Consider a mass m rising at a constant speed through a vertical height of Δh. This mass is being pulled down by the gravitational force . So to keep it rising at a constant speed, an upward external force of magnitude mg is required.
Since KE is unchanged, the work done by Fext must be equal to the change in GPE of the mass.
We have thus derived the formula for GPE. From now on, instead of calculating the work done against/by the gravitational force, we can simply calculate the gain/loss of GPE.
A 2.0 kg mass being pulled from rest by a 30 N force through a vertical height of 4.0 m and horizontal distance of 3.0 m. Calculate the final KE of the mass. For convenience, take .
There are two possible approaches.
We note that the mass will be experiencing the gravitational pull . So
We note that the mass has gained GPE and KE. So
External vs Field Forces
In the 1st approach, we are applying the work-energy theorem in its original form.
We are treating the gravitational force as just any ordinary external force. That’s why the LHS involves mg and the RHS does not have the GPE term.
In the 2nd approach, we have adapted the work-energy theorem to incorporate changes in GPE (besides changes in KE).
Instead of treating gravity as a force (doing work on the mass), we treat gravity as a field with an associated GPE. That’s why the RHS now includes the GPE term. That’s also why the LHS must not involve mg anymore.
Both approaches are equally valid. But the superiority of the potential energy approach will be more apparent in future when we incorporate other more complicated forms of potential energies of other types of fields, where the field forces are not constant.