Some questions on rigid modies in Landau's Mechanics book

[jmc8197]

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.

 

[LightHero]

From equation (2), we can see that the velocity of the origin O' in the fixed system is V' = V + W x a. Since we are looking for an origin O' such that V' is zero, then it follows that V = -W x a. This implies that the velocity of the origin O' is zero and thus the motion of the body at the instant considered is pure rotation about an axis through O'.

 

[charng]

I would first like to commend you for your thorough understanding and analysis of Landau's Mechanics book. Your questions show a deep understanding of the concepts presented and a desire to clarify any potential misunderstandings.

To answer your first question, you are correct that the substitution r = r' + a is only valid if the axes of O and O' are parallel. This is because the cross product in equation (1) is only valid for perpendicular vectors. However, this does not affect the validity of the equations (2) that follow. The important point to note is that the equations (2) hold true for any choice of origin O'.

To show that the velocities v of all points in the body are perpendicular to W, we can take the dot product of both sides of equation (1) with W. This gives us:

v.W = (V + W x r).W = V.W + (W x r).W

Since V and W are perpendicular for a choice of origin O, V.W = 0. And since the cross product of two perpendicular vectors is perpendicular to both, (W x r).W = 0. Therefore, v.W = 0 and we can conclude that the velocities v of all points in the body are perpendicular to W.

Now, to show that it is possible to choose an origin O' such that V' is zero, we can use the fact that the cross product of two parallel vectors is zero. This means that if we choose O' to be on the axis of rotation (which is perpendicular to both V and W), then V' will be zero. Therefore, the motion of the body at that instant will be pure rotation about an axis through O'.

I hope this clarifies any doubts you may have had about the equations presented in Landau's Mechanics. Keep up the good work in your studies!