Springs

One end of a spring is attached to a wall. The other end of the spring is attached to a block. The block is on a flat, frictionless surface. The block is pulled out a little ways. This stretches the spring. The block is then released. The spring starts to contract, pulling the block back to where it started. What is the acceleration of the block?

Their Way

The formula for spring force is:

F
s = −kx

k is a constant. x is how far a spring has been stretched.


The force acting on the block is the spring force of the spring. We have:

F = F
s

ma = −kx

a = −kx ⁄ m


My Way

A stretched spring has energy. The energy a stretched spring has is elastic potential energy. The formula for elastic potential energy is:


EPE = ½ kx ²


The derivative of elastic potential energy is:


DEPE = kx


As the block slides back, the kinetic energy of the block goes up, the elastic potential energy of the spring goes down, and the total amount of energy remains the same. The rate of change of energy is zero. We have:

DKE + DEPE = 0


ma + kx = 0


a = −kx ⁄ m