Direction of instantaneous axis in rigid body dynamics

AI Thread Summary
The discussion focuses on the dynamics of rigid bodies, particularly the concept of instantaneous axes of rotation. It explains that the motion of a rigid body can be viewed as a combination of the motion of its center of mass and its rotation about an instantaneous axis through the center of mass. An example involving a thin uniform disc connected to a massless stick illustrates how these axes can change orientation during rotation. The cases presented show the disc's orientation affecting the direction of the instantaneous axis when rotated about the z-axis. The complexity of these concepts suggests that they are more suited for an in-depth textbook treatment rather than a brief forum post.
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The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of
mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed)
horizontally at its rim to a massless stick, as shown in the figure. When the disc-stick system is rotated
about the origin on a horizontal frictionless plane with angular speed w, the motion at any instant can be
taken as a combination of (i) a rotation of the centre of mass of the disc about the z-axis, and (ii) a rotation
of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the
changed orientation of points P and Q). Both these motions have the same angular speed w in this case.
Now consider two similar systems as shown in the figure: Case(a) the disc with its face vertical and
parallel to x-z plane; Case (b) the disc with its face making an angle of 45° with x-y plane and its
horizontal diameter parallel to x-axis. In both the cases, the disc is welded at point P, and the systems are
rotated with constant angular speed wabout the z-axis.what is the direction of instantaneous axis (passing through the centre of mass in case (a) and (b).please explain all the concepts in detail.i will be highly thankful.
 
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