Work Done Doing Push-ups on an Incline

[bob012345]

Push-ups take a certain amount of work. If starting from a 45 degree angle instead of from a zero degree angle (the ground), my calculation shows it requires 1/√2 the amount of work. If the relative movement of the body, remaining stiff, is the same, the center of mass moves a certain height. Starting from 45 degree incline and going through the same relative motion, the center of mass goes up 1/√2 as much. So the net work is 1√2 m g h. Note, doing 45 degree push-ups seems a lot easier which makes me wonder if there might be other factors involved not directly related to work done in physics terms. Please correct me if I've missed something obvious in this simple calculation. Thanks.

 

[jbriggs444]

Your calculations are correct. The work done for a [small angle] deflection from 45 degrees should be $\frac{1}{\sqrt{2}}$ times as much as for a similar deflection from 0 degrees.

I can think of several confounding factors. First, it is not clear how accurately the 45 degree angle was measured. Second, it is not clear whether the small angle assumption is upheld. Third, perceived effort does not necessarily scale proportionately with work done.

If you have a bathroom scale, you could measure force for both push-up types and see whether the calculated ratio holds up.

 

[russ_watters]

Push-ups take a certain amount of work. If starting from a 45 degree angle instead of from a zero degree angle (the ground), my calculation shows it requires 1/√2 the amount of work. If the relative movement of the body, remaining stiff, is the same, the center of mass moves a certain height. Starting from 45 degree incline and going through the same relative motion, the center of mass goes up 1/√2 as much. So the net work is 1√2 m g h. Note, doing 45 degree push-ups seems a lot easier which makes me wonder if there might be other factors involved not directly related to work done in physics terms. Please correct me if I've missed something obvious in this simple calculation. Thanks.

It's probably because the geometry is more complex than just horizontal or at a 45; your body creates a trapezoid, not a triangle, and its a different trapezoid when you are in the down position than if you are in the up position. 3 of the 4 sides are fixed; the length of your foot length of your torso to the shoulder for two of them. The fourth is your arm length at the shoulder or height of the down position.

Because for a normal pushup the "up" position is itself at a positive angle, but the down position is at a negative angle, it makes sense that doing one on a 45 degree incline is much easier than 1/√2.

 

[etotheipi]

I am unclear on how you define 'work' here; the muscle fibers do work as they contract (they exert a force through a distance), and that is the process which is converting chemical energy locked up in ATP to eventually mechanical energy in the increase in your GPE. The contact forces on your hands on the other hand are not doing any work on you (i.e. we are dealing with a closed system). So if you know the increment in GPE with a little bit of trigonometry then you can deduce the work done by the muscle cells, yes, like you have done. But I have never heard of 45 degree pushup (what is this)?

 

[jbriggs444]

I am unclear on how you define 'work' here; the muscle fibres do work as they contract (they exert a force through a distance), and that is the process which is converting chemical energy locked up in ATP to eventually mechanical energy in the increase in your GPE. The contact forces on your hands on the other hand are not doing any work on you (i.e. we are dealing with a closed system). So if you know the increment in GPE with a little bit of trigonometry then you can deduce the work done by the muscle cells, yes, like you have done. But I have never heard of 45 degree pushup (what is this)?

My wife's physical therapist has her do "wall push-ups" which I expect are the same thing. You lean against the wall and push off from it. She does these at about a 30 degree angle from the vertical, so about half the force required versus a horizontal push-up. [Talk about OMG easy]

 

[bob012345]

It's probably because the geometry is more complex than just horizontal or at a 45; your body creates a trapezoid, not a triangle, and its a different trapezoid when you are in the down position than if you are in the up position. 3 of the 4 sides are fixed; the length of your foot length of your torso to the shoulder for two of them. The fourth is your arm length at the shoulder or height of the down position.

Because for a normal pushup the "up" position is itself at a positive angle, but the down position is at a negative angle, it makes sense that doing one on a 45 degree incline is much easier than 1/√2.

Thanks. I think you are suggesting that the start from horizontal push-up is actually harder than if the body were in a perfect stiff triangle which it more resembles in the 45 degree case. I wonder if it has to do with the fact that effort is being expended to hold the body up and level as a starting position. That effort expends physiological energy but does not do work in physics terms. One exercise people do is hold a partially raised position without locking elbows for support.

