Question about angular momentum conservation

[Banker]

Homework Statement

Question - Models of global warming predict that large sections of the polar ice caps will melt. Explain what effect this will have on the rotation of the Earth, however slight.

Homework Equations

L = Iw

The Attempt at a Solution

Assuming polar ice caps protrude the earth, if they melt, then their height will diminish and radius will decrease. So then moment of inertia should decrease. To maintain angular momentum, then angular velocity must increase, right?

The answer says the opposite. Inertia will increase as 'more mass moves further from the axis of rotation, as ice caps as close to the axis of rotation.'

 

[SteamKing]

Homework Statement

Question - Models of global warming predict that large sections of the polar ice caps will melt. Explain what effect this will have on the rotation of the Earth, however slight.

Homework Equations

L = Iw

The Attempt at a Solution

Assuming polar ice caps protrude the earth, if they melt, then their height will diminish and radius will decrease. So then moment of inertia should decrease. To maintain angular momentum, then angular velocity must increase, right?

The answer says the opposite. Inertia will increase as 'more mass moves further from the axis of rotation, as ice caps as close to the axis of rotation.'

Well, think about what happens after the ice cap melts. Where does all the melt water go? Does it hang around the poles? Or is it able to go elsewhere?

 

[Banker]

Well, think about what happens after the ice cap melts. Where does all the melt water go? Does it hang around the poles? Or is it able to go elsewhere?

Hmm.
So with unmelted ice caps, there are fewer areas where radius is higher than the rest of the earth.
With melted ice caps, the height of water that is melted is redistributed across the earth, so the overall radius increases?

My issue with this is that if this were the case, how would you be certain that the loss of moment of inertia from the ice caps is exactly replaced(also with more) moment of inertia from the redistribution of the water? What if they only cancel each other out? Then L - Iw will be unchanged on the RHS.

 

[SteamKing]

Hmm.
So with unmelted ice caps, there are fewer areas where radius is higher than the rest of the earth.

I don't know what this means. The ice caps are not "higher" than the rest of the earth.

With melted ice caps, the height of water that is melted is redistributed across the earth, so the overall radius increases?

Perhaps you don't realize it, but the shape of the Earth is not a perfect sphere. The polar radius is slightly smaller than the equatorial radius, and the Earth is described as having a slight equatorial 'bulge'.

https://en.wikipedia.org/wiki/Earth

My issue with this is that if this were the case, how would you be certain that the loss of moment of inertia from the ice caps is exactly replaced(also with more) moment of inertia from the redistribution of the water? What if they only cancel each other out? Then L - Iw will be unchanged on the RHS.

Your initial assumption, that the polar ice caps 'protrude' the earth, whatever that means, is somewhat shaky.

Remember, the north polar cap is sitting in the Arctic Ocean, which means it's floating. Ice is only slightly less dense than seawater, so most of the ice is going to be submerged in the water, and there won't be much 'protruding' going on. Also, when water freezes, it expands slightly, which is why ice floats.

The ice cap at the South Pole is a little more complicated to assess, since a lot of it is sitting atop the continent of Antarctica, and the rest is arranged in various ice shelves placed around the periphery of that land mass. The weight of all that ice sitting directly on the continent of Antarctica has depressed the surface of the land somewhat. The ice cap averages 1.6 km in thickness, and much of Antarctica lies more than 3000 meters above sea level.

https://en.wikipedia.org/wiki/Antarctica

 

[haruspex]

Assuming polar ice caps protrude the earth,

Not relevant. The Earth spins about an axis through the poles. The moment of inertia of a given mass that's part of the Earth only depends on its straight line distance from that axis. For the purposes of the problem, you can safely treat all the ice as being located right at the pole, so currently has no moment at all. Whether that's at sea level or 100 miles high makes no difference.

 

[Banker]

Not relevant. The Earth spins about an axis through the poles. The moment of inertia of a given mass that's part of the Earth only depends on its straight line distance from that axis. For the purposes of the problem, you can safely treat all the ice as being located right at the pole, so currently has no moment at all. Whether that's at sea level or 100 miles high makes no difference.

So how would that fit in with the question? Wouldn't Iw stay the same then?

 

[Banker]

I don't know what this means. The ice caps are not "higher" than the rest of the earth.

I mean like they are above sea level, so when calculating moment of inertia, the radius would be larger than it would be if the ice caps were at sea level.

 

[SteamKing]

I mean like they are above sea level, so when calculating moment of inertia, the radius would be larger than it would be if the ice caps were at sea level.

But since the axis of rotation for the Earth goes thru the poles, the location of the ice caps relative to sea level has a minimal effect on the moment of inertia of the earth.

See haruspex' Post #5. The radius measured normal to the axis of rotation is what determines the moment of inertia of a body.

 

[HallsofIvy]

You seem to be thinking that the distance from the center of the Earth is important. It is not- it is distance from the axis of rotation that is relevant. Ice at the north pole is right on the axis so does not contribute much to the angular momentum. Water that is farther from the north pole is farther from the axis of rotation so contributes more to the angular momentum.

 

[haruspex]

So how would that fit in with the question? Wouldn't Iw stay the same then?

Stand at one pole. What is your moment of inertia about an axis through the poles?
Now move to the equator. Answer the same question there.

 

[Banker]

You seem to be thinking that the distance from the center of the Earth is important. It is not- it is distance from the axis of rotation that is relevant. Ice at the north pole is right on the axis so does not contribute much to the angular momentum. Water that is farther from the north pole is farther from the axis of rotation so contributes more to the angular momentum.

Stand at one pole. What is your moment of inertia about an axis through the poles?
Now move to the equator. Answer the same question there.

Ahh, now I get it. That's where I was getting it wrong. Yeah so the moment of inertia of the polar ice caps at the axis of ratation is small. But when it melts, water is redistributed away from the pole and r increases, hence moment of inertia increases. So then to keep angular momentum constant, angular velocity would decrease!

 

[haruspex]

Ahh, now I get it. That's where I was getting it wrong. Yeah so the moment of inertia of the polar ice caps at the axis of ratation is small. But when it melts, water is redistributed away from the pole and r increases, hence moment of inertia increases. So then to keep angular momentum constant, angular velocity would decrease!

That's it.