$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Newton