1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question about angular momentum conservation

  1. Jan 28, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    L = Iw

    3. 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.'
     
  2. jcsd
  3. Jan 28, 2016 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    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?
     
  4. Jan 28, 2016 #3
    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.
     
  5. Jan 28, 2016 #4

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    I don't know what this means. The ice caps are not "higher" than the rest of the earth.
    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
    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
     
  6. Jan 28, 2016 #5

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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.
     
  7. Jan 29, 2016 #6
    So how would that fit in with the question? Wouldn't Iw stay the same then?
     
  8. Jan 29, 2016 #7
    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.
     
  9. Jan 29, 2016 #8

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    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.
     
  10. Jan 29, 2016 #9

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  11. Jan 29, 2016 #10

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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.
     
  12. Jan 29, 2016 #11
    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!
     
  13. Jan 29, 2016 #12

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    That's it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Question about angular momentum conservation
Loading...