# Newton's 3rd law

#### Prerequisites

The last of Newton's laws deals with the conceptual issue of reciprocity. It's pretty clear that when two objects interact they do things to each other. So whenever you have an $\overrightarrow{F}_{B \rightarrow A}$ (object B exerting a force on object A), then there is also an $\overrightarrow{F}_{A \rightarrow B}$ (object A exerting a force on object B. The question N3 asks is: How do they compare? We can get the answer two ways: by looking at the internal consistency of the theory (!) and by doing an experiment. If our theoretical framework is any good, these two approaches better agree!

## Internal consistency

One of the requirements of our theoretical framework is that it should be internally consistent; that is, if we can look at something two ways, we ought to get the same answer. One of the ideas in the Newtonian framework is that we can choose anything as our object — including a part of an object or a collection of objects. Let's consider the example of the hand pushing on the box that we discussed in Free-body diagrams, but let's now push hard enough to get the two boxes accelerating. And let's simplify the situation by assuming that we have oiled the table so we can ignore the friction. Then our figure and free body diagrams look like this.

Both of the boxes will accelerate together. We can see that object B has an unbalanced horizontal force acting on it, so since the two boxes have to move together (and therefore have the same accelerations), box A also has to have an unbalanced horizontal force acting on it. That tells us that $N_{hand \rightarrow A}$ has to be bigger than $N_{B \rightarrow A}$. The up-down forces balance on both boxes.

But how does $N_{A \rightarrow B}$ compare to $N_{B \rightarrow A}$? To see that, consider the pair of boxes as a single object as in the figure below.

The free-body diagram for the combined object is

## Inferring Newton's 3^{rd} law

Let's just think about the horizontal components since that's where the AB objects interact with each other. We can make a number of observations.

- All the objects, A, B, and AB, have to have the same acceleration since the boxes are moving together and it shouldn't matter how we describe it — as one box or two.
- The force of the hand on the combined object is that same as that on box A (since the hand doesn't know whether we are describing the boxes as separate or combined).

This implies that if we write the horizontal part of the N2 equations for AB, A, and B respectively, we get:

$$(m_A + m_B) \overrightarrow{a} = \overrightarrow{N}_{hand \rightarrow A}$$

$$m_A \overrightarrow{a} = \overrightarrow{N}_{hand \rightarrow A} + \overrightarrow{N}_{B \rightarrow A}$$

$$m_B \overrightarrow{a} = \overrightarrow{N}_{A \rightarrow B}$$

Adding the last two equations together gives

$$(m_A + m_B) \overrightarrow{a} = \overrightarrow{N}_{hand \rightarrow A} + \overrightarrow{N}_{A \rightarrow B} + \overrightarrow{N}_{B \rightarrow A}$$

Now compare the first line (the N2 equation for AB) and the last line (the sum of the N2 equations for A and for B). If our theory is to give the same result no matter how we describe it, we have to have these equations be the same. This means that we have to have

$$\overrightarrow{N}_{A \rightarrow B} + \overrightarrow{N}_{B \rightarrow A} = 0$$

or

$$ \overrightarrow{N}_{A \rightarrow B} = - \overrightarrow{N}_{B \rightarrow A}$$

Since we could do this for any forces whenever two objects are interacting, we must demand for consistency (our ability to treat objects as pieces or combined) the following.

**Newton's 3rd law**: Whenever any two objects interact with a particular type of force, the forces of that type that they exert on each other are equal and opposite:

$$ \overrightarrow{F}^{type}_{A \rightarrow B} = - \overrightarrow{F}^{type}_{B \rightarrow A}$$

Careful! This is quite tricky -- as well as unexpected. The superscript "type" on this equation means that the SAME type of force has to be on both sides. And the "cause-feel" labels have to be flipped. There are other situations where forces are equal and opposite -- for example, the vertical forces for the box sitting on a table. But this is **NOT** an example of N3 because the normal force and the weight are DIFFERENT TYPES of forces, and they both act on the same object.

## Doing the experiment

One way to check this out is to measure the forces with Force Probes. A force probe looks like the figure at the right. It plugs into a USB port in the computer and measures the force that is being exerted on it. (It basically has a spring inside with some electronics to detect how much it is bent.) An experiment to test N3 would use two force probes and look something like this.

We can do the experiment with different carts, banging them together, moving both together, moving one fast and the other away, whatever! It doesn't matter. Strikingly enough you get graphs that look equal and opposite no matter what you do!

Joe Redish 9/22/11

#### Follow-ons

Last Modified: July 12, 2019