# Using system schemas for Newton's 3rd law

#### Prerequisites

Now that we've spent some time working with free-body diagrams and Newton's 3rd law, let's return to the concept of system schemas and see how this all fits together. Recall that system schemas help us identify the objects of interest in a system as well as the interactions between those objects. Let's return to the example of pushing two boxes and consider the system schema for this situation.

## Describing the system:

What would the system schema look like for this situation? Let's start by defining the objects of interest, in this case we'd need to include the Hand, Box A, Box B, and the Table/Earth. Similarly we notice that the interactions in this system are primarily contact interactions along with gravitational interactions, so the system schema would look like this:

Now, looking at the system schema, I can see that when drawing a free-body diagram for Box A, I need to include 4 interactions - 3 contact and 1 gravitational. I know I need at least one force for every interaction, so that would mean 4 forces on Box A. Similarly, for Box B I have 3 interactions - so at least 3 forces in the free-body diagram for Box B.

We found in the previous reading on Newton's 3rd law that the two forces $\overrightarrow{F}^c_{A\rightarrow B}$ and $\overrightarrow{F}^c_{B\rightarrow A}$  have the same magnitude, but opposite directions.

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

Where is this information in the system schema? Let's start with the "type" of force in this expression. That corresponds to the "type" of interaction. So in this case that's the contact interaction between Box A and Box B.

The equal magnitude part is inside the idea that the interaction is represented by a single arrow - this arrow represents one single magnitude, not multiple. Where's the opposite direction part? Well, let's look closer at those arrows that represent the interaction. Notice that the line has two arrowheads, and those arrows point in opposite directions to one another. So re-writing Newton's 3rd law from the system schema perspective:

Newton's 3rd law: For each interaction there are two forces, one on each object, and the forces are equal in magnitude and opposite in direction.

Now in the reading on Newton's 3rd law it was emphasized that the normal force and the gravitational force that the box sitting on the table feels are not an example of Newton's 3rd law, even though they are equal and opposite. In this re-statement it's easy to see why. The normal force in the system schema is represented by the contact interaction, and the gravitational force is a completely different interaction! So they cannot be examples of Newton's 3rd law, which requires that we start with a SINGLE interaction and represent the two forces that describe that interaction.

## Combining Objects:

One last thing we should mention about using the system schema to unpack Newton's 3rd law is this business with combining the boxes. We know that the boxes must accelerate together (how else would Box B move?), so that means we can combine Box A and Box B into a single object.

In the system schema this looks like combining Box A and Box B into a single object and combined all the interactions of the same "type".

We should point out that we haven't changed the phenomenon in any way, we have only changed our representation of that phenomenon. In this new representation we consider Box AB as a combination of Box A and Box B, and the gravitational and contact interactions between Box AB and the Table/Earth as a combination of those same interaction between Box A and Box B. We've also absorbed the contact interaction between Box A and Box B. Looking at the free-body diagram of Box AB we see the only force left of interest is the $\overrightarrow{F}^{c}_{hand\rightarrow AB}$, which tells us all we need to know about how Box AB accelerates as a whole unit.

This is worth noticing because we do this all the time when describing the interactions with objects. We combine atoms into molecules and molecules into objects, and then describe the forces on the compound objects. The system schema and Newton's 3rd law allow us to make sense of why we can do this in so many situations.

Vashti Sawtelle 9/18/12

Article 356