A disc rotating about its axis with angular speed
Question:

A disc rotating about its axis with angular speed $\omega_{0}$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is $R$. What are the linear velocities of the points A, B and C on the disc shown in Fig. 7.41? Will the disc roll in the direction indicated?

Solution:

$v_{A}=R \omega_{0} ; v_{B}=R \omega_{0} ; v_{c}=\left(\frac{R}{2}\right) \omega_{o} ;$ The disc will not roll

Angular speed of the disc $=\omega_{0}$

Radius of the disc $=R$

Using the relation for linear velocity, $v=\omega_{0} R$

For point A:

$v_{\mathrm{A}}=R \omega_{0} ;$ in the direction tangential to the right

For point B:

$V_{B}=R \omega_{0} ;$ in the direction tangential to the left

For point $C$ :

$v_{c}=\left(\frac{R}{2}\right) \omega_{o} ;$ in the direction same as that of $v_{\mathrm{A}}$

The directions of motion of points $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$ on the disc are shown in the following figure

Since the disc is placed on a frictionless table, it will not roll. This is because the presence of friction is essential for the rolling of a body.