Work done running on an inclined treadmill

[PeroK]

Do the math.

Do the exercise!

 

[jbriggs444]

Do the exercise!

Do the math. The work being done by the legs is indeed zero and not mgv. See if you can find out why.

Hint: There are two interfaces.

 

[PeroK]

Again, this is completely irrelevant. The work performed by your muscles is the work performed by your muscles.

Your muscles do little or no work to stand up. Your body can produce a force of $mg$ downwards without any effort. It's the force over and above $mg$ that is provided by your muscles.

On a treadmill (or step machine), the additional force is to keep lifting your legs.

On a real hill the additional force is to lift your whole body.

If you go to the gym and experiment, you will see this immediately. The difference is fundamental.

 

[jbriggs444]

Your muscles do little or no work to stand up. Your body can produce a force of $mg$ downwards without any effort. It's the force over and above $mg$ that is provided by your muscles.

On a treadmill (or step machine), the additional force is to keep lifting your legs.

On a real hill the additional force is to lift your whole body.

If you go to the gym and experiment, you will see this immediately. The difference is fundamental.

Do the math. Here. I'll do it for you. There are two interfaces. Hips and feet. You are standing on a treadmill in motion.

Hips: $(mg) \cdot -v$ [Upward force on downward-moving hips]
Feet: $(-mg) \cdot -v$ [Downward force on downward-moving track]
Total: $0$

 

[russ_watters]

Completely irrelevant. The work done by feet on belt is the same regardless of what other forces act on the belt.

I'll add a +1, but elaborate:

Consider a zero friction inclined plane. It requires a constant force to hold stationary against gravity's tendency to accelerate the object (you). Now move your legs against that force, creating a distance moved(the moving treadmill). That's the work, and it is exactly the same as on a hill (neglecting wind).

You guys are letting the fact that the treadmill moves itself confuse you. In fact, the fact that the treadmill maintains a constant speed against your pushing on it means it is absorbing power from you, not powering itself when you are pushing on it. We recently had a thread about non-powered treadmills, where we discussed the fact that they required brakes.

 

[russ_watters]

The treadmill moves itself. There is no need to do any work to move the treadmill. To make it go faster, you press a button!

Let me try a different way: since the treadmill moves itself, you have to apply a force to it to keep from falling off the back of it. That force is larger the higher the slope. And that force, times the speed of treadmill, is the power you apply, and it is exactly the same as climbing a hill.

 

[PeroK]

Let me try a different way: since the treadmill moves itself, you have to apply a force to it to keep from falling off the back of it. That force is larger the higher the slope. And that force, times the speed of treadmill, is the power you apply, and it is exactly the same as climbing a hill.

Imagine a flat treadmill. You need to walk to stop going off the end of it, but once you have attained the required speed, you only need the biomechanical energy to keep walking.

On an inclined but stationary treadmill, friction keeps you from sliding down.

On an inclined moving treadmill, you need a combination of these: You need to walk. Friction still holds your weight in place, but your legs are pulled down and you need extra energy to keep lifting them up.

The evidence, however, is clear: you can do speeds and "height" gains on a treadmill that are impossible on a real hill.

 

[jbriggs444]

Yes, but most of that force is provided by gravity, not by your muscles. That's the difference.

The muscles go through the same motion either way. The notion that the force is "provided by gravity" is not well posed and is irrelevant.

Anyway, a force is only needed to accelerate; not to stay stationary.

A force is needed to resist gravity. If you prefer we can treat gravity as a fictitious force and adopt a free fall inertial frame, but that seems pointless.

Imagine a flat treadmill. The same applies. You need to walk to stop going off the end of it, but once you have attained the required speed, you only need the biomechanical energy to keep walking.

Yes. Walking or running on a flat treadmill is equivalent to walking or running on a springy sidewalk. However, walking up an incline requires force to maintain one's state of motion in spite of gravity.

On an inclined but stationary treadmill, friction keeps you from sliding down.

Yes, friction provides a force. But only as long as it is resisted.

On an inclined but stationary treadmill I would think that we are all in violent agreement that the leg muscles require no power. One could sit on a chair and achieve a similar effect.

On an inclined treadmill, you need to walk. Friction still holds your weight in place, but your legs are pulled down and you need extra enrgy to keep lifting them up.

