Running Machines: Real Incline vs. Fake Conveyor Incline

[pikpobedy]

Increasing the incline on a belt type running machine increases the expenditure of energy of the runner. However the runner is not increasing his/her potential energy in the gravitational field. The only change appears to be in the manner the leg joints and muscles bend, flex, stretch and contract.

My question becomes what is the difference between jogging on a real incline and on a fake conveyor incline?http://www.bing.com/images/search?q...m=QBIR&pq=running+machine&sc=1-15&sp=-1&sk=#a

 

[A.T.]

Increasing the incline on a belt type running machine increases the expenditure of energy of the runner. However the runner is not increasing his/her potential energy in the gravitational field.

When running up a real incline with constant slope at constant speed, you can also pick an inertial frame of reference where the runner doesn't gain any height.

My question becomes what is the difference between jogging on a real incline and on a fake conveyor incline?

For constant conveyor speeds there is none, except maybe aerodynamics.

 

[Brinx]

When running up a real incline with constant slope at constant speed, you can also pick an inertial frame of reference where the runner doesn't gain any height.

Hmm, I'm not sure about this. A simplified case: sliding a mass across a frictionless flat surface at constant velocity requires only the initial impulse and no subsequent work, but sliding the same mass up a frictionless slope (i.e. partly against the force of gravity) does require a steady input of work and hence a steady power.

Of course, actual running motion involves the 'bobbing' up and down of the center of mass of the runner, an energy exchange which should be fairly efficient because tendons can temporarily store potential energy as tension. But the above simplification does indicate that running up a slope requires more power because you are raising your center of gravity against the Earth's pull. And you do notice the difference quite clearly if you try it...

On a treadmill with an incline, it depends on how much you have to elevate your center of mass with every stride. It might be the case that the motion of your center of mass does not differ significantly from what happens when you run on a level surface, but as pikpobedy said there is a difference in the range of motion of your legs.

 

[A.T.]

Hmm, I'm not sure about this. A simplified case: sliding a mass across a frictionless flat surface at constant velocity requires only the initial impulse and no subsequent work, but sliding the same mass up a frictionless slope (i.e. partly against the force of gravity) does require a steady input of work and hence a steady power... the above simplification does indicate that running up a slope requires more power because you are raising your center of gravity against the Earth's pull.

Work and power are frame dependent. In the inertial rest frame of the upper belt surface the runner is also raising his center of gravity against the Earth's pull.

but as pikpobedy said there is a difference in the range of motion of your legs.

If there is a difference in the frame invariant range of motion (joint angles, oscillation of mass center around its average) then because of psychological effects. People on treadmills get different visual cues and are constrained in space, so they move differently.

 

[stevenjones3.1]

When you run on a treadmill (flat or incline) it is much easier because you do not have to push yourself forward, you basically just hop in place. You do have to push yourself a little bit to counter the effect of the track moving underneath you but it is not as difficult.

So for a treadmill on an incline vs running up a hill, in addition to not having to change your potential energy, you more or less just have to hop in place and the track will continue to pass underneath you.

 

[A.T.]

When you run on a treadmill (flat or incline) it is much easier because you do not have to push yourself forward, you basically just hop in place. You do have to push yourself a little bit to counter the effect of the track moving underneath you but it is not as difficult.

This is only true in the acceleration phase of the belt. Once the belt is moving at a constant speed, there is no difference in the forces that the legs have to produce. This is a direct consequence of Galilean invariance:

http://en.wikipedia.org/wiki/Galilean_invariance

So for a treadmill on an incline vs running up a hill, in addition to not having to change your potential energy, you more or less just have to hop in place and the track will continue to pass underneath you.

Not if the belt speed is constant (and gravity can be assumed to be uniform). Galilean invariance doesn't care if the inertial frames move horizontally or vertically.

Again, the differences in motion between ground and a constant speed treadmill come from psychological/motion control reasons (constrained space, no visual motion of surroundings), and not from a physical need to do something differently.

To see why the argument about potential energy is bogus, consider an elevator going down at high constant speed. Raising slowly from a squat in this elevator is exactly the same, as raising slowly from a squat on the ground. Despite the fact, that in the frame of the ground the elevator guy losses potential energy, while the ground guy gains potential energy.

 

[Brinx]

A.T., the fact remains that running up a hill takes a lot more energy than running on a flat surface - this is something that is easily experimentally verified. If I'm reading your words right, you seem to think that elevating your center of mass does not require energy. In that case, you'll also have a problem explaining (for instance) why hydro-electric turbines provide power from water that flows down.

On performing squats in the elevator: if you do this while the elevator is moving up, you are simply making a small change in the rate at which you are gaining potential energy. The motor powering the elevator takes care of hoisting the elevator cage and you (partly offset by the counterweight), and this takes power. That small difference in the rate of potential energy gain you provide yourself (i.e. the power you generate) is equal to the rate at which you gain it when doing squats standing on solid ground. The same reasoning applies when the elevator is moving down: you are making yourself lose potential energy a little less quickly.

