Work done running on an inclined treadmill

[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.

 

[FactChecker]

A table holding a book up is exerting force on the floor. That is not work.

 

[jbriggs444]

A table holding a book up is exerting force on the floor. That is not work.

Irrelevant.

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

Edit: Fair is fair. I've given you mine (see #67). Now you give me yours.

 

[FactChecker]

Force times distance at the point of contact. At the point of contact with the treadmill, here is no difference in force between: 1) a person walking on the treadmill, 2) a person standing on the treadmill, and 3) a lead block of the person's weight sitting on the treadmill.

 

[A.T.]

A table holding a book up is exerting force on the floor. That is not work.

In a frame where the floor moves downwards, the table is doing work on it.

Force times distance at the point of contact. At the point of contact with the treadmill, here is no difference in force between: 1) a person walking on the treadmill, 2) a person standing on the treadmill, and 3) a lead block of the person's weight sitting on the treadmill.

If the treadmill is inclined they all do work on the belt, in the frame of the gym.

 

[FactChecker]

There is a big difference between work and expenditure of energy. It is possible to expend a great amount of energy while doing no work. If we try to make a consistent definition of work that will distinguish between a person hanging from a bar with his arms extended versus a person hanging from a bar in a "pull up" position, we will fail. They both apply the same force over the same (zero) distance, but one will expend a lot more energy.

 

[Grinkle]

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.

That is the design goal of a treadmill. The treadmill motor does different work to ensure the belt speed stays as constant as the motor is able to keep it. The motion of the belt is not perfectly constant - an underpowered treadmill will exhibit profound belt velocity variance.

 

[A.T.]

If we try to make a consistent definition of work that will distinguish between a person hanging from a bar with his arms extended versus a person hanging from a bar in a "pull up" position, we will fail.

Comparing different joint positions is irrelevant for walking on the treadmill vs. ground, which both can be achieved with the same joint kinematics.

 

[FactChecker]

Well, I will be happy to accept your expert consensus that walking on an inclined treadmill is work. I often wanted to say that but I convinced myself that it could not be consistently done. I will accept your opinions and think about it some more. Thanks.

 

[jbriggs444]

Force times distance at the point of contact.

Given a runner on a moving treadmill belt, viewed from a frame of reference in which the exercise room is at rest, the dot product of the force of shoes on belt times distance moved by belt under those shoes is non-zero.

QED.

 

[Grinkle]

walking on an inclined treadmill is work.

The work is being done against the treadmill motor. Without the motor, the belt is loose, and one cannot advance along the incline. Try to take a step and the belt slips backwards without moving your body center of mass at all forwards, so one cannot step onto the belt of the treadmill. Ones foot just pulls the belt backwards and ones foot falls on the floor instead of ones body advancing to stand on the treadmill belt.

 

[russ_watters]

Not true. Gravity is the only force being opposed.

You're applying a force to the treadmill.

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.

Yes he is. Imagine the treadmill could free-spin or imagine it was ice. If you tried to run on it you'd fall on your face and slide off the back. The treadmill must apply a force forward to hold you up against gravity.

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.

You are letting that confuse you: the fact that the treadmill moves on its own doesn't tell you anything about the forces on it.

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.

We're discussing the standard definition, you're just applying it to the wrong thing.

 

[FactChecker]

If the treadmill is inclined they all do work on the belt, in the frame of the gym.

I stand corrected. I could not accept that a block of lead being lowered on a treadmill was doing work. I guess I was not correctly applying the definition of work.

 

[jbriggs444]

I stand corrected. I could not accept that a block of lead being lowered on a treadmill was doing work. I guess I was not correctly applying the definition of work.

For what it's worth, that block of lead can be seen as exerting two forces: a contact force on the treadmill belt and a gravitational force on the Earth. The contact force is doing positive work -- it is a downward force on a downward moving belt. The gravitational force is zero work -- it is an upward force on a motionless Earth.

These two numbers for work (or, more properly, power) are not invariant. They can change depending on what frame of reference one adopts. However, perhaps surprisingly, the sum of the two is invariant. It is the same no matter what frame you choose.

If you adopt a frame of reference in which the belt is stationary and the Earth is moving upward, the contact force does no work, but the gravitational force does work on the rising Earth.

If you adopt a frame of reference in which the treadmill is falling at 100 m/s (e.g. if the frame is anchored to an elevator rising at a steady rate of 100 m/s and the treadmill is on the ground) then the contact force is doing lots of positive work on the rapidly falling belt and the force of gravitational attraction on the Earth is doing slightly less negative work on the slightly less rapidly falling Earth so that the sum still comes out the same.

Edit: The elevator scenario was badly worded and has been updated -- twice. I knew how it had to come out but did not back it up with matching words. Hopefully it is sensible now.