Yes indeed. You need to power your legs in order to maintain the strain that allows the friction to exist. That's power required on the downstroke.

The evidence, however, is clear: you can do speeds and "height" gains on a treadmill that are impossible on a real hill.

Unfortunately, the proffered explanation for this "evidence" is just plain wrong.

 

[PeroK]

Unfortunately, the proffered explanation for this "evidence" is just plain wrong.

You dismiss the evidence too easily, by the way. Why not do the experiment yourself or get a friend to do it? You ought to be willing to test your theory by experiment.

a) How fast can you run up a hill?

b) How fast can you run up a moving treadmill of the same gradient?

You may deduce theoretically that a) and b) are equal, but you shouldn't be unwilling to put this to the test. You will, in fact, undoubtedly find that b) is far greater than a).

It may turn out that my evidence is "plain wrong", but you shouldn't dismiss evidence simply because it doesn't concur with your theorectical conclusions.

I can simplify my explanation (in the ground frame):

When the treadmill is stationary, there is a force of $F = mg \sin \theta$ acting down the treadmill. This is balanced by an equal and opposite friction force. Normal forces are of course balanced. There is no motion.

When the treadmill is moving, there is an additonal tangential force generated by the person's leg muscles of $f$ and the friction force increases to balance this. Again, the centre of gravity of the person remains constant. $f$ is required for walking at the required speed and to lift the legs/moving parts against gravity.

The work done/power by the person's muscles is $fv$. Where $v$ is the speed of the treadmill. There is no need for the muscles to assume the additional burden of providing the force $F$ when the treadmill is moving. $F$ continues to be provided by gravity.

 

[jbriggs444]

It may turn out that my evidence is "plain wrong"

I made no judgement about the evidence. It is the explanation that is incorrect.

 

[jbriggs444]

The work done/power by the person's muscles is $fv$. Where $v$ is the speed of the treadmill. There is no need for the muscles to assume the additional burden of providing the force $F$ when the treadmill is moving. $F$ continues to be provided by gravity.

This notion that a force of one object on another being "provided" by some other source is decidedly unphysical.

 

[FactChecker]

CORRECTION: I have been corrected about this. I no longer believe this is a good way to analyse work. Why? At the point of contact between foot and treadmill, work is being done (force times distance). If the man is not walking, gravity is doing the work and his mass loses altitude. If the man is walking, the man is doing the work and his mass altitude remains constant.

Using anything except the official definition of "work" opens a can of worms. But you can still talk about expending energy while no "work" is officially being done. Your zero-energy path on an inclined treadmill is the downward trajectory you would take while standing still without walking on the moving treadmill. By walking, you expend the same amount of energy to stay in place as if you were climbing a hill. No work is being done, but energy is being expended.

The calculation of the difference in potential energy between the zero-energy trajectory and the "walk to stay in place" is easy to do.

 

[A.T.]

Your muscles do little or no work to stand up. Your body can produce a force of $mg$ downwards without any effort. It's the force over and above $mg$ that is provided by your muscles.

On a treadmill (or step machine), the additional force is to keep lifting your legs.

On a real hill the additional force is to lift your whole body.

If you go to the gym and experiment, you will see this immediately. The difference is fundamental.

When the treadmill runs at constant speed, the only difference is air resistance. Just analyse the treadmill from the inertial frame, where the upper belt surface is at rest. Here you continuously move upwards, just as you would on a real hill.

 

[russ_watters]

On an inclined moving treadmill, you need a combination of these: You need to walk. Friction still holds your weight in place, but your legs are pulled down and you need extra energy to keep lifting them up.

Ok...so that's the work, right?

 

[FactChecker]

My two cents:
Stopping something from falling (either free fall or any other downward trajectory) can not be considered work. It would not be possible to say that a person on an inclined treadmill is doing work while at the same time saying that a table that keeps a book from falling is not doing work. The standard definition of work (force times distance) can not be changed without opening a can of worms.

But you can say that the man on the treadmill is expending energy and the table holding up a book is not. That can be done easily without any tricky definitions or controversy.

 

[PeroK]

When the treadmill runs at constant speed, the only difference is air resistance. Just analyse the treadmill from the inertial frame, where the upper belt surface is at rest. Here you continuously move upwards, just as you would on a real hill.

