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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?

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It takes no energy to hold a weight at a fixed height. A rope can do that. Ropes don't burn calories.

Lowering a weight slowly also takes no energy. This can be done easily with a number of devices, such as your proposed piston. A piston is typically powered by hydraulic or pneumatic pressure. Release the pressure slowly through a valve and the weight will gradually drop. You can lower a 10 ton weight as slowly as you please, with no more energy input than it takes to turn a valve a quarter turn.

Raising a weight does require energy, but raising it slowly requires no more energy than raising it quickly. A simple example is using a hand-cranked jack to lift a car. The task is not made any easier if you rush. You can turn the jack as lazily as you like, or take a break halfway through for a sandwich (don't actually walk away from a jacked-up car though). As long as your muscles aren't directly supporting the car, there's no penalty for being slow.

Of course your muscles take quite a lot of energy to do any of these tasks directly. Skeletal muscles are incredibly inefficient at holding things in place. They waste huge amounts of energy doing absolutely no mechanical work at all. And the longer you use them, the more energy they waste -- this is why you have the intuition that operations that take longer require more energy. But you've explicitly asked to use a simplified model where all that muscle-specific stuff is ignored.

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