Potential Energy Based On Order Of Force Applications?

[DocZaius]

An elevator cable pulls an elevator up with a force of (g+a)*m for t seconds where m is the mass of the elevator, accelerating it upwards. Then, for t seconds (just as many as the first time interval), it applies an upward force of (g-a)*m, decelerating it to a stop (the net force here is downwards because of g-a).

Let's put the potential energy reference point at the beginning of the situation. The elevator moved up. The elevator thus gained potential energy.

Now consider that the order had been reversed. First (g-a)*m then (g+a)*m, each for the same time intervals. The total force applied to the elevator would have been the same, yet the elevator would have lost potential energy in the latter case.

Is that correct? Did I do something wrong? It seems odd to me that two situations with identical force applications but differing orders of those applications would result in differing total energy for each case.

Note: this is not a homework question.

 

[DocZaius]

Maybe I should rephrase my scenarios:

Consider an elevator at height h. If that elevator's cable first applies a force of (g+a)*m for y meters, then applies a force of (g-a)*m for y meters, the elevator's height is h+2y meters.

Now consider that same elevator back at height h. This time though, the elevator is first applying a force of (g-a) * m for -y meters, then applying a force of (g+a)*m for -y meters, the elevator's height is h-2y

In the first case, the elevator accelerated up, then accelerated down, and gained height (at the time it accelerated down, it had a positive velocity and so the acceleration down stopped it)
In the second case, the elevator accelerated down, then accelerated up, and lost height. (at the time it accelerated up, it had a negative velocity and so the acceleration up stopped it)

First case elevator's final PE > Second case elevator's final PE

The only difference between the first case and the second case was the order in which the forces of the elevator cable were administered.

Force order therefore matters when it comes to potential energy. Do you agree with this, and if not, where did I go wrong?

Again, not a homework question. I created this problem to show that order in which forces are administered has an effect on an object's total energy.

 

[ibc]

Of course the order which you apply the forces on the body determines the direction in which the body moves, but you must agree that you case is equivalent to the case with no gravitational field, when the force is ±ma.
The potential energy here has no meaning, since the field is uniform, by definition it is infinite, so the fact that you choose some surface in that infinite uniform field, and say all objects must stop there, has no physical or mathematical meaning (the distance from this chosen surface).