# Total energy "exertion" raising and lowering a weight

I'm thinking of the example of somebody raising and lowering some kind of weight, say a barbell and what effect the total time takes to perform the whole raising and lowering action would have on the energy expenditure of energy required from the person. Assume that we are just using a simplified model where we don't consider anything to do with energy metabolism or mechanical efficiency of a person's joints/muscles or perceived difficulty. Just the amount of "output" from the muscles. We can assume this person is a machine with a piston that moves it up and down if that makes it simpler. Also we can ignore air resistance and friction. I'm interested in the acceleration that the person puts out, and the acceleration due to gravity and how they interact. We can assume that the speed is linear at each point.

If we had a specific weight, say 10kg and it is moved up 1meter and then down 1meter. We could calculate the force at any given point on the arc with F = mA which will be relative to the speed, so the more speed the more "A". But if the rep is completed in a longer period of time then if we integrate over the time of the motion, then even though A is smaller the total integrated should come to the same value.

However once we think of gravity, this exerts a constant downward force (ignore the slight difference in gravitation field strength at different heights). And there must be some energy required simply to work against this. For example if you held a weight at a set height without it going up or down then some energy would have to be exerted.

My assumption is that this total energy (due to gravity resistance) would be higher if we did the rep slower because of the greater time for which gravity is acting. So doing a rep more slowly would require a greater amount of total energy, but for some reason I'm not sure about this.

I know that the total "work" done would be the same regardless of speed, but that assumes that gravitational energy is being "recovered" in some way?