Work-Energy Theorem

Let's review what we have done. Work was defined as  and by putting in F = ma we found that the total work is always ΔK where the kinetic energy is defined as . Thus

So far so good. Note carefully what we did to get this result. We put in the right hand side of F = ma to prove W = ΔK. What we actually did was

Now let's not put  but just study the integral  by itself. Before we do that, we must recognize that there are two types of forces called conservative and non-conservative. Anyway, to put it briefly, conservative forces ''bounce back" and non-conservative forces don't. Gravity is a conservative force. If you lift an object against gravity and let it go then the object falls back to where it was. Spring forces are conservative. If you pull a spring and then let it go, it bounces back to where it was. However friction is non-conservative. If you slide an object along the table against friction and let go, then the object just stays there. With conservative forces we always associate a potential energy. Thus any force  can be broken up into the conservative piece _C and the non-conservative piece _NC, as in

and each piece corresponds therefore to conservative work W_C and non-conservative work W_NC. Let's define the conservative piece as the negative of the change in a new quantity called potential energy U. The definition is

where -ΔU = -(U_f - U_i) = -U_f + U_i. Now we found that the total work W was always ΔK. Combining all of this we have

or

which is the famous Work-Energy theorem.