Question

why the potential energy increases when a spring is compressed

Answer 1

Hi,

See the following figure:

In the first case, the spring is compressed, in the second case , the spring is stretched. The arrows above each case show the direction of restoring force . hooke’s law gives us this restoring force that is the force which is applied by the spring to get back to its original position.

Now , in case of compression, see that the restoring force is acting in the positive x direction hence :

Restoring force F = kx where k is the spring constant and x is the displacement

Potential energy is the work done to compress the spring. Now we know that

work done = force * displacement

Earlier before the spring was compressed work done was 0 or no work was done since displacement of spring was 0 but when it is compressed ,some amount of work will be done.

force here is the average of external F_ext and restoring force F

= ½ (F_ext +F)

Here F_ext = 0

Therefore,

Force = ½ F= ½ kx

Potential energy = work done = ½ kx*x = ½ kx²

Which is the required expression which also shows that potential energy has increased.