Yes, but so does everything else. The person standing next to the treadmill is also continuously moving upwards in this frame. Are they moving just as they would on a real hill?

 

[PeroK]

Ok...so that's the work, right?

Yes, the work is only the work needed to keep moving the legs up and down. It's not the work needed to move your whole body mass continuously against gravity.

A treadmill, muscularly, feels like fast stepping only - not much harder than a flat treadmill. It doesn't feel like walking fast uphill. In fact, as already mentioned, the speeds I can do comfortably on a treadmill are practically impossible on a real hill.

There is an aspect of being on the treadmill that I don't understand and can't explain in terms of forces.

Is it simply that the operation of the treadmill's internal engine prevents an analysis using free-body diagrams of the person and the treadmill?

 

[A.T.]

When the treadmill runs at constant speed, the only difference is air resistance. Just analyse the treadmill from the inertial frame, where the upper belt surface is at rest. Here you continuously move upwards, just as you would on a real hill.

Yes, but so does everything else. The person standing next to the treadmill is also continuously moving upwards in this frame.

The person standing next to the treadmill is irrelevant, because it is not interacting with you.

Only the upper belt surface interacts with you. Your gain in potential energy in the inertial rest frame of the supporting surface is the same in both scenarios, and so is the work your muscles have to do (ignoring air drag, psychological effects etc.).

 

[PeroK]

Here's a link that backs up what you guys are saying:

https://physics.stackexchange.com/q...ng-up-a-hill-and-running-up-an-inclined-tread

And yet, there is a massive difference between the two - despite the physics concluding that the runners are making the same biomechanical motions, my experience is clear that the biomechanics are very different in the two cases.

Also, there's a new descending step machine at the gym. It's like a flight of stairs that continuously moves downwards. I've used it a couple of times. It seems to me that you can climb these steps simply by an appropriate movement of the legs. The standing leg does nothing except straighten as one step descends and the free leg comes up to the next step. In this way, you can climb the steps without any sensation of moving your body weight.

The raw physics would be the same as the treadmill: just the same as going up a real flight of stairs. But, by "clever" movements of the legs you avoid the configuration of forces that are being analysed. I wonder if something similar is happening on the treadmill, where there is certainly the sensation (perhaps illusion) of really only moving your legs and not having to move your whole mass at any time?

If, on the other hand, you allow the steps to descend, then ascend a step quickly that more accurately mimics going up stairs.

But, again, in any case, I can set these steps to a speed (about 30m of ascent a minute) that would be unsustainable on a real set of stairs.

Perhaps, alternatively, these machines simply allow you to spread the load and the overall work is the same, but it's easier to sustain because it's all evened out.

I must admit I'm still puzzled about why it is so easy to run up a treadmill way faster than I could ever do on a real hill.

 

[A.T.]

And yet, there is a massive difference between the two

Do you know of any objective proof for this massive difference? Like measurements of used oxygen, etc?

Aside from air drag, you have different visual queues on a treadmill. This could explain some difference, but hardly a massive one.

 

[PeroK]

Do you know of any objective proof for this massive difference? Like measurements of used oxygen, etc?

Aside from air drag, you have different visual queues on a treadmill. This could explain some difference, but hardly a massive one.

Just raw speed. I can do over 6 km/h at 15° on a treadmill, which is at least 1,500m of ascent an hour. And this is just a moderate work out. Next time I'll see just how fast I can go (I just walk fast, but I can try running!).

But, sustained 1,500m of ascent an hour is elite mountain athlete, which I'm certainly not, fit though I am. See my previous link about the "vertical km" in 30mins. Or here:

http://www.irunfar.com/2012/04/the-30-minute-kilometer-a-look-at-the-vertical-kilometer-record.html

I could just about (not quite) do a vertical km in 30mins on a treadmill. Even at my normal moderately hard workout pace it's only 40 mins. On a real hill it would take me at least 75 minutes at lung-busting speed. And, to be honest, I've never tried to go that fast. It's hard enough at 750m of ascent an hour, which is the fastest I've ever done; and harder than 1,500m of ascent on a treadmill.

