Non-conservation of mechanical energy

We saw that when we have just conservative forces, that energy is conserved. But what happens if a non-conservative force such as friction is in a physical situation? Can we still get something out of energy arguments? Well the answer is yes we can. Things aren't quite as powerful as before, but valuable information can still be obtained.

Let's try to go through the derivation of conservation of mechanical energy and see what happens when non-conservative forces are included.

Let's separate out work done on an object into two parts, those derived from conservative forces, such as gravity, and those from non-conservative forces such as friction. The total work $W$ is the some of the conservative and non-conservative contributions,

$$W ~=~ W_c + W_{nc}$$ (1.34)

For the conservative part $W_c$ it is possible to express it in terms of a potential $U$, because just as before $W_c ~=~ -\Delta U$. So $W ~=~ -\Delta U + W_{nc}$. But also we recall that $W ~=~ \Delta K$. Together this gives

$$\Delta (K+U) ~=~ W_{nc}$$ (1.35)

So the change in mechanical energy is equal to the work done by non-conservative forces.