Question

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 cofficient of friction between the ring and the finger is $\mu$ and the acceleration due to gravity is $g$

• A. $M{\omega }_0^2(R-r{)}^2$
• B. $M{\omega }_0^2{R}^2$
• C. $\frac{3}{2}M{o}_0^2(R-r{)}^2$
  
  
• D. $\frac{1}{2}M{c}_0^2(R-r{)}^2$

Answer 1

The correct option is B $M{\omega }_0^2{R}^2$



$k=\frac{1}{2}M{v}_c^2+\frac{1}{2}{I}_c{\omega }^2$

$=\frac{1}{2}M{\omega }_0^2(R-r{)}^2+\frac{1}{2}M{R}^2{\omega }_0^2$

$=\frac{1}{2}M{\omega }_0^2{R}^2{(1-\frac{r}{R})}^2+\frac{1}{2}M{\omega }_0^2{R}^2$

$r<<R$

$\frac{r}{R}\to 0$

$k=\frac{1}{2}M{\omega }_0^2{R}^2+\frac{1}{2}M{\omega }_0^2{R}^2$

$=M{\omega }_0^2{R}^2$