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 total kinetic energy of the ring is

• A

  $M{\omega }_0^2{R}^2$
• Correct answer
  B

  $M{\omega }_0^2{\left(R-r\right)}^2$
• C

  $\frac{1}{2}M{\omega }_0^2{\left(R-r\right)}^2$
• D

  $\frac{3}{2}M{\omega }_0^2{\left(R-r\right)}^2$

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