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Some questions on rigid modies in Landau's Mechanics book

  1. Feb 17, 2005 #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 the "fixed" system, V = velocity of
    O in "fixed" system , W is the particle's and body's angular velocity in O,
    x is a
    cross product and r the particle's radial vector in 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)

    ???But the earlier substitution r = r' +a is only correct if the axes of O
    an O' are parallel and not for some arbitary orientation of O', yes?

    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'. Formula (1) shows
    that in this case the velocities v of all points in the body are
    perpendicular to W.

    This is easy to prove by taking the dot product of both sides with W and
    using the fact that V.W=0 if perpendicular.

    Next he says it then follows from (1) that in this case the velocities v of
    all points in the body are perpendicular to W.
    This is easy to show since v.W = V.W + (Wxr).W = 0.

    He says it is then possible to choose an origin O' such that V' is zero so
    that the motion of the body at the instant considered is pure rotation about
    an axis through O'.
    How do you show this to be true?

    Thanks.
     
  2. jcsd
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