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Question:

The general motion of a rigid body can be considered to be a combination of (i) a motion of its center of mass about an axis, and (ii) its motion about an instantaneous axis passing through the center 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 ω, the motion at any instant can be taken as a combination of (i) a rotation of the center of mass of the disc about the z-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its center of mass (as is seen from the changed orientation of points P and Q. Both these motions have the same angular speed ω in this case. Now consider two similar systems as shown in the figure: Case (a) the disc with its face vertical and parallel to the x-z plane; Case (b) the disc with its face making an angle of 45° with the x-y plane and its horizontal diameter parallel to the x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angular speed ω about the z-axis. Which of the following statements about the instantaneous axis (passing through the center of mass) is correct?

It is horizontal for case (a); and is at 45° to the x-z plane and is normal to the plane of the disc for case (b).

It is vertical for case (a); and is at 45° to the x-z plane and is normal to the plane of the disc for case (b).

It is vertical for both the cases (a) and (b).

It is vertical for case (a); and is at 45° to the x-z plane and lies in the plane of the disc for case (b).

Solution:

When the complete system turns by 180°, the disk also turns by 180° about its vertical axis. Hence angular speed is ω for both cases since the body is rigid. Axis of rotation is vertical in both the cases, as ω is same in both the cases.