# Superposition

#### Prerequisites

Although we are simplifying, simplifying, simplifying, in order to build a framework in which to create models of motion, we can't let ourselves get too simple. In no real situation do we only have a single object interacting with another single object. When you are simply standing quietly and not moving, you are touching the ground, the earth is pulling on you, and you are surrounded by air. As you move — or don't move — all of these are interacting with you, possibly exerting forces. So in order to make sense of motion we have to figure out how to deal with more than one object and more than one force.

Since forces have both a direction and a magnitude — you can push something harder and you can push in different directions to get it moving — we will model force mathematically as a vector. (We'll see that this works well.) From the math of vectors, we inherit addition and subtraction. A plausible assumption — which will have to be tested in a lot of different situations — is that if an object feels a lot of different influences that are trying to change its velocity, we might just add those forces up as vectors.

We call this the idea of *superposition:*

*If there are a lot of different objects that are interacting with the object we are considering, the overall result is the same as if we add up all the forces as vectors and produce a single effective force -- the net force.*

Since we have a notation for force that specifies the object and the cause, we can write this mathematically as follows:

$$\overrightarrow{F}_A^{net} = \overrightarrow{F}_{B\rightarrow A} + \overrightarrow{F}_{C\rightarrow A} + \overrightarrow{F}_{D\rightarrow A} + ... = \sum_j{\overrightarrow{F}_{j\rightarrow A}}$$

where the "Σ" (capital Greek sigma) notation stands for "sum". The little "j" under the sum means sum over all the different values of j (all the other objects interacting with A: objects B, C, D, ....

Joe Redish 9/13/11

#### Follow-ons

Last Modified: August 9, 2018