Question

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 $\omega$, 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 $\omega$ in this case.

Now consider two similar systems as shown in the figure.
Case (a): The disc with its face vertical and parallel to $x-z$ axis;
Case (b): The disc with its face making an angle of $45^{\circ }$ with $xy$-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 $\omega$ about the $z$-axis.

Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

• A. It is $\sqrt{2}\omega$ for both cases.
• B. It is $\omega$ for case (a); and $\frac{\omega }{\sqrt{2}}$ for case (b).
• C. It is $\omega$ for case (a); and $\sqrt{2}\omega$ for case (b).
• D. It is $\omega$ for both cases.

Answer 1

Answer: (D)

As depicted in the figure shown here, when the system, as a whole, turns by $180^{\circ }$, the disc also turns by $180^{\circ }$ about vertical axis. Hence, for both Cases (a) and (b), the angular speed about the instantaneous axis that passes through the centre of mass is $\omega$.