 

[bob012345]

I am unclear on how you define 'work' here; the muscle fibers do work as they contract (they exert a force through a distance), and that is the process which is converting chemical energy locked up in ATP to eventually mechanical energy in the increase in your GPE. The contact forces on your hands on the other hand are not doing any work on you (i.e. we are dealing with a closed system). So if you know the increment in GPE with a little bit of trigonometry then you can deduce the work done by the muscle cells, yes, like you have done. But I have never heard of 45 degree pushup (what is this)?

I don't include that. I only compute work as defined in physics terms. Of course, the body is expending more energy than that to some degree because of physiological processes. A 45 degree or any angle push-up is simply leaning against something, a wall, a counter or whatever as a starting position. It's a good way for older folks to do some push-ups without having to get down on the ground.

 

[etotheipi]

I don't include that. I only compute work as defined in physics terms. Of course, the body is expending more energy than that to some degree because of physiological processes.

OK sure I agree with the last part since there will be significant dissipation (e.g. via heat), but what in your view is "work as defined in Physics terms" in this scenario?

 

[bob012345]

OK sure I agree with the last part since there will be significant dissipation (e.g. via heat), but what in your view is "work as defined in Physics terms" in this scenario?

How much mass is raised against gravity. One could integrate the entire body shape at the starting and ending positions but I think a center of mass approximation is fine if the body is kept stiff.

 

[bob012345]

Makes me think about how to do push-ups in space...with a spring against a surface.

 

[etotheipi]

How much mass is raised against gravity. One could integrate the entire body shape at the starting and ending positions but I think a center of mass approximation is fine if the body is kept stiff.

Gravity would do negative work on our person, the modulus of this (in the ideal case) would indeed be the value we're after for 'work done by the man'. Maybe this comes down to biophysics, but I think we must be careful when speaking of 'work done by the man'; what you have is a bunch of cells which are all exerting forces on each other and contracting and relaxing, making a very complicated system of lots of things doing work on lots of other things. Maybe it is better to keep it vague and just speak of '(chemical) energy expended', converted to GPE.

 

[jbriggs444]

Gravity would do negative work on our person, the modulus of this (in the ideal case) would indeed be the value we're after for 'work done by the man'. Maybe this comes down to biophysics, but I think we must be careful when speaking of 'work done by the man'; what you have is a bunch of cells which are all exerting forces on each other and contracting and relaxing, making a very complicated system of lots of things doing work on lots of other things. Maybe it is better to keep it vague and just speak of '(chemical) energy expended', converted to GPE.

If you want to be able to measure it without doing gas analysis on a respirator, you want to stick to measurable physical outputs. e.g. dynamometers on treadmills or rep counts with free weights.

Not necessarily reflective of the quantity you care about, but measurable is better than not.

 

[bob012345]

Gravity would do negative work on our person, the modulus of this (in the ideal case) would indeed be the value we're after for 'work done by the man'. Maybe this comes down to biophysics, but I think we must be careful when speaking of 'work done by the man'; what you have is a bunch of cells which are all exerting forces on each other and contracting and relaxing, making a very complicated system of lots of things doing work on lots of other things. Maybe it is better to keep it vague and just speak of '(chemical) energy expended', converted to GPE.

Well, it doesn't have to be that complicated. If I lift two 5 kg masses by 0.5 meters, one in each arm, I've done at least 50 J of work which is about 12 calories of energy. Note, calories as in food content is actually kilocalories so lifting those two weights once is only 0.012 'calories' if I'm trying to lose weight.

 

[etotheipi]

Well, it doesn't have to be that complicated. If I lift two 5 kg masses by 0.5 meters, one in each arm, I've done at least 50 J of work which is about 12 calories of energy. Note, calories as in food content is actually kilocalories so lifting those two weights once is only 0.012 'calories' if I'm trying to lose weight.

Well yes but this is conceptually simpler, because we can just take the contact force of your hand to be doing positive work on the body you are moving upward against gravity, so it is easy to analyse the thing being lifted.

During a pushup the points of contact with the ground are not moving, so there is no power flowing into the system (you) at the contact point.

