## 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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