Discussion Forum



1)

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 ω, 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 .vertlcal axis passing through its centre of mass (as is seen from the changed orientation of poinis P and Q). Both these motions have the same angular speed $\omega$ in this case.

 231120213_u7.PNG

 

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 $45^{0}$ with x-y plane and its horizontal diameter parallel to x-axis. In both cases, the disc is welded at point P, and the systems are rotated with constant angular speed to about the z-axis.

9112021607_k23.PNG

Which of the following statemenrs about the instantaneous axis (passing
through the centre of mass) is correct?


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

B) It is vertical for case (a); and is at $45^{0}$to the x-z plane and lies in the plane of he disc for case (b)

C) It is horizontal for case (a); and is at $45^{0}$ to the x-z plane and is normal to the plane of the disc for case (b)

D) It is vertical for case (a); and is at $45^{0}$ to the x-z plane and is normal to the

Answer:

Option A

Explanation:

(i) Every particle of the disc rotating in a horizontal circle.
(ii) Actual velocity of any particle horizontal.
(iii) Magnitude of the 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 the actual velocity of any particle is
the resultant of two velocities

 $V_{1}=r_{1}\omega_{1}$  and $v_{2}=r_{2}\omega_{2}$

Here,

$r_{1}=$perpendicular distance of the centre of mass from z-axis.

$\omega_{1}$=angular speed of rotation of centre of mass from z-axis.

$r_{2}$= distance of the 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 wiil be horizontal, if $v_{1}$ and $v_{2}$ both are horizontal. Further, v, is

already horizontal, because the centre of mass is rotating about vertical z-axis.
To make v2, aiso horizontal, second axis should also be vertical.