Work done running on an inclined treadmill

[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.

 

[Grinkle]

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

Agreed.

Moving the discussion what is the simplest way to look at it, I claim that the work being done is most easily calculated by looking at the torque vs radial displacement of the treadmill motor. This motor is moving the runners center of mass down the belt as the runner moves themself back up the belt. The motor does work in preventing the belt from slipping.

Many people simply don't think this is the case - they have never seen a person too heavy for a given treadmill try to run on an underpowered treadmill. The belt slips and they fall forward as their feet move backward out from under them. Like @russ_watters said, it becomes trying to run on ice.

In these discussions, I tend to get very tied up in the runner's biomechanics for which the kinetmatics are very complex, but the kinematics of the treadmill motor are very simple. Neglecting heat dissipation in the friction of the treadmill (not trivial for most treadmills I suppose), the motor work must be the runners work.

 

[Roger Chase]

Awesome discussion! I work in the exercise equipment industry as a test engineer with a physics background and this topic comes up regularly for various pieces of equipment, one of which is the TM. The human body is actually pretty complex mechanically and we do see some surprising results sometimes...if you examine the current wave forms going into the drive motor when a person is on a TM (we call this an active load) it drops with every footstep by quite a bit because the person is actually pushing the walk belt back which "assists" the drive motor temporarily. I tried to dig through some of the O-Scope captures we did in previous testing for TMs and the one below seemed pretty cool. This is at 0° incline for demonstration purposes. The Yellow waveform is the brushed DC motor current, Blue (which is kind of obscured in the background) is the PWM'ed motor voltage and the Red is their product (power in Watts). On an incline this effect is amplified because you have more mechanical advantage i.e. it's easier to push the belt since you're trying to keep upright and because the normal force mgCosθ is less since θ>0 so the peak current is less because overall friction is less and the dips are more pronounced. I'll see if I can find some comparative incline/non-incline examples or actually it might take less time if I just went out to the lab and took new data...I could use the exercise anyways :)

 

[jbriggs444]

Agreed.

Moving the discussion what is the simplest way to look at it, I claim that the work being done is most easily calculated by looking at the torque vs radial displacement of the treadmill motor. This motor is moving the runners center of mass down the belt as the runner moves themself back up the belt. The motor does work in preventing the belt from slipping.

The motor absorbs work in preventing the belt from slipping.

Neglecting heat dissipation in the friction of the treadmill (not trivial for most treadmills I suppose), the motor work must be the runners work.

The motor's work must be the additive inverse of the runner's work [ignoring frictional losses].

 

[rude man]

I've thought about this for a long time and discussed it with a professor of mechanical engineering. We both say the apparent gain in height on a treadmill is far greater than that of climbing an actual mountain. PeroK seemd to have the most credible answer. I wil probably continue to mull over it for some time yet.

 

[A.T.]

We both say the apparent gain in height on a treadmill is far greater than that of climbing an actual mountain.

What is "apparent gain in height", and why is it relevant?

 

[A.T.]

The Yellow waveform is the brushed DC motor current, Blue (which is kind of obscured in the background) is the PWM'ed motor voltage and the Red is their product (power in Watts).View attachment 212748

The crucial parameter would the speed of the belt. If there is only negligible variation of belt speed, you can define an inertial frame where the support surface is at rest. Per Galilean Invariance this frame is equivalent to the rest frame of a hill incline (ignoring air drag). So there is no physical reason to walk differently and do different work, just psychological ones.

 

[lesaid]

I wonder if a major contributor to the perceived greater effort required to climb a hill is simply down to the rough ground. I know personally, it takes substantially more effort to hike up a rough, uneven slope than a smooth steady gradient, and found a treadmill easier. On smoother surfaces, it is natural to fall into a regular rhythmical gait which (presumably) is more efficient that the constant, step-by-step adjustments required when going up a real hill.

When on a treadmill, there is also the temptation to rest one's hands on the rail - while serious users presumably may not do this - doing that could help a lot with maintaining balance, increasing the efficiency of the walking/running?

Has anyone compared the perception of running up a long steady incline on a road or tarmac path against a treadmill of similar slope? I'm guessing it may still seem harder due to psychological differences, but perhaps much less different? Perhaps hard to come up with a fair comparison with all the subjective influences at work!

 

[A.T.]

I know personally, it takes substantially more effort to hike up a rough, uneven slope than a smooth steady gradient, and found a treadmill easier.

Yes, of course. The equivalence is to a smooth constant slope hill.

When on a treadmill, there is also the temptation to rest one's hands on the rail - while serious users presumably may not do this - doing that could help a lot with maintaining balance, increasing the efficiency of the walking/running?