Strictly speaking, and to be pedantic, a reference frame at rest with respect to the Earth's surface is not an inertial frame (an inertial frame would have to be accelerating with g towards the Earth's center when at the surface). We're using apparent forces here so we can pretend it is one.

 

[A.T.]

running up a hill takes a lot more energy than running on a flat surface

And running on an inclined treadmill takes more physiological energy than running on a level treadmill.

you seem to think that elevating your center of mass does not require energy.

"Elevating your center of mass" is a frame dependent statement. The physiological energy expended by the body is a frame independent quantity. You cannot relate the two directly.

On performing squats in the elevator: if you do this while the elevator is moving up, you are simply making a small change in the rate at which you are gaining potential energy. The motor powering the elevator takes care of hoisting the elevator cage and you (partly offset by the counterweight), and this takes power. That small difference in the rate of potential energy gain you provide yourself (i.e. the power you generate) is equal to the rate at which you gain it when doing squats standing on solid ground. The same reasoning applies when the elevator is moving down: you are making yourself lose potential energy a little less quickly.

All of that is completely irrelevant to the forces required to do the squat, and thus irrelevant to the physiological energy expended by the body. Doing a squat in a constant speed elevator requires exactly the same forces, and thus physiological energy from the body, as a squat on the ground. The same applies to a constant speed inclined treadmill vs. inclined ground.

an inertial frame would have to be accelerating with g towards the Earth's center

Yeah, let’s bring General Relativity into it, because you are not confused enough already.

 

[sophiecentaur]

So for a treadmill on an incline vs running up a hill, in addition to not having to change your potential energy, you more or less just have to hop in place and the track will continue to pass underneath you.

Work is force times distance. The force in that situation is still your weight (plus a bit) and the vertical distance is the distance moved downward by the belt plus your normal hopping height. If you don't hop higher, you will hit the belt early and end up further back down the slope. Alternatively, you could be hopping faster to stay still - either way you need to be making up for the net downward velocity of the belt by putting in more power.
It's always easier to assume that there's an explanation than that you've found something that doesn't obey 'the rules'.

 

[russ_watters]

Work is force times distance. The force in that situation is still your weight (plus a bit) and the vertical distance is the distance moved downward by the belt plus your normal hopping height. If you don't hop higher, you will hit the belt early and end up further back down the slope. Alternatively, you could be hopping faster to stay still - either way you need to be making up for the net downward velocity of the belt by putting in more power.
It's always easier to assume that there's an explanation than that you've found something that doesn't obey 'the rules'.

Another way of saying the same thing:

On a flat surface, the ground is moving perpendicular to the net applied force (your weight), so there is no work done for anything but the "hopping". On an incline, there is a vertical component of motion that multiplied by the net force gives us work against gravity.

 

[sophiecentaur]

Another way of using more energy than you'd expect is to run on muddy / soggy ground. Your legs need to move further for a required bounce height, which requires more work on each step.
You could imagine a hime gym which consisted of a tray of mud and you run on the spot. Cheaper than a treadmill - and a beauty treatment at the same time.

 

[pikpobedy]

The power used by the treadmill? At constant speed with the same user, is the power consumption a function of inclination? How so?

 

[Brinx]

On a flat surface, the ground is moving perpendicular to the net applied force (your weight), so there is no work done for anything but the "hopping". On an incline, there is a vertical component of motion that multiplied by the net force gives us work against gravity.

Right, that makes sense. So running on a treadmill with a given slope will have you expend about as much energy as running up an actual hill with the same slope. Both of those situations require that you expend more energy per unit time than if you were running along flat ground.

 

[A.T.]

So running on a treadmill with a given slope will have you expend about as much energy as running up an actual hill with the same slope.

For a real human it is about as much. For a simple walking robot or a toy car it is exactly the same. I neglect aerodynamics here, but you can also recreate the relative headwind with a big fan.

 

[sophiecentaur]

The power used by the treadmill? At constant speed with the same user, is the power consumption a function of inclination? How so?

The inclination and speed tells you how much more the runner has to raise himself every step, to avoid going back down. Force times more distance for a steep slope. Just think in terms of running up the down escalator - just to stay still - compared with running on the spot on one stair. (I keep thinking of the Charlie Chaplin escalator sequence (in "The Floorwalker" I seem to remember).

 

[Fantasist]

If you would not be running on the incline, the belt would drag you down, so you would be losing potential energy. So you have to do work in order to keep your potential energy constant i.e. to stay at the same height.

It is somewhat similar to an airplane that also has to do work just to stay at the same height.

 

[Chestermiller]

Imagine a man running up a hill oriented upward and to the right. Now imagine somehow making the road move downward and to the left so that it translates parallel to itself at the same speed that the man was moving upward and to the right. What do you see? Now imagine being on a bike riding up the hill next to the man, with the road appearing to move downwards and to the left relative to yourself (and the man). What do you see? The man is expending the same amount of energy in all three cases.