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Coriolis Torque On a Spinning Object

  1. Jan 26, 2015 #1
    1. The problem statement, all variables and given/known data
    The Coriolis force can produce a torque on a spinning object. To illustrate this, consider a horizontal hoop of mass m and radius r spinning with angular velocity w about its veritcal axis at colatitude theta. Show that the Coriolis force due to the earth's rotation produces a torque of magnitude mwWr^2sin[theta] directed to the west, where W is the earth's angular velocity.

    2. Relevant equations
    Coriolis force = 2mr' X W
    Torque = F X d


    3. The attempt at a solution
    r'=rXw

    Fcor= 2m(rXw) X W
    = 2mrwW

    Torque = Fcor X r
    = 2mr^2wWsin[theta]

    I'm supposed to get half of this value somehow, and the direction is not at all clear and from my diagram is constantly changing...
     
    Last edited by a moderator: Jan 27, 2015
  2. jcsd
  3. Jan 26, 2015 #2

    TSny

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    This equation is for a particle of mass m moving with linear velocity ##\dot{\mathbf{r}}##. Each mass element of the hoop should be treated as a particle with its own velocity vector. (Different mass elements have different position and velocity vectors.)
     
  4. Jan 26, 2015 #3

    I thought that rewriting r' as w X r with both w and r being vectors, would take care of this? If not, how would I go about treating each point independently?
     
  5. Jan 27, 2015 #4

    TSny

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    You'll need to introduce a coordinate system and express the position and velocity vectors of a mass element with respect to the coordinate system. For example, you could introduce Cartesian axes with origin at the center of the hoop and with the z axis along the axis of rotation of the hoop, the x axis pointing east, and the y axis pointing north.

    A mass element of the hoop will be at some angle ##\phi## to the x-axis. How would you express the position and velocity vectors of this mass element?
     

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    Last edited: Jan 27, 2015
  6. Jan 27, 2015 #5
    [r cos[phi] x direction + r sin[phi] y direction] X w z direction
     
  7. Jan 27, 2015 #6

    TSny

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    I'm guessing that this is an expression for the velocity vector of the mass element located at angle ##\phi##. Does the expression have the correct overall sign? Can you simplify by carrying out the cross product?

    Note that math symbols are available by clicking on the ##\Sigma## tab.
     
  8. Jan 27, 2015 #7
    Thank you for that, I couldn't remember how i had made symbols before. Anyway, I understand it now, thanks a lot!
     
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