[Physics] Conceptual question on work and potential energy

newtonian-gravitynewtonian-mechanicspotential energywork

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:

  1. Potential energy is always described as the potential energy of the system. For example, the gravitational potential energy of the Earth-Moon system, belongs to the system as a whole, not the Earth or the Moon individually. So for your example, if you are for instance throwing a brick upwards, it would be the potential of the brick-Earth system.
  2. The Work-Energy Theorem can be written (in terms of conservative, non-conservative and other, external forces) as:

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

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.