Question

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 the 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 the cases

Answer 1

Answer: (D)

(i) Every particle of the disc is rotating in a horizontal circle.
(ii) Actual velocity of any particle is horizontal.
(iii) Magnitude of velocity of any particle is
$v=r\omega$
where, r is the perpendicular distance of that particle from actual axis of rotation (z-axis).
(iv) When it is broken into two parts then actual velocity of any particle is resultant of two velocities
$v_1=r_1{\omega }_1$ and $v_2=r_2{\omega }_2$
Here,
$r_1$= perpendicular distance of centre of mass from z-axis.
${\omega }_1$= angular speed of rotation of centre of mass from z-axis.
$r_2$= distance of particle from centre of mass and
${\omega }_2$= angular speed of rotation of the disc about the axis passing through centre of mass.
(v) Net v will be horizontal, if v, and v2 both are horizontal. Further, v, is already horizontal, because centre of mass is rotating about a vertical z-axis. To make v2 also horizontal, second axis should also be vertical.