# Newton's Laws of Motion/Third Law

## Contents

## Physical Law

**Newton's third law of motion** is one of three physical laws that forms the basis for classical mechanics.

### Statement of Law

- To every force there is always an equal and opposite force. That is, the forces of two bodies on each other are always equal and are directed in opposite directions.

As Isaac Newton himself put it:

*Whenever one body exerts a force on a second body, the second body exerts a force on the first body. These forces are equal in magnitude and opposite in direction.*

#### Proof of third law in the special case of one-dimensional motion

It is actually possible to *almost* prove^{[1]} the third law, provided we restrict the discussion to one dimension, and provided we accept two axioms:

**Axiom 1:** Newtonian mechanics applies not only to particles, but to bodies made up of particles in such a way that can be easily understood for a nearly infinite number of connected particles.

This is where the *almost* comes in: If Newtonian physics is to be useful for entire planets, the forces on these planets need to be understood without understanding all the internal forces between the many, many particles involved. For that reason, the next axiom is *almost* a corollary to the first axiom:

**Axiom 2:** When Newton's second law is applied to collections of particles, only external forces need be considered.

Without this second axiom, Newtonian physics would be so complicated that it is virtually useless; planets would go where their internal forces send them.

Consider two bodies $m_1$ and $m_2$ that are held at a fixed distance from each other.

It is assumed that they exert "internal" forces on each other, but we do not assume that these forces are equal and opposite.

Let $F_{12}$ be the internal force on $m_1$ by $m_2$.

Let $F_{21}$ be the internal force on $m_2$ by $m_1$.

Let a force $F_{\text{ext} }$ act only on $m_2$.

Since the bodies $m_1$ and $m_2$ are held at a fixed distance from each other, they both have the same velocity $v$ and the same acceleration $a$.

Note that the following three equations hold:

- $(1): \quad F_{\text{ext} } = \left({m_1 + m_2}\right) a$

- $(2): \quad F_{\text{ext} } + F_{21} = m_2 a$

- $(3): \quad F_{12} = m_1 a$

Adding equations $(2)$ and $(3)$ and comparing with equation $(1)$ yields Newton's third law:

- $F_{12} + F_{21} = 0$

$\blacksquare$

#### Footnotes and References

- ↑ This was submitted to Physics Teacher a few years ago, but was rejected. It is virtually certain that this idea has been put forth somewhere, and a proper reference to it would be appreciated.

## Also known as

This law is also referred to as the **law of action and reaction**.

It is also often referred to as just **Newton's third law**.

## Also see

## Source of Name

This entry was named for Isaac Newton.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1966: Isaac Asimov:
*Understanding Physics*: $\text{I}$: Chapter $3$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Newton's laws of motion**