## Walking on an escalator

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### Walking on an escalator

I was watching the national science quiz on TV yesterday (which is broadcast every year with christmas here in The Netherlands), and one of the question they asked was: If you walk upon an escalator, does the escalator use more, less or an equal amount of energy per second, compared to where you are standing still on it.

The answer they gave as correct was that it uses an equal amount of energy. They demonstrated this with an experiment that I did not find particularly convincing (their model escalator was powered by hand, and the people operating it said they did not feel a significant difference in strength required. Which is hardly very accurate). They also put down two scales on a stairs and showed that the total weight measured by both scales doesn't change much if you step up slowly from one to the other. Again not very convincing, imho.

I've been thinking about this since, and it seems to me that if you walk on an escalator you push it down. It has to do more work then, right? As the escalator transports you upwards it does work. If you walk on the escalator you go upwards faster, so more work has to be done (per second). No escaping that. But at least part of this extra work is done by you, and on tv they said all the extra work is done by you. But it seems to me the escalator at least has to do some of the work. As you step up you push it downwards, imparting momentum on the escalator. This translates to kinetic energy which has to be overcome to keep the escalator's velocity constant.

Anyone here who can shed some light on this.
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### Re: Walking on an escalator

Well, at the same time as you're using more power, you also get off the escalator sooner, which might balance out the total energy used.
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### Re: Walking on an escalator

gmalivuk wrote:Well, at the same time as you're using more power, you also get off the escalator sooner, which might balance out the total energy used.

Yeah, but the question was not about total energy usage, but energy usage per second. That was explicitely mentioned in both the question and the official answer.
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### Re: Walking on an escalator

gmalivuk wrote:Well, at the same time as you're using more power, you also get off the escalator sooner, which might balance out the total energy used.
That doesn't apply if you're walking the wrong way on the escalator.

Your total weight does not change when you are walking, but the impact force provided by the energy of your muscles muscles does create more force on the escalator. You said they used the caveat of walking slowly, which I guess would mitigate most of this impact, but I believe it still exists.

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### Re: Walking on an escalator

It might be easier to model if you have Alice on a platform with a ladder on it, and the platform is being raised by Bill pulling on a pulley. If Alice just stands still on the platform then Bill has to do all the work to raise Alice 10m. If Alice instead climbs 5m up the ladder while Bill is pulling the pulley, then Bill can stop after raising the platform 5m which takes the same amount of energy per meter as before because the platform/ladder/Alice system isn't changing weight. Bill has only used half as much energy as before to raise alice but in half the amount of time so energy usage per second of Bill is the same as the stationary scenario.

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### Re: Walking on an escalator

It does indeed exist. If you want to do it experimentally, get a large person to run up their escalator; doing it theoretically is, of course, much easier. A hand-wavy (yet precise) argument:

1. Neglecting friction, the net work done in each case (i.e., walking vs. standing still) is the same: the initial and final states of the system are identical (you're at the bottom and then at the top).
2. When walking up the escalator, you reach the top faster than when you are standing still.
3. The rate at which work is done (the "amount of energy [used] per second") is therefore greater when you are walking up the escalator.

I would be extremely surprised if frictional effects turn out to negate statement 3.

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### Re: Walking on an escalator

All of that is true, but that still leaves the question of where the additional power comes from; the climber, the escalator mechanism, or both (and in what proportion if both).
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### Re: Walking on an escalator

Mr_Rose wrote:All of that is true, but that still leaves the question of where the additional power comes from; the climber, the escalator mechanism, or both (and in what proportion if both).

As long as you are on the escalator and not accelerating the engines running it is working equally hard no matter if you are moving and in which direction.

But if you are moving upwards then you are spending less time there at the expense of your own energy. If you are walking downwards then you spend more time there while absorbing energy in your legs (going to heat, not useful and still fatiguing you).

In any case the energy budget fits.

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### Re: Walking on an escalator

The extra power comes from the motors - they're putting out a lot more torque than is needed just to raise a weight, so applying additional opposing force won't slow them down. They are indeed exerting more power when a person walks up.
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### Re: Walking on an escalator

If you think in terms of force applied by you on the escalator, it gets clearer.

If you stand still, these forces are your weight.

If you are walking at constant speed, and if we neglect the impacts of your feet, there is no acceleration, and the only force you apply is your weight. When you start climbing, you accelerate, which causes an additional downward force on the escalator, and increases the power needed. When you stop climbing, you decelerate, which cuases an additional upward force, and decreases the power between. In between, it's exactly the same as if you were standing still.

