Question

Why is only half of the work done in compressing a spring stored as potential energy?

- Is it just coincidental that it is only half of the work done in charging a capacitor is stored as potential energy or is this an overall limit to the amount of work that can be stored as potential energy no matter what?

-If only half of the energy can be stored as potential energy how come a ball that is dropped can bounce up to higher than 1/2 of it's original height?

Answer 1

The easiest way to show this is with a graph.

For a spring, we know:
F = k * x

We also know that work is defined as:
W = F * x

So a plot of F vs. x would look like this:


Now I'm not sure what mathematics you know at this point. If you do not know calculus, then I will simply let you know that work is only the area under the curve (line in this case). You'll notice this area is a triangle. Triangle area is given by: 1/2 * b * h. So that's where the 1/2 comes from.

If you do know calculus, then you can show that Work is more appropriately defined as an integral of the force over a specific displacement. The definition of an integral is the area under a curve.

If you would like a much more in-depth explanation similar to this, see:
https://www.khanacademy.org/science/physics/work-and-energy/hookes-law/v/potential-energy-stored-in-a-spring

Answer 2

haha, my apologies I realized I just colored in the entire rectangle, it should only be the triangle under the curve.