Angular momentum question

[Matty1973]

Saw this cartoon on www.xkcd.com and am not sure if it is true.

The idea is that a person spinning round gives/takes angular momentum to the planet. Is this really true? Would the amount given when the person speeds up be taken back again when slowing down? Is there a way round this?

Also is it correct that this would not work at the equator and would work one way in the Northern hemisphere and the other way in the Southern hemisphere?

 

[nathan]

Saw this cartoon on www.xkcd.com and am not sure if it is true.

The idea is that a person spinning round gives/takes angular momentum to the planet. Is this really true? Would the amount given when the person speeds up be taken back again when slowing down? Is there a way round this?

Also is it correct that this would not work at the equator and would work one way in the Northern hemisphere and the other way in the Southern hemisphere?

Angular momentum is a conserved quantity. Thus if something increases its angular momentum, something else must be decreasing its.

if I spin round in place -- around a (locally) vertical axis -- the Earth will rotate around the same axis in the opposite direction. Of course because Earth is much more massive, its rate of rotation due to this will be miniscule.

As the Earth's major axis of rotation is around a polar axis, the analysis gets complicated, but one can decompose my local rotation into a polar and equatorial components. The polar component will change the day length, the equatorial component will, I think, change the precession.

The size of the effect is similar to how the center of Earth moves when I jump up in the air -- less than an atom I'd guess.

 

[Ziggurat]

The size of the effect is similar to how the center of Earth moves when I jump up in the air -- less than an atom I'd guess.

Let's do the numbers. Suppose we take a 75 kg person, have then jump 0.5 meters up. How much does the earth move? Well, the combined center of mass remains in the same place. The person's mass displacement is 37.5 kg m, so the earth's mass displacement will be 37.5 kg m in the other direction. The earth itself is 5.97x10²⁴ kg, so the displacement would be 6.3x10^-24 meters. So... considerably less than an atom (~10^-10 m). In fact, that's considerably smaller than a proton (~10^-15 m).

Calculating the difference a human body's rotation would make in the rotational period of the earth should produce similarly absurdly small numbers.

 

[Matty1973]

Angular momentum is a conserved quantity. Thus if something increases its angular momentum, something else must be decreasing its.

This is what I'm getting at - It's agreed that when the person starts spinning they take some of the angular momentum from the Earth thus slowing it down - but does that mean in order to keep the Earth at it's new slowed down speed the person has to keep spinning?

It's a bit like saying in the second example that when the person jumps the Earth moves away, but when the person falls back to the ground does the Earth moves back again to it's original position?

 

[dasmiller]

Calculating the difference a human body's rotation would make in the rotational period of the earth should produce similarly absurdly small numbers.

Actually, it's worse than that. When you're jumping up & down, you're competing with Earth's mass. When you're spinning, you're competing with Earth's inertia, which is proportional to mass and radius^2.

So, according to the back of my excel sheet, the Earth should have an MOI of ~8E37 kg-m^2. The girl is more like 25 kg-m^2. If the Earth wasn't spinning, and we put her at the north pole, had her spin at 2 rev/sec for the entire history of the universe, then the Earth's equator would have moved about 1E-11 meters by now, or something less than the diameter of a hydrogen atom. Unless my math is off, which is possible.

 

[dasmiller]

This is what I'm getting at - It's agreed that when the person starts spinning they take some of the angular momentum from the Earth thus slowing it down - but does that mean in order to keep the Earth at it's new slowed down speed the person has to keep spinning?

Yes.

It's a bit like saying in the second example that when the person jumps the Earth moves away, but when the person falls back to the ground does the Earth moves back again to it's original position?

Yes, if we look at just the 2-body problem. In theory, things get more complicated when we take into account the other bodies in the universe, but since the first-order effect is so tiny, it's hard to picture a significant second-order effect.

 

[ImaginalDisc]

Yes, it's true.

No, it's not actually remotely significant to the Rotation of the Earth.

It's kind of like asking if going for a swim makes sea levels rise. Yes, you are displacing water and thus making sea levels rise but the amount you contribute compared to the total is insignificant.

I'm pretty sure that just ruins the joke which is true.

 

[nathan]

This is what I'm getting at - It's agreed that when the person starts spinning they take some of the angular momentum from the Earth thus slowing it down - but does that mean in order to keep the Earth at it's new slowed down speed the person has to keep spinning?

yes. when they reduce their angular momentum, something else must increase its.