There are variations of this question on here but I was unable to find exactly what I was looking for so please bear with me if this is old hat.
If I start with a mass m at rest at height H1 and raise it through a height of h to finishing height H2 what force is required? I often see the simple answer that the upward force required to raise the mass m at constant speed is simply mg, and that the work done with be mgh - equal to the gain in gravitational potential energy of the mass in rising from H1 to H2. The problem I have with this is that it seems to ignore the fact that I have to accelerate the mass from rest - if I simply apply an upward force mg to a resting mass m will it not simply remain at rest support by that force?
It therefore seems I must apply a force somewhat greater than mg, call it (F+mg, to get the mass moving upward. This then seems to imply that the work done on the mass is (F+mg)h which is greater than the increase in gravitational potential energy mgh. What has happened to the extra energy transferred ie Fh?
If at height H2 the mass is placed on a plunger which then falls (with some resistance) under the weight of the mass m as it returns from height H2 to H1, what is the work done by the mass? It would seem to be mgh - ie the change in gravitational potential energy as it falls through height h (the extra energy Fh is still missing). However, given the plunger is offering some resistance to the fall of the mass it seems it must be applying an upward force to the mass which is something less than the weight mg of the mass (otherwise it wouldn't fall). But doesn't the principle of action and reaction then mean that the force applied to the plunger by the mass is similarly less than its weight mg? I guess not - otherwise the work done on the plunger by the mass as it falls would be less than mgh and I would have some more missing energy to deal with!
Could you (gently) show me where I am going wrong - or if by some miracle I am on the right track, help me find the "missing" energy?