Question

A uniform circular disc of radius r is placed on a rough horizontal surface and given a linear velocity $v_{0}$and

angular velocity $\omega_{0}$ as shown. The disc comes to rest after moving some distance to the right. It follows that

• A. $3v_{0} = 2\omega_{0}r$
• B. $2v_{0} = \omega_{0}r$
• C. $v_{0} = \omega_{0}r$
• D. $2v_{0} = 3\omega_{0}r$

Answer 1

Answer: (B)

$2v_{0} = \omega_{0}r$

Answer 2

Since the disc comes to rest. It stops rotating & translating simultaneously ⇒ v = 0 & ω = 0. That means, the angular momentum about the instantaneous point of contact just after time of stopping is zero we know that, the angular momentum of the disc about P remains constant because frictional force f. N & mg pass through the point P, thus produce no torque about this point.

⇒ $L_{initial} = L_{final}$ ⇒ $mvr - I_{0}\omega_{0} = 0$

⇒ $mvr = I_{0}\omega_{0}$ ⇒ $mvr = \frac{1}{2}mr^{2}\omega_{0}$

⇒ $2v_{0} = \omega_{0}r$