Ah, AuraSeer, you have seen my aura truly–but, ummmm… you have the sign reversed. Long, long ago in a galaxy far, far away, I was a physics professor and I did spend quite a bit of time in Physics 101. I liked it, too, and was good at it. But, alas, I was incompetent at academic politics and had to go and make my way in the Real World ™. Nowadays I spend way too much time in the Real World building machines that raise and lower things, and not enough engaged in the elegance of pure physics discussing tricky concepts with bright young physics majors like our friend Konrad. You read me right, and I thank you for it.
Interestingly, the main point in my first post was that a pure basic physics argument could not answer the question at hand. You have to know how real muscles work to know if they need to expend energy (ie, burn fuel) in order to lower a mass through gravity. Konrad’s point is that, with suitable idealization, pure physics can give an answer–and that the answer is that it takes equal energy to raise and lower a mass. He’s now given us a lucid and elegant explanation of that assertion.
I’m very interested in this question because I keep seeing it pop up, usually in the form, “Did you know that, as strange as it sounds, it actually takes more energy to descend a staircase (mountain, hill, ladder, etc) than it does to climb it? Isn’t that cool?” Usually, however, the respondent cannot justify the assertion because he or she lacks the facility with physics. Now here’s Konrad, clearly bright and articulate, a physics major, making a somewhat more nuanced claim: it takes exactly the same energy to descend as to climb. I want to know how he draws this conclusion.
So… Konrad. If you’ll permit me, let’s parse your theory.
- | |As far as the block and the finger goes, A doing work on B means to me that B is absorbing energy.| |
Yes, it means the same to me, too. Your earliest few posts indicated that we agreed on this, but later material gave me pause. I hope you see my point about the sign. Note that at constant velocity, B is not speeding up, but it is still absorbing energy (and dissipating it as heat). Hold that thought.
- | |Yes, it does take energy to lower something down.| |
This is the nub of the discussion. My claim is that it can take energy, but it does not necessarily. It depends on the mechanism employed. These fall into three classes: regenerative (reversible processes required), friction (simplest irreversible process), or “clutch slipping” (energy required to regulate the amount of friction).
- | |How? Well look at it this way. Suppose instead of doing it slowly we let it speed up. Now we have something moving with a speed relative to us. What’s one way to stop it? Well we could launch another object at it of the same mass going the same speed. The two will collide and stop. (Without it doing any work on your arm. In fact your arm does work on the second object.)| |
Very nice! This description seems to me to be perfectly rigorous. It’s just that it isn’t general. It describes one method of the class of methods I’ve called “clutch-slipping.” Suppose instead of launching an interceptor of equal mass and speed, you launch one of much less mass–a bullet to stop a brick, as it were. With no bouncing, the interceptor must have the same momentum as the object you are trying to stop, so as mass goes down, speed goes up proportionally. But energy goes up as the square of the speed, so the lightweight interceptor needs to be supplied with much more energy to get up enough momentum to do its job. At the other extreme, suppose you use an interceptor that is much more massive than the object you are trying to stop–a brick to stop a spitball, as it were. Now to match momenta, the brick hardly needs any speed at all, and has near zero energy to bring to the party. This is “clutch slipping” at the limit of pure friction.
- | |So we have used energy to stop an object, and we have not gained any ATP from this. Of course all the energy goes into heat. (The amount of heat energy will be twice what you would absorb if you could somehow extract all the energy from the first object.)| |
Right. But perhaps a more rigorous statement would be that we have used momentum to stop an object. The energy used depends on the mass of the interceptor object chosen. By the way, nobody ever claimed that real muscles were reversible and you could gain ATP from descending under muscle power. If I gave this impression, I apologize for being unclear.
- | |If you are decreasing the speed of something relative to you, you can theoretically absorb the energy instead of using more energy to slow it down. That is what is meant by the object doing work on whatever is slowing it down.| |
Yes. Except that in your example of “using more energy to slow it down” you still end up absorbing the kinetic energy of the object being stopped. The object being stopped does work on the interceptor. Since you specified an inelastic collision, all that work goes straight to heat, which is absorbed by the matter of the two objects now in contact. I’d say that the energy of the object being stopped was absorbed by the object doing the stopping at the incidental expense of the original energy of the object doing the stopping.
- | |But that doesn’t mean you have to absorb it, a human doesn’t.| |
This I don’t understand. In your early posts you indicated that you do absorb it, but that it costs you energy to do so. In your discussion of paragraph 3 you tell what you mean. If something in the problem doesn’t absorb the energy of the object being stopped (or slowed, or lowered), where does the energy go? You say it goes into heat. I agree. Does that not count as being “absorbed” by the body being heated? Do we have a definition-of-terms problem here?
Or is the issue that the energy is absorbed (as heat) by the interceptor particles, but not by the arm that launched them. If so, I’d have to point out that in a muscle, the interceptor particles and the agency that launches them are in thermal contact.
Finally, it’s a non-sequitur to go from “doesn’t mean you have to” to “doesn’t.” To assert that the human doesn’t absorb energy because it doesn’t have to does not follow. You have to look at the mechanism of real muscle contraction to decide the case.
- | |When you are walking down stairs, gravity isn’t doing any work on you unless you are speeding up and therefore absorbing its energy.| |
I don’t see how this statement follows. If gravity makes you move, it is doing work on you no matter what your speed profile. Work represents the conversion of gravitational potential energy into kinetic energy if you just fall, or heat if you lower yourself gradually. Either way, you are absorbing energy under the terms of paragraph 1. (There’s also the third possibility of absorbing energy and storing it in springs or batteries or whatever, but we’ve agreed that that is an irrelevancy for muscles since they are irreversible machines.)
Furthermore, you very astutely used the abstraction that gradual lowering consists of an infinite number of infinitesimal drops and catches. So even in your example, gravitational potential is first converted to KE, and then reduced to heat by collision with your interceptor particle. Gravity (or more precisely, the gravitating bodies) are most certainly doing work on each other in raising and lowering processes.
- | |You are using equal but opposite energy instead.| |
Fundamentally, the energy used is more or less irrelevant. It is momentum that is doing the heavy lifting, so to speak. If the mass of the interceptor is arbitrary, the energy of the interceptors could be anything–and whatever it is, it all gets absorbed as heat during the collision.
Konrad, I think I understand