There's also the lack of engagement of my upper legs on the treadmill. It's really all calves on the treadmill. On a real hill you need to power up using the whole leg. I tend to use the bike or a step machine to get the upper legs working.

 

[A.T.]

Perhaps, alternatively, these machines simply allow you to spread the load and the overall work is the same, but it's easier to sustain because it's all evened out.

Yes, if the slope of the real hill varies a lot, that could explain why it's more exhausting.

 

[jbriggs444]

Yes, the work is only the work needed to keep moving the legs up and down. It's not the work needed to move your whole body mass continuously against gravity.

Again, this is incorrect. The effort to pull the legs upward is not the only work being done by the hip and knee extensors.

 

[A.T.]

Again, this is incorrect. The effort to pull the legs upward is not the only work being done by the hip and knee extensors.

I think the simplest way to avoid such confusions is to look at both scenarios from the rest frame of the support surface. Here the work done on the surface is zero, so all the gain in potential energy comes from muscles.

 

[PeroK]

Okay, so I went to the gym and checked this out. First, I have to admit that @jbriggs444 (and the rest of you) were right all along. The raw physical work is the same.

Also, the angle is 15%, not 15°, which was a major blunder on my part. And makes a big difference to my figures.

Nevertheless, I can do speeds on the treadmill that I cannot do on a real hill. To explain this I did note, however, three major biomechanical differences:

The treadmill greatly increases your stride length. By the time your foot moves forward the treadmill has moved and this was increasing my stride length from about 0.9m to 1.2m. This partly explains the extra horizontal speed that can be attained.

As you push with your foot, your foot is being drawn back towards your body. This allows you to push largely from directly below you. On a real hill, a lot of the work is done by the foot in front of you. This also partly explains why the treadmill focuses on your calves. It's impossible to engage your upper legs fully, because your foot is being sucked under you. This has pros and cons I would imagine. Perhaps pros for speed and cons for endurance.

There is a definite and significant spring in the surface. On the treadmill you are definitely getting some spring out of it and not losing all your energy at every stride. It's very different from walking on a road surface.

So, it's clearly easier on the body to run/walk on a treadmill (both on the flat and on an incline) but that is all biomechanical.

 

[jbriggs444]

I think the simplest way to avoid such confusions is to look at both scenarios from the rest frame of the support surface. Here the work done on the surface is zero, so all the gain in potential energy comes from muscles.

Agreed. Like doing deep knee bends in an elevator. It does not matter whether the elevator is moving upward or downward in the shaft or even whether it is stopped. As long as it moves at a steady velocity, its motion is irrelevant to the work done by the muscles.

 

[A.T.]

As you push with your foot, your foot is being drawn back towards your body. This allows you to push largely from directly below you. On a real hill, a lot of the work is done by the foot in front of you.

Mechanically, there is nothing different about the legs compared to a static incline.

Psychologically, on the treadmill you know that you have to keep a constant speed. And keeping a constant speed might of course improve the efficiency.

 

[Grinkle]

@PeroK Let the treadmill become longer, say 100m long or 1000m long or whatever length is helpful to make it seem the same as no treadmill mentally - and install a fog machine at the end so you lose sight of the end. Install whatever other visual queues you might think of to remove any visual evidence that you are not running on the ground. Let the speed of treadmill automatically adjust such that no matter what your speed, you stay at the same spot relative to whatever surface is supporting the treadmill. Let the treadmill become wider - as wide as it is long. What experiment will you be able to do on the treadmill that will prove it is easier for you to walk a km uphill on the treadmill than on the ground next to the treadmill? I think no experiment would show this, because it isn't the case.

What changes your bio-mechanics are you not wanting to fall of the end of the treadmill and the treadmill moving a constant speed that it dictates as opposed to your body dictating. Remove these factors and you won't find it any different running on the treadmill than on the road. Given a big enough treadmill one should see that it becomes equivalent to arguing its easier to run E to W than W to E because the rotation of the Earth is helping in one direction and not the other.

