Question

The common motion of a rigid body can be assumed to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion along an instantaneous axis which is passing through the centre of mass. These axes need not be stationary. Let us assume one example. A thin uniform disc is welded or rigidly fixed at its rim to a mass less stick horizontally, as shown in the figure. If the disc-stick system is rotated about the origin on a horizontal frictionless plane having an angular speed ω, the motion at any instant can be considered as the combination as (i) disc rotating at the centre of mass of the disc with respect to the z-axis, and (ii) a disc rotating at the disc through an instantaneous vertical axis passing through its centre of mass, as is seen from the varied orientation of points P and Q. Both these motions have the identical angular speed $\omega$ in this case.
Now let us assume two identical systems as shown in the diagram: Case (a) the disc with its face vertical and parallel to {it x-z} plane; Case (b) the disc with its face creating an angle of $45{}^{\circ }$ with {x-y} plane and its parallel horizontal diameter to the x-axis. The disc is welded at point P, and the systems are rotated with a fixed angular speed $\omega$ with respect to the z-axis in both the cases.
Which of the statements given below will be correct about the angular speed along the instantaneous axis (passing through the centre of mass)?

A. the value is $\sqrt{2}\omega$ for both the cases.
B. for the case (a) the value is $\omega$ and for case (b), it is ${\displaystyle \frac{\omega }{\sqrt{2}}}$
C. the value is $\omega$ for case (a) and $\sqrt{2}\omega$ for case (b)
D. the value is $\omega$ for both the cases.

Answer 1

Hint: Angular velocity of an object undergoing any rotational motion with respect to any of the axes around the object will be the same as that of angular velocity about its centre of mass. These all will help you in finding the answer.

Complete answer:
Angular velocity of an object undergoing any rotational motion in accordance with any of the axes around the object will be identical as that of angular velocity about its centre of mass.
Here we can see that for both cases, angular velocity about the z-axis is the same. That is $\omega$. Therefore the angular velocity about their centre of mass will be also $\omega$ for the two.

Note:
Angular velocity is defined as the measure of how fast a body can rotate or revolve relative to another point. Which means how fast the angular position or the orientation of an object varies with time. Orbital angular velocity and spin angular velocity are the two types of angular velocity. Spin angular velocity is defined as the measure of how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity is basically the measure of how fast a point object revolves about a fixed origin. This means that the time rate is a variation of its angular position in accordance with the origin.