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Insights Frequently Made Errors in Mechanics - Moments - Comments

  1. May 10, 2015 #1

    haruspex

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    Last edited: Dec 25, 2016
  2. jcsd
  3. May 10, 2015 #2

    Drakkith

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    Nice post, Haruspex. I'll have to come back to this when I start my physics and mechanics classes.
     
  4. May 10, 2015 #3
    Nice work haruspex!
     
  5. Jul 30, 2015 #4

    andrewkirk

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    Having read this, my future calcs involving torques and moments of inertia will be much faster and easier.
     
  6. Jun 15, 2016 #5
    Nice article. I just want to add some comment on the equation of momentum for rigid body and on some very frequent errors that arise in this regard.
    The equation of momentum is
    $$J_A\dot{\boldsymbol \omega}+\boldsymbol\omega\times J_A\boldsymbol\omega=\boldsymbol M_A.\qquad (*)$$ Here ##J_A,\boldsymbol\omega## are the operator of inertia about the point ##A## and the angular velocity of the rigid body respectively; ##\boldsymbol M_A## is the torque about the point ##A## applied to the rigid body.

    But what is the point ##A##? If ##A## is a stationary point of the rigid body or its center of mass then equation (*) is correct.
    In general, it is incorrect to use formula (*) for ##A## to be instantaneous centre of rotation; it is incorrect even for planar problems.

    Another frequent error is concerned to the term ##\boldsymbol\omega\times J_A\boldsymbol\omega##. This term is equal to zero identically in planar problems. But one can not forget it in essentially 3D problems.
     
    Last edited: Jun 15, 2016
  7. Jun 15, 2016 #6

    vanhees71

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    One should also note that the tensor of inertia, ##J_A##, must refer to the body-fixed point ##A##.
     
  8. Jun 29, 2016 #7
    sure


    accidentally I came across an article
    https://www.jstor.org/stable/2973359?seq=1#page_scan_tab_contents
    perhaps It should be noted about a general formula. Let a point ##A## be any point of the rigid body. Then
    $$J_A\dot{\boldsymbol\omega}+\boldsymbol\omega\times J_A\boldsymbol\omega+m\boldsymbol{AS}\times \boldsymbol a_A=\boldsymbol M_A;$$
    where ##S## is the center of mass, ##m## is the mass of the rigid body, ##\boldsymbol a_A## is the acceleration of the point ##A##.
     
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