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Question concerning rigid bodies

  1. Aug 20, 2004 #1
    The following comes from Landau's Mechanics, pages 97 - 98.

    For a particle in a rigid body, v = V + W x r -- (1)

    where for some origin O of the moving body measured in the "fixed" system of
    co-ordinates, v = particle's velocity in body in the "fixed" system, V = velocity of O in "fixed" system , W is the body's angular velocity in *fixed system", x is a cross product and r the particle's radial vector within body measured from O.

    For another origin, O' distance a from O, r = r' + a, and substituting in
    (1) gives:

    v = V + W x a + W x r'. The definition of V' and W' shows that v = V' + W' x
    r' and so

    V' = V + W x a, W' = W -- (2)

    He then says that the first part of (2) shows that if V and W are
    perpendicular for a choice of origin O, then V' and W' are also
    perpendicular for O'. Why?

    Thanks in advance.
  2. jcsd
  3. Aug 20, 2004 #2


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    If you dot (*) equation (2a) with W, you get

    W*V' = W*(V + W x a)=W*V +W*(Wxa) = W*V + 0 , since W is orthogonal to any vector crossed with W.

    So, W*V' = W*V.

    Since W'=W, by equation (2b), the left hand side is rewritten so that
    W'*V' = W*V

    So, if W*V=0, then W'*V'=0.

    I think this is correct.
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