# Newton's Law of Universal Gravitation

## Physical Law

Let $a$ and $b$ be particles with mass $m_a$ and $m_b$ respectively.

Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by **Newton's Law of Universal Gravitation**:

- $\mathbf F_{a b} \propto \dfrac {m_a m_b {\mathbf r_{b a} } } {r^3}$

where:

- $\mathbf F_{a b}$ is the force exerted on $b$ by the gravitational force on $a$
- $\mathbf r_{b a}$ is the displacement vector from $b$ to $a$
- $r$ is the distance between $a$ and $b$.

Thus it is seen that the direction of $\mathbf F_{a b}$ is specifically **towards** $a$.

By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction, that is, towards $b$.

### Universal Gravitational Constant

The **universal gravitational constant** is the physical constant which is the constant of proportion in Newton's Law of Universal Gravitation.

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:

- $\mathbf F_{a b} = \dfrac {G m_a m_b \mathbf r_{b a} } {r^3}$

## Also presented as

Newton's Law of Universal Gravitation can also be presented in the form:

- $\mathbf F_{a b} \propto \dfrac {m_a m_b \hat {\mathbf r}_{b a} } {r^2}$

where $\hat {\mathbf r}_{a b}$ is the unit vector in the direction from $b$ to $a$.

Some presentations do not consider the vector aspects, merely expressing the relationship in terms of scalars:

- $F = \dfrac {k m_a m_b} {r^2}$

where the constant is variously specified.

## Also known as

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

**Newton's Law of Gravitation**- the
**Inverse Square Law of Gravitation** **Newton's Inverse Square Law of Universal Gravitation**

and so on.

Some sources refer to it as **Newton's Law** but there are a number of these.

## Also see

## 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.

The children's version of the tale has it that the apple hit him on the head.

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*... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: $(11)$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
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*Electromagnetism*(2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation - 1998: David Nelson:
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*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Newton - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**inverse square law of gravitation**