Work by Variable Force, and Spring Force

When a force varies as it pushes or pulls an object, one cannot simply calculate work as the product
```
      work  =  (force) * (distance)
```
Instead, one must integrate the force through the distance over which it acts
```
               /
      work  =  |  (force) * (dx)
               /
```
As before, if the force and displacement are not in exactly the same direction, one must take the dot product within the integral.
Springs are very important because they serve as simple models for lots of complicated physical systems. Objects which behave like springs behave in a manner which is described as simple harmonic motion; you will see SHM over and over as you continue in physics.
The defining character of a spring is that it resists displacement from its rest position with a force which increases linearly:
```
       restoring force  =  - k * (displacement)
```
where k is called the spring constant. It has units of Newtons per meter.
When a spring pulls something, or pushes something, over a distance x, it does work
```
                           2
       work  =  1/2 * k * x
```
If a spring is compressed (or stretched) it stores energy equal to the work performed to compress (or stretch) it. We might call this spring potential energy.

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