Paragraph: One twirls a circular ring (of mass $M$ and radius $R$ ) near the tip of one's finger as shown in Figure 1 . In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is $r$. The finger rotates with an angular velocity $\omega_{0}$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is $\mu$ and the acceleration due to gravity is $g$.


Question: The minimum value of $\omega_{0}$ below which the ring will drop down is

• A

  $\sqrt{\frac{3g}{2\mu \left(R-r\right)}}$
• Correct answer
  B

  $\sqrt{\frac{g}{\mu \left(R-r\right)}}$
• C

  $\sqrt{\frac{2g}{\mu \left(R-r\right)}}$
• D

  $\sqrt{\frac{g}{2\mu \left(R-r\right)}}$

Hence, the correct answer is option: 'B'.