# Reading the content in Newton's 3rd law

#### Prerequisites

Newton's 3^{rd} law confuses many students. In part, because it contradicts many of our natural "more is more" intuitions. ("Surely a bigger object interacting with a smaller one must exert more force." Nope.) But in my opinion the biggest problem is the absolutely deadly phrase often used in high school physics texts: "To every interaction, there is an equal and opposite reaction." I've crossed this out since it is so easy to remember and so easy to misinterpret. There are lots of actions that cause reactions, and sometimes they may be equal and opposite, but often they are not. (If I punch a boxing champ, he may punch me back but my action is unlikely to be equal to his reaction.)

Newton's third law is much more specific -- and much more precise than the silly "action-reaction" rule. The best way to keep it straight is to start with the equation that expresses N3 and learn how to interpret it.

What N3 is about is the interaction between a single pair of objects. A nice way to picture it is in terms of our System Schemas. When two objects interact they do something to each other that could potentially change each of the object's velocities.

When a bowling ball hits a 10-pin trying to resolve a 7-10 split, the pin and the ball *interact* through making contact. If we want to focus on one of the object's in the interaction -- say the bowling pin -- then the interaction expresses itself as a force on that object: a contact force caused by the bowling ball and felt by the pin. If we want to focus on the other object in the interaction -- say the bowling ball -- then the interaction expresses itself as a force on *that* object: a contact force caused by the pin and felt by the bowling ball. These forces are the ways that the two objects see the interaction differently. And of course by N0, it's the forces each object *feels* that appears in N2 and tells the object how to change its velocity. (Image source: http://www.slate.com/articles/sports/sports_nut/2015/02/hardest_shot_in_bowling_it_s_not_the_7_10_split_it_s_the_greek_church.html)

Newton's 3rd law expresses how these two parts of the *same* interaction resolve themselves as forces:

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

Although this equation looks fairly simple on the surface, it contains 7 distinct symbols carrying lots of information about what the law is saying. Let's unpack what each symbol is telling us.

1. *F* -- The two F's on either side of the equation tell us this law is about forces. To make sense of them, we need to understand what a force is. As in our discussion of N2, this means we need to understand that it is about the interactions with other objects that cause objects to change their motion (accelerate).

2. **→** --The little arrows on top of the acceleration and net force remind us that Newton's third law is a vector equation (as is the second). It tells us something about both the magnitude and direction of the two forces we are considering.

3. *type* -- What matters here is that the SAME superscript "type" appears on both of the forces. This tells us that Newton's 3rd law is about the same type of force. Better, it tells us that the two forces we are equating are part of the *same* interaction between the two objects (the opposite ends of a single connection in our System Schema diagram). This means that the *same* superscript must be on both forces. So both forces have to be a normal force, or a friction force, or a weight. N3 is NEVER about the equality of two different kinds of forces. Such forces might in fact be equal in some problems, but N3 is never the reason!

4. **A→B and B****→A -- **The key here is that we have the SAME pair of objects but in opposite directions. So one of the forces in N3 is caused by A and felt by B, and the other is caused by B and felt by A. This is very natural if you are thinking of N3 as telling you something about two ends of a single interaction as shown in the System Schema diagram above. If you have an equation showing two forces are equal, and those forces have 3 different objects in their "causing/feeling" label you can be sure that the reason they are equal is NOT N3 (even if their "types

are the same).

5.** = and -** -- The equal and minus sign for an equation setting two vectors equal tells us the magnitudes of the two vectors are the same and the point in opposite directions.

Although it is easy to misuse N3, if you start from the equation and check items 3 and 4 above, you should have no trouble getting it right every time!

{* Technical note*: N3 is true as long as we are only talking about objects. In that context it is equivalent to momentum conservation. But when we begin to talk about electric and magnetic forces, it becomes convenient to discuss the forces between object through an intermediate concept -- the

*field*. Once we have introduced fields, it becomes clear that fields can carry momentum. Light, for example, is an electromagnetic field and it carries both energy and momentum. Momentum conservation still holds, but it has to be modified to include not just the momentum of objects but of fields as well -- and this can result in some violations of N3.}

#### Follow-on

Last Modified: March 3, 2019