I'm confused by classic description of work and negative work. If someone pulls (slowly) on a rope to lift a bucket in a well, I understand that the person is doing work on the bucket and gravity is doing negative work on the bucket. So, the net work is zero.

But if the net work is zero, where is the energy coming from that increases the bucket's potential energy?

## Best Answer

Here are a few points to keep in mind:

$$W_{tot}=W_{cons}+W_{non-cons}+W_{ext}=\Delta K = K_{f}-K_{i}$$

But for a conservative force, the definition of the associated potential energy is

$$W_{cons} = -\Delta U = -(U_{f}-U_{i})$$

and so our previous equation becomes: \begin{align*} W_{tot}=W_{cons}+W_{non-cons}+W_{ext}&=\Delta K = K_{f}-K_{i}\\ -(U_{f}-U_{i})+W_{non-cons}+W_{ext}&=\Delta K = K_{f}-K_{i}\\ K_{i}+U_{i}+W_{non-cons}+W_{ext}&= K_{f}+U_{f} \end{align*} If there are no external forces or non-conservative forces, then: $$K_{i}+U_{i} = K_{f}+U_{f}$$ So we see that we can either use the concept of the work done by gravity, OR we can use the concept of gravitational potential energy. But we don't want to do both at the same time, as then we would count the influence of gravity twice.