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I am trying to figure out how to calculate the amount of work done for two people of differing sizes (160lb man which is me and 200lb man) in reference to varying exercises we do in the gym.

All this to see which person is in fact doing more workload per pound. I have ideas, but don't know where to start.

For example, if he and I do a clean and press workout at 50% of body weight and both do the same amount of reps in one minute, did we both do equal work?

Additionally, if we both do box jumps, and I do 25 of them at 20 inches high, and he does 20 of them at 20 inches high, same amount of time, are we doing the same work?

Third, pull-ups. If I do 20 pull ups (160lb man) and he does 16 (200lb man), is that the same amount of work, assuming same time they are done in?

Last of all, running. How in the world can we determine who is doing more work when calculating one mile run times? Any help would be appreciated, as I didn't study this in school. It is very intriguing to us to get the science of this so we can determine who is in fact does more work in each exercise.

BRM
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3 Answers3

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"Work" in physics and "effort" are very different things. Work is a very useful notion in fundamental physics, but not at all a useful notion in human motion! The heavier person in all of your examples will be doing more mechanical work, even if they both put the same amount of mental effort in. If they both lifted a 50lb weight, they would do the same amount of mechanical work on the weight, but depending on the biomechanics of it the heavier person would probably still burn more calories! It's complicated and isn't well described by simple mechanics concepts.

Your running example is a good one. Fundamental physics tells us that it is possible to travel a mile, so long as the height doesn't change, while doing 0 (or almost 0) work. Yet the human body burns ~100 food calories (=100 kilocalories) in running a mile! We're limited by the engineering of our body, not by physics, so the physics laws don't really apply.

Another example highlighting the absurdity of some things: say you lift ten 700lb weights, and put them each on shelves six feet in the air. Totally superhuman, but how much work have you done? Fundamental physics tells you that the energy done is 10 * (700 pounds) * (9.81 meters/second^2) * (6 feet). Plugging this in to wolframalpha, we see you've expended about 14 food calories of energy (14 kilocalories). That's about the energy in a potato chip.

These are just the limits imposed by fundamental physics. It tells you that any lifting machine must spend at least 14 kcal of energy to lift those weights. It tells you that you can spend as little energy as you want travelling a mile. How to actually achieve those things is a matter of engineering!

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You can make a rough comparison of who is doing more work (as defined by physics) in a particular exercise by using the formula : work done = weight lifted x distance. This does not tell you who is burning more calories during the exercise, which depends on how efficiently you are using energy. To do that you need to measure how much oxygen is being used.

If the exercise depends on raising different weights through the same height then the person lifting the greater weight is doing more mechanical work. This clearly applies to the clean and press and pull ups. The person who does more repetitions does proportionally more work.

It also applies to running and box jumps, during which you are raising and lowering your body slightly. If you are approximately the same height then you are raising your bodies through the same distance with each step or jump. If you run the same distance you probably take the same number of steps, so you do the same number of repetitions. Again, the heavier person does more work.

The time which the exercises take does not affect the work done. If you do the same work in a quicker time (or more work in the same time), you are more powerful. That is a different comparison, and may not give you the same answer. For example, your friend may lift 200lb three times in a minute, while you lift 100lb seven times a minute. Your friend does more work on each lift, but you do more work in the same amount of time, so you are more powerful. However, this may not be a fair comparison : your friend may lift 100lb eight times per minute.

Deciding "who is fitter" is not the same as deciding who is doing more mechanical work, or even who is doing it more quickly. Comparison between different exercises is not easy.

sammy gerbil
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If you're after the simple physics answer, and in the interest of keeping it simple, let's use the formula for work. Work = Force x Distance. Force = Mass x Acceleration (which we use as gravity's acceleration that we are overcoming. So for each exercise (and I have to convert to metric):

First Assumptions (not too practical but not going down a rabbit hole)

  1. Each rep takes the weight or body 1 meter in distance. So a pull-up from bottom to top is one meter traveled. Also you're going to jump on a one meter box. You are the same height and your cleans are moving the weight one meter
  2. The first rep and last rep are done at same speed. Each rep takes one second. So for Force we will simply use mass x acceleration ( Zero to 1G = 1) So again you weight 70kg x 1 = 70 Newtons.

OK Cleans

You: 50% x 70 Newtons = 35N x 1 meter x 10 reps = 350 joules Him 50% x 90 Newtons = 45N x 1 meter x 10 reps = 450 joules (divide by kg of body weight if wish)

Box Jumps

You: 70N x 1 meter x 25 reps = 1750 joules Him: 90N x 1 meter x 20 reps = 1800 joules

Pull Ups

You: 70N x 1 meter x 20 reps = 1400 joules Him: 90N x 1 meter x 16 reps = 1440 joules

So he is always doing more work but you more work per kg of bodyweight.

Anthony Pape
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