That can make it completely different, depending on how much force you put on the rails.

 

[Sherwood Botsford]

Consider this thought experiment:

I stand on the belt, and move backward dropping 10 cm.
I then take two steps forward rising 10 cm.

In this case I'm raising my center of mass (cm) each time. It should be the same as going up a real hill.

Now if I do the old calculus limits bit, and do smaller and smaller steps back and forward, what changes?

This gets messy because people don't roll forward, but bounce all over the place. As the steps decrease, I can keep my torso at a constant level, by extending my leg to make it longer. Force times distance. I'm not really climbing, but the grade is more than it would be if the treadmill were level. This would support the argument above that it's about half the effort.

consider another experiment: What about biking on a treadmill? This isolates the user part of the system (cyclist + bike) from the treadmill in a more tractable way. Does biking on a sloped treadmill take more energy than biking on a flat treadmill? Yes. The wheel meets the treadmill on an angle. This will produce a torque on the wheel trying to roll the cyclist backward. He has to overcome that torque in addition to the other work.

However he is not gaining potential energy.

Comments?

 

[Grinkle]

@Roger Chase Its really generous of you to go to the trouble to share that data. If you have a way to add the radial velocity of the motor shaft or the belt speed (they should be proportional I think so it doesn't matter much which you plot if one or the other is available) like @A.T. was saying, that would be interesting to look at.

If one can calibrate out the unloaded work a treadmill motor does and if the addition of a runner does not change the dynamic frictional losses in the system significantly, then a treadmill with instrumentation to record the motor currents / voltages should be able to report very precisely the effective work a runner is doing, similar to work computers on bicycles. I am curious to know if any treadmills do this, if happen to know. Not sure I can ask this on a runner forum without folks thinking I am asking about heart rate calorie burn computations.

 

[jbriggs444]

This would support the argument above that it's about half the effort.

What argument that what is half the effort of what?

 

[votingmachine]

I think the effort being ignored is the "ride" on a treadmill. Consider if you put one foot on the side rail, and move the other thru a walking motion. That foot mimics walking up a hill. But it is clearly not moving the body mass up a hill. You do lift your foot to move it up the treadmill. But then it just rides down. Moving two feet can seem to be that same motion. You lift your foot and move it up, then let it ride down, barely working to keep your mass centered. Then you lift the next foot and repeat.

When you walk up a real hill, you never get the "ride". You push thru the entire stroke. If you walk up a flight of 20 stairs, you use muscles the entire stroke. If you walk 20 steps on an escalator moving against you, you get opportunities to rest. You lift your foot a single stair height, but as you extend your leg, the stair moves down, rather than lifting your body up. If you move up one step on stairs, you are higher at the end. If you move one step on the escalator, you are the same height. It seems obvious that there was less work. I feel that I work harder, walking UP an UP escalator, and work less hard, walking UP a DOWN escalator,

When you stand in place, you are working, even though the physics says you are not. There is constant muscle use. Don't ignore the inefficiency of the human body as though it is a "body" at rest. It could be that the difference is just in inefficiencies.

 

[jbriggs444]

When you walk up a real hill, you never get the "ride".

You are going to have to make this argument more detailed. As it stands, it is not even wrong.

Let me see if I can fill in the blanks where the details should be.

You do lift your foot to move it up the treadmill. But then it just rides down

During the downstroke on a moving treadmill, your muscles are contracting and your leg is extending. The "just rides down" verbiage suggests that you believe that no effort is expended and no work is being done.

Please clarify.

Edit: found it.

It seems obvious that there was less work

The word obvious is the clue that this is the error.

 

[Grinkle]

Moving two feet can seem to be that same motion.

No, it is not that same motion.

If you are standing on an incline and you drag one foot backwards along the ground while your weight is supported by your other foot its not hard. If you start to walk up the incline, its hard.

If you are standing on a treadmill and you let one foot travel freely on the belt while supporting your weight on the other foot, its not hard. If you start to walk up the inclined belt with both feet, its hard.

 

[lesaid]

I stand on the belt, and move backward dropping 10 cm.
I then take two steps forward rising 10 cm.

Each time you are carried back and down, you lose potential energy, which ends up in the motor-drive assembly - either as a reduction in energy supplied to the motor, or as in heat from a brake, perhaps.

Each time you step up, you are regaining that potential energy, this time from the chemical energy reserves in your body, via the actions of your legs. If you step up the same distance as you were previously carried down, you end up with exactly the same amount of potential energy.

Over time, the net effect is to transfer energy from your body to the motor/drive system.

At least, that's my perception.