 

[FactChecker]

I think I get it now. The same amount of work is being done by the foot on the treadmill whether the man is standing still or walking. When the man is standing still and his CG is losing altitude, gravity is doing the work. When the man is walking and his CG is remaining still, the man is doing the work. (Or should I say supplying the energy for the work?)

 

[phinds]

jbriggs, I usually find your comments spot on but I think you seriously missed the boat on this one. (1) The treadmill moves exactly the same whether you are on it or not and (2) yes it DOES matter that you are not raising your center of mass.

 

[lesaid]

Can one not simplify the thought process by imagining a sealed corridor with no windows - set on the side of a hill, and someone walking up it. He will do a certain amount of work to get from the bottom to the top of the corridor.

Now, keeping the corridor as the frame of reference - imagine it in a spaceship accelerating at 1g, maintaining the acceleration at the same angle to the slope as gravity was in the first version.

Or have it suspended from a rope attached to a crane at that angle, and lowered or raised at constant speed in a 1g field - or sitting on a very long treadmill belt carrying it up or down - doesn't matter.

From inside the corridor, surely there would be no way of telling which scenario was true - and the energy required to get the man up the corridor has to be the same in all cases, as measured from inside the corridor.

And in each case, the force from his feet causes a reaction force from the floor of the corridor - and then into the earth, the rope, the spaceship, the treadmill motor/brake or whatever that balances the man's force and prevents the corridor itself from accelerating downwards.

But why it would 'feel' to the man to be more effort on a real hill than inside the 'corridor' at the same angle, regardless of which of these scenarios was true, I don't know.

Make sense?

 

[jbriggs444]

Can one not simplify the thought process by imagining a sealed corridor with no windows - set on the side of a hill, and someone walking up it. He will do a certain amount of work to get from the bottom to the top of the corridor.

At this point, I think that we are all on the same page and are preaching to the choir.

In order to correct a misunderstanding, one ought to first understand it. That is not an easy task. An incorrect understanding is or should be hard to state in an understandable way. It should be impossible to state rigorously and coherently.

One of the themes that I think I heard was the notion that no work was being done on the treadmill because the treadmill's motion was unaffected by the runner's footsteps. This would imply either an incorrect understanding of the definition of work or a too-hasty application of the work-energy theorem: "If work is being done, kinetic energy should increase; since kinetic energy does not increase, no work is being done". Unfortunately, none of the correspondents went down to this detailed level of argumentation. That meant that the flaws in such an argument could not be attacked and revealed.

Possibly there was an idea that the motor in a treadmill supplies power (a very plausible fiction and even true in many circumstances). If the track is moving but not accelerating and motor is already supplying power then "obviously" the runner cannot also be doing work on the track. Surely that'd be silly?

One way to find errors in an argument: Look for the word "obviously"​

Another incorrect understanding that I expected but do not recall anyone falling for was the difference between center-of-mass work and real work. The runner is exerting force on the treadmill, but the treadmill (as a whole) is not moving, therefore no work is being done on the treadmill as a whole. No one made such a mistake obvious. There was no detailed and erroneous argument to attack.

My suspicion is that a meta-problem was confirmation bias. With an experimentally confirmed effect in hand and a semi-plausible explanation, one is not going to be very receptive to someone saying that it's all wrong.

 

[phinds]

My suspicion is that a meta-problem was confirmation bias. With an experimentally confirmed effect in hand and a semi-plausible explanation, one is not going to be very receptive to someone saying that it's all wrong.

Yeah, I think that's where my head was. The runner's center of mass didn't move upward on the treadmill so I looked no further. Thank you for your extensive and lucid discussion in this thread.

 

[Ron G]

Your question compares work done -- what work? The treadmill is an approximation of the hill climb but in so many ways is only a simplistic approximation. How shall we compare miles covered and elevation conquered against the number of revolutions of a rubber belt? However, we really are not looking for the best way to perform some work task; we are just trying to expend energy using two different devices.

A better question would be: "Given the same speed and incline, does a treadmill match the effort needed for running a hill?"

Ask a runner of sufficient training, and he or she will always perceive the treadmill as easier, because there is so much more energy being expended during a free run having little to do with distance covered or elevation conquered. The same applies to free-weights vs machine weights. You will have the same problem comparing tread mill effort with cycling, swimming, rowing, or jungle ball.

Training for a sport, such as running is so much more than conditioning. It is also intense energy management training, that is learning to expend less energy doing the same thing. Results will vary from athlete to athlete. I have trained for and raced marathons, ultras, and triathlons; the gym was always a holiday from the concentration needed "out there." The treadmill was easier even at a higher pace; but, @#$%, it's boring. :)

 

[marcpollard]

take an extreme example:climbing a vertical ladder versus climbing a waterwheel. Body mass is accelerated against the force of gravity either way. energy requirement is the same.