Possible reasons you find it so much easier in the gym include -

Its cooler in the gym / noticeably windy outside
You get more immediate feedback on your speed in the gym and it motivates you
- You do not run with a partner outside / away from the treadmill who pushes your pace the way a treadmill does by contruction
- The experience of others in the gym working while you are working is beneficial your perceived effort
The treadmill in the gym is not properly calibrated.
The hill outside is steeper than you believe it to be.
Your warm up routing in the gym vs outside is not identical - you do a more effective warm up on the treadmill before inclining it vs what you do outside.
You were in better shape when you ran on the treadmill vs outside.A popular rule of thumb many runners use is 1% additional incline on the treadmill results in the same perceived effort as not running on a treadmill - most runners perceive as you do, that the treadmill is less effort. There are many different explanations that float around the running community to explain this discrepancy - I have yet to see a single one that starts from a properly defined force diagram.

The anecdotal finding is not universal - some folks find it mentally harder to run on a treadmill - their perceived effort increases.

 

[russ_watters]

My two cents:
Stopping something from falling (either free fall or any other downward trajectory) can not be considered work. It would not be possible to say that a person on an inclined treadmill is doing work while at the same time saying that a table that keeps a book from falling is not doing work. The standard definition of work (force times distance) can not be changed without opening a can of worms.

But you can say that the man on the treadmill is expending energy and the table holding up a book is not. That can be done easily without any tricky definitions or controversy.

Sorry, but you are glossing over the issue being discussed. When not moving up or down against a gravitational field, you can say no work is done against gravity, but that doesn't mean no work is being done against anything. A common similar example is a helicopter doing work against the air when hovering. Again, the power is force times [air] velocity.

 

[A.T.]

@PeroK Let the treadmill become longer, say 100m long or 1000m long or whatever length is helpful to make it seem the same as no treadmill mentally - and install a fog machine at the end so you lose sight of the end. Install whatever other visual queues you might think of to remove any visual evidence that you are not running on the ground.

Or the other way around: Imagine you walk up a hill, enclosed in an opaque box that that has a hole in the floor (of same size as the treadmill), The box drives up the incline at constant speed, and you have to keep up, while walking on the ground within the floor hole.

 

[FactChecker]

Sorry, but you are glossing over the issue being discussed. When not moving up or down against a gravitational field, you can say no work is done against gravity, but that doesn't mean no work is being done. A common similar example is a helicopter doing work against the air when hovering. Again, the power is force times [air] velocity.

Not true. Gravity is the only force being opposed. This is not similar to a hovering helicopter because the helicopter is pushing the air around and a person on an inclined treadmill is not pushing the treadmill down. The treadmill surface is rotating down on its own and would do that if no one was on it. I have not "glossed over" the problem. I have thought about this several times over decades and could not come up with a consistent definition of "work" other than the standard one.

 

[jbriggs444]

But you can still talk about expending energy while no "work" is officially being done.

I disagree with the claim that no work is officially being done. The official definition of work involved the force that is applied and the motion of the target object at the point where the force is applied. There is an alternate definition of "center-of-mass" work which involves the motion of the center of mass of the target object.

No "center-of-mass" work is being done on the runner as a whole by treadmill's belt. That's because the center of mass of the treadmill is stationary.
No "center-of-mass" work is being done on the treadmill as a whole by the runner's feet. That's because the center of mass of the runner is stationary.

Work is being extracted from the runner's feet by the treadmill belt. This is possible because the belt is moving.
Work is being provided to the treadmill belt by the runner's feet. This is possible because the runner's feet are moving.

 

[jbriggs444]

Not true. Gravity is the only force being opposed. This is not similar to a hovering helicopter because the helicopter is pushing the air around and a person on an inclined treadmill is not pushing the treadmill down. The treadmill surface is rotating down on its own. I have not "glossed over" the problem. I have thought about this several times over decades and could not come up with a consistent definition of "work" other than the standard one.

Are we really going to have to engage in this whole debate again? Do you really want to claim that a person on a treadmill is not exerting a downward force on a moving treadmill belt?

Can you recite for us your understanding of the standard definition of work?

 

[FactChecker]

A person on a treadmill is not changing the motion of the treadmill. The treadmill surface would be moving the same way if no one was on it.
Here I am assuming that the treadmill is motorized. If, in fact, it is not and the person is forcing the treadmill surface to rotate, that is different.

 

[jbriggs444]

A person on a treadmill is not changing the motion of the treadmill. The treadmill surface would be moving the same way if no one was on it.

Irrelevant.

Again, please recite your understanding of the definition of work.