If we consider your speed is not strictly constant, because you accelerate and decelerate at each step, things get hairier, especially if we consider a non linear friction coefficient.

Any way, unless you are jumping on the stairs, your acceleration will be so low, the additional forces will be small compared to your weight, which is pretty small compared to the other forces applied to the escalator.
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### Re: Walking on an escalator

Diadem wrote:I've been thinking about this since, and it seems to me that if you walk on an escalator you push it down. It has to do more work then, right? As the escalator transports you upwards it does work.

If you are climbing the escalator with a constant velocity of your centre of mass, you do not exert more force on the escalator than when you are standing still. In reality, you are continuously accelerating and decelerating vertically when you walk or climb, but on average you are still not transferring momentum to or from the escalator. The escalator's engine cannot tell the difference between someone climbing and someone jumping on the same step.

So the escalator is on average feeling your weight, but no more. The work it does is that weight, times its own vertical velocity. The work you do is your own weight times the extra velocity you have relative to the escalator.

lamemaar
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### Re: Walking on an escalator

I agree with Zamfir, but I would like to clarify his last two sentences. At first I had the impression that he was completely wrong in writing here about work being equal to weight times velocity, and wanted to replace this with 'weight times increased height'. Then I realized that the vertical component of this velocity is equal to the increased height per second. In other words those two sentences are correct if work per second is intended.

By the way, at first I also thought that the escalator should provide extra energy, but I now see that this is not the case.
If the upward force of the escalator to the walker were greater than his weight, there would have been acceleration.
So this force is exactly equal to this person's weight as long as he does not accelerate or slow down.
Last edited by lamemaar on Tue Jan 04, 2011 1:16 pm UTC, edited 2 times in total.

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### Re: Walking on an escalator

lamemaar wrote:By the way, at first I also thought the the escalator should provide extra energy, but I now see that this is not the case.
If the upward force of the escalator to the walker were greater than his weight, there would have been acceleration.
So this force is exactly equal to this person's weight as long as he does not accelerate or slow down.

That works in theory for a self-powered ball on an inclined plane with a belt going around it, but it ignores how humans actually move, which is not at constant velocity, but in jerky bursts.

Now, that does not necessarily mean that the escalator is the source of the extra power. I'd bet it's the person's muscles. That's still ignoring impact force, which is probably what we're looking for in the first place.
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### Re: Walking on an escalator

whereswalden90 wrote:Now, that does not necessarily mean that the escalator is the source of the extra power. I'd bet it's the person's muscles. That's still ignoring impact force, which is probably what we're looking for in the first place.

If the human is walking at a quite constant speed, it's average acceleration is null, so the average force it exerts on the escalator is its weight.
In the perfect world of frictionless escalators in a void, the average force exerted by the escalator is the same as when the human is still, and so is the total work.
If you want to play with solid friction, you need a model of the escalator and of the human movement. Neither are trivial.
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### Re: Walking on an escalator

Should air resistance be included? Because air resistance increases much faster than your speed (speed cubed? I forget) by walking on the escalator you have increased the total amount of work that needs to be done. I would assume that at least some of that additional work would have to be done by the escalator, but I've no clue how to calculate it.

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### Re: Walking on an escalator

It seems to me that we need to break this into a smaller iterative problem. If we consider each step to be to 2 impukses than we can count the energy expenditure this way.

Assumptions.
v1 = 2 units = the speed of the escalator
v2 = 2 units = the peak speed of the pedestrian relative to the escalator
m = mass of pedestrian.

impact 1 : the pedestrian gains Delta-v = to v2 and follows a parabolic arc to the next step.

Energy expenditure by pedestrian = .5m(v2^2 - v1^2) = .5m(16-4) = 6m

Now the fiddly bit comes in. If the pedestrian perfectly selects his trajectory he meets the next step with a velocity = to v1 and repeats the motion. In this scenario the escalator has been slowed by an equal amount of energy and risen to meet the pedestrian having spent an equal amount of energy. the energy spent by the pedestrian and the escalator is equal. but I doubt that. I'm betting that the pedestrian over estimates by a safe and comfortable margin and meets the next step with a velocity less than v1. Now in order to complete the iterative loop the escalator must now work to speed the pedestrian back up to v1. at this point the pedestrian is causing the total amount of work done by the escalator to increase so that the total amount of energy spent by the sum of the two parties increases to greater than the cost of walking. This is equivalent to pogo-sticking up the escalator while riding up the same stair the whole time.