Both are sort of similar in that the muscle cells are doing work in both scenarios.

 

[bob012345]

Well, it doesn't have to be that complicated. If I lift two 5 kg masses by 0.5 meters, one in each arm, I've done at least 50 J of work which is about 12 calories of energy. Note, calories as in food content is actually kilocalories so lifting those two weights once is only 0.012 'calories' if I'm trying to lose weight.

That 500 calorie TV dinner you ate last night actually gives the body over 2 million joules of chemical energy to work with.

 

[jbriggs444]

During a pushup the points of contact with the ground are not moving, so there is no power flowing into the system (you) at the contact point.

One might think of it as the upper arm (humerus) doing work on the shoulder, lifting your body. The shoulder does move, so there is a positive power flow there -- during the up stroke.

Sadly, we get no power regeneration during the down stroke.

 

[bob012345]

One might think of it as the upper arm (humerus) doing work on the shoulder, lifting your body. The shoulder does move, so there is a positive power flow there -- during the up stroke.

Sadly, we get no power regeneration during the down stroke.

But if we did, that would partially negate the value of the exercise. In jogging however, the spring action of the leg muscles give the benefit of recycled power for gaining distance.

 

[etotheipi]

One might think of it as the upper arm (humerus) doing work on the shoulder, lifting your body. The shoulder does move, so there is a positive power flow there -- during the up stroke.

I did consider something like this, but you have to choose to draw the line somewhere for the model (i.e. what about the increase in height of the elbow?). You mention taking this cut-off to be the shoulder; that is a good idea, this will probably give a good approximation.

Maybe you could imagine a heavy plank of wood with one vertical spring at each corner, initially compressed. Then we have a complete physical model to work with , and we could build complexity from there.

 

[jbriggs444]

I did consider something like this, but you have to choose to draw the line somewhere for the model (i.e. what about the increase in height of the elbow?). You mention taking this cut-off to be the shoulder; that is a good idea, this will probably give a good approximation.

Agreed, 100%.

What I have in mind is not so much a model to calculate against as a warm and fuzzy mental picture to reassure ourselves that "work done by the person" is not a complete categorical mistake because "but the ground isn't moving!".

Once reassured that we are calculating something real we can relax and see that the body does increase in potential energy, that the muscles are busy doing something and take the reasonable step of equating the two. (With all the known caveats about inefficient biochemical and biomechanical processes).

 

[bob012345]

Agreed, 100%.

What I have in mind is not so much a model to calculate against as a warm and fuzzy mental picture to reassure ourselves that "work done by the person" is not a complete categorical mistake because "but the ground isn't moving!".

Once reassured that we are calculating something real we can relax and see that the body does increase in potential energy, that the muscles are busy doing something and take the reasonable step of equating the two. (With all the known caveats about inefficient biochemical and biomechanical processes).

You could define a 'method of images' where the person is working against a virtual, mirror image person of identical mass, shape and posture.

 

[hutchphd]

Oh please not another endless discussion about who does work on what by whom...

Energy is conserved and the gravitational potential energy of you is being cyclically changed by your muscles. That change gets smaller as $cos{(\theta)}$

 

[etotheipi]

You could define a 'method of images' where the person is working against a virtual, mirror image person of identical mass, shape and posture.

I'm not sure I understand this part; as far as I can tell, their hands would still be fixed in place and they'd do no work on each other. It would just end up looking like a weird cirque-du-soleil-style interpretive dance and I don't know what further insight it gives us? Maybe I have misinterpreted this.

Anyway I think I had better stop now since if I say another word about who does work on what, @hutchphd is going to faint.

 

[bob012345]

I'm not sure I understand this part; as far as I can tell, their hands would still be fixed in place and they'd do no work on each other. It would just end up looking like a weird cirque-du-soleil-style interpretive dance and I don't know what further insight it gives us? Maybe I have misinterpreted this.

Anyway I think I had better stop now since if I say another word about who does work on what, @hutchphd is going to faint.

It was actually an attempt at a bit of humor speaking of the problem as it were an electrostatic problem...

 

[etotheipi]

It was actually an attempt at a bit of humor speaking of the problem as it were an electrostatic problem...

It would seem that Physics has destroyed my sense of humour... not that I had any in the first place, though