Figure skaters increase spin rpm by pulling in their arms. How ?

The other day I was chuckling at a spoof Newtonian mechanics theory that the earth may spin faster on its axis due to deforestation. Just as a figure skater’s rate of spin increases when the arms are brought in close to the body, the cutting of tall trees may cause our planet to spin dangerously fast.

Then I realized that I don’t really understand the physics behind why figure skaters DO speed up - but I know a bunch of people that would.

I guess the final sub-question has to be, “Is there any, even to the smallest unmeasurable degree, an element of truth behing the joke?” I suspect not as the skater thing is probably something to do with her doing work in pulling her arms in which converts into something or other but happy to get the straight dope.

It’s easier to think about these physics using the reverse process. Let’s say you are spinning a sturgeon on the end of a pole around your head. You put X amount of momentum into it to give it Y speed and the resulting Z revolutions per minute. If you allow the line to real itself out, without adding to X, your sturgeon will still be travelling no more than Y. Since the line is longer, the circumference is longer, and the sturgeon will circle your head less than Z in a given minute. Reeling your fish in will increase its rpm.

The same is true with the skater or the earth. Not all of the mass is changing distance from the center of mass, but some of it is. The skater spinning example would be more dramatic if she were holding a barbel in each hand; sometime when you’re on a spinning chair you should try it - spin yourself with your legs extended, and then pull them back before you knock over any potted plants. Your rpm should increase.

The cutting trees example is a joke only because it’s an insignificant fraction of the earth’s mass, but some folks have theorized that, since dams hold tons and tons of water, and they tend to be located closer to the tropics than to the poles, they will “reel out the sturgeon”, i.e., increase the amount of mass near the equator, slowing the earth down. Even the difference caused by these tons and tons of water is a matter of an almost immeasurably small amount of time, but scientists like to deal with immeasurably small stuff, I guess.

I guess though the dams and water thing slowing the earth is only to the extent that the radius of the earth is slightly greater at the equator that at the Poles.

“Let’s make this less complicated by thinking about it in terms of a sturgeon”.

The effect is actually greater than implied by Boris’s post. He stated that the speed of the mass (I refuse to bring fish into this!) stays the same when the radius changes. In fact, if the radius decreases, the speed will actually increase. There’s a conserved quantity called angular momentum, which, for the case of circular motion, can be calculated by multiplying the massspeedradius. Since the product of these three must remain constant, and the mass isn’t changing, if the radius decreases, the speed must increase, and vice versa. Hence, if you had a spinning apparatus with two arms, and all of the mass at the ends of the arms, and you cut the length of the arms in half, the angular speed (rpm) would quadruple, not double.

I’ve decided to put everybody’s mind at ease, about the deforestation thing. In yet another effort to avoid doing any real work, I computed how much the Earth’s rotation would be affected by, say, the felling of 10,000,000 Giant Sequoias at the Equator (where it would make the most difference). Anyway, I had to make up a couple of numbers, but the answer I got states that if some psychopathic logger were to pull off this feat, days would become a whopping 0.036 nanoseconds shorter. If you thought there wasn’t enough time in the day now

I knew something was fishy.

And the day is getting longer, around 5 nanoseconds every day, on average. Unless my fingers^Wcalculator is in error.

[singing] 'Cause that’s what Brian Boitano’d do! [singing]

Thanks, Chronos. It looks like I had forgotten yet another bit of physics and geometry.

Now that I think about it, I’m worried that I have the dam thing backwards. Maybe they tend to be located nearer the poles than the tropics, and will speed up the Earth’s rotation by moving liquid filled with trout, salmon, shad, and crawdads closer to the polar axis.

Of course the fun thing that no here has considered is that all those cut down trees and new lakes are going to change the planet’s center of gravity…Our spinning is gonna start getting more offbalance and the planet’s gonna start flying around.

PS. If you listen closely at the north pole, you can hear a squeaking noise…thats not the ice settling, its the planets ball bearings starting to go from the strain. Scientists don’t like to talk about it because they want to avoid panic.

PPS. If you want to help, I have some of the special oil that is used to lubricate the bearings, only 19.99 an oz.

Chronos
Unless I am confused (and that’s certainly possible), Boris B is just using linear momentum to explain angular momentum. If the linear monentum is constant, the angular momentum will also be constant as the radius changes. I think you must mean angular velocity when you say that the “speed will actually increase”. I don’t think that the linear speed (=angular velocity * radius) changes. Does it?

Chronos
Unless I am confused (and that’s certainly possible), Boris B is just using linear momentum to explain angular momentum. If the linear monentum is constant, the angular momentum will also be constant as the radius changes. I think you must mean angular velocity when you say that the “speed will actually increase”. I don’t think that the linear speed (=angular velocity * radius) changes. Does it?

Yeah, he was using conservation of linear momentum, and his solution works if you let the mass continue in a straight line, but he was applying it to circular motion. In that case, linear momentum of the mass is not conservered, as there’s a force on it, but it’s a central force, so angular momentum is still conserved.

Yes, it does, as Chronos pointed out, and Boris B admitted. In the reference frame of the rotation, there is a centrifugal force that has to be overcome as the sturgeon is pulled inward. That is work, which is converted to kinetic energy of the fish–thereby increasing its velocity.