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11/25/08 3:14 PM
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groundfighter2000
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If all the polar bears on earth migrated to the north pole how would this affect the length of the day? Im not sure but I thought it would increase rotational inertia and therefore decrease angular velocity so the length of the day would increase but Im not sure
12/8/08 1:51 AM
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AlbertEinstein
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You need to figure out how this would affect the moment of inertia of the planet. The moment of inertia is defined as an integral, but you also need to realize that a lot of mass is leaving other areas of the Earth and concentrating at the poles. In short The moment of inertia will decrease and the Earth will spin faster, hence a shorter day.
12/8/08 2:02 AM
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AlbertEinstein
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Actually, I'm not sure. The moment of inertia depends of the square of the distance from the center of mass. As the polar bears move along the Earth, that distance should remain constant (assuming a perfectly spherical Earth), so it shouldn't change anything. I don't know. I'm too drunk right now to be on the Edcutional Ground. Sorry, wish I could help more.
12/8/08 2:17 AM
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AlbertEinstein
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Ok I thought about it for a little bit, but like I said I've been drinking all day. Under the assumption of a perfectly spherical Earth, the moment of inertia shouldn't change. But if you take the more accurate description of an elliptical Earth, then the moment of inertia depends on how far the bears are from the center of the Earth. Need more details.
12/8/08 5:42 PM
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AlbertEinstein
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Ok I've sobered up now and I have an answer for you. The moment of inertia depends on the square of the distance from the axis of rotation, not the distance from the center of mass. So my original answer was correct. As the bears move toward the poles, the moment of inertia of the Earth will decrease, and it will spin faster, i.e. shorter days.
12/8/08 11:24 PM
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groundfighter2000
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well the test already happened, thank you for the response, i answered wrong according to ur answer ( and ur answer makes sense) getting the test back tommorow night so hopefully i didnt fuck up the rest of it
12/10/08 7:41 AM
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Willybone
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The moment of inertia depends of the square of the distance from the center of mass.

Axis of rotation, IMO.
Bears move toward the axis, rotational intertia decreases, earth spins faster.
Think of a skater spinning, and then she pulls in her arms. Her speed increases. This happens even if she pulls her arms in and over her head, away from her center of mass but closer to her axis of rotation.
12/11/08 2:46 PM
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Voolf
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Edited: 12/11/08 2:52 PM
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Ok guys, I've figured it out. Angular momentum of the Earth/Bear system is conserved:

L = L_e + L_b = constantwhere the L's are vectors.

Picking an appropriate coordinate system, assuming a spherical Earth, we'd get that the L's point up, along the axis of rotation. L_e is much larger than L_b due to the difference in mass; whatever L_b does it won't affect the basic orientation of L.

L_b = m*w*R^2*sin(theta)

where m is mass of bears, w is angular frequency of rotation, R is the radius of the Earth, and theta is the angle of the bears to the axis of rotation.

At the north and south poles, where theta is 0 or pi, L_b=0. So L_e (and thus the frequency of rotation) is the same when the bears are at either pole.As the bears travel from the south pole, L_b increases as theta goes from 0 to pi. As total angular momentum is conserved, L_e must decrease accordingly, thereby decreasing the frequency of rotation, if only so slightly.

This effect has its maximum at the equator, which corresponds to theta = pi/2, where sin has its maximum, and then decreases as theta approaches pi.

Smarty Powntsed!
12/12/08 4:19 AM
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Briscoe
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That's what I was going to say but you beat me to it.

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