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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.

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    $\begingroup$ Are you looking for the physics answer or the biological one? The physics one has very clear answers, but the biology of how the human body lifts things is very complicated. This is why we have so many different workout types. $\endgroup$
    – Cort Ammon
    Aug 27, 2017 at 19:31
  • $\begingroup$ Honestly, not sure where to start on it, as I don't know how complex it could get. I'm physics dumb. Is it more a question of efficiency I should be asking? There has to be some way of calculating across all workout types (strength, endurance, etc) who is 'fitter' than another on a lb per lb basis. I'm not wanting to oversimplify and say, "yes, he deadlifts more so he is the fitter person, or "I can jump higher so I am the fitter person." How do we compare across all frontiers? $\endgroup$
    – BRM
    Aug 27, 2017 at 23:58
  • $\begingroup$ Others have touched on this, but you're asking for a simple answer to an insanely complex question. Take pull-ups for example. The naive physics answer would be to multiply your weight by the vertical distance you traverse to calculate work. However, your technique and the structure of your arms, shoulders, and back will determine the mechanical advantage of the movement, your fitness level will determine how efficiently calories are converted into useful work, etc. $\endgroup$
    – J. Murray
    Aug 28, 2017 at 11:13

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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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  • $\begingroup$ So, is it more a question of who expends less energy to do said workout? It seems to me like in your example it takes the same energy to lift whatever weights, whatever workout, so how in the world could one calculate who is more efficient (fitter, so to speak) than another? $\endgroup$
    – BRM
    Aug 27, 2017 at 23:54
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    $\begingroup$ @BRM I don't know what you mean "is it more a question of [...]". Fitness could have to do with strength, or flexibility, or weight, or health, or BMI, or maybe efficiency if you really want to use that as a metric. Those are all totally different things. I don't think there's any easy physics answer to what you're getting at. $\endgroup$
    – user12029
    Aug 28, 2017 at 0:00
  • $\begingroup$ If we disregard flexibility, health, bmi, then how would you measure such things as strength or efficiency? Trying to find a starting point for comparison. Doesn't feel like it should be this difficult since we have so many knowns (weight of person, weights used, distances used, times, etc). Most everything we do is vertical movements. I guess I'm just really confused... $\endgroup$
    – BRM
    Aug 28, 2017 at 2:04
  • $\begingroup$ From quick google searches, human efficiency is calculated from work output (literally force times distance pushing a pedal) and oxygen consumed, being about 20% efficient (in the same way that your car's engine is 20% efficient). I don't know how you define "strength", but if it is force exerted then it could just be measured by a spring scale! In general, yeah, anything to do with the human body is going to be very confusing and complicated. $\endgroup$
    – user12029
    Aug 28, 2017 at 5:11
  • $\begingroup$ @BRM the things that AREN'T complicated are the ones that are procedurally and mathematically well-defined! Work, power, efficiency, and force are all procedurally/mathematically well-defined for physicists, but the physicist definition doesn't always match the plain english meanings of the words. The most important thing for you right now might be to figure out the real question you're trying to ask! $\endgroup$
    – user12029
    Aug 28, 2017 at 5:17
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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.

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    $\begingroup$ While it is as simple as saying $W=Fd$ in a mechanics problem, it is not so simple with the human body. Counterexample: Pick up something heavy, and hold it there. $d=0$, but you you will feel that you are certainly continuing to do work. $\endgroup$
    – CDCM
    Aug 27, 2017 at 19:52
  • $\begingroup$ Sammy, how would we begin the process of assessing each one of these movements? Their has to be a calculable formula... seems like back in my college days they were real big on VO2 max, etc, which is great on some levels, but poor on others. If given a time limit on a set exercise and set proportional weights, is that a fair comparison of those type of movements? What does "expending more power" mean? Is 'power' the metric we need to use when considering these movements, then dividing that by body weight? I just need a starting point that would be fair...ty $\endgroup$
    – BRM
    Aug 28, 2017 at 1:55
  • $\begingroup$ You said that you want to determine who is in fact doing more work in each exercise. I have suggested a simple way of doing that. Trying to calculate the exact amount of physical work done in each exercise, and to compare one exercise with another, is far more difficult, not something which a 'newbie' could tackle. Also, are you trying to compare work done according to physics, or compare calories used? Or are you trying to find who uses energy more efficiently? $\endgroup$
    – sammy gerbil
    Aug 28, 2017 at 7:42
  • $\begingroup$ Power tells you who is doing the work faster. If you lift the same weights through the same height then you are doing the same work, but if you can do it faster than your friend then you are more powerful. That is a different comparison. ... The difficulty is that you are not being clear about what exactly you want to compare. Comparing across all frontiers adds another level of difficulty, because each of you may do more work, or be more efficient or more powerful, in different exercises. The overall winner then depends on which exercises are compared, and what 'weight' is given to each. $\endgroup$
    – sammy gerbil
    Aug 28, 2017 at 7:55
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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.

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