Question

A flat surface of a thin uniform disk $A$ of radius $R$ is glued to a horizontal table. Another thin uniform disk $B$ of mass $M$ and with the same radius $R$ rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of $B$ has fixed angular speed $\omega$ about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $n M \omega R^2$ with respect to the center of $A$. Which of the following is the value of $n$ ?

• A. 2
• B. 5
• C. $\frac{7}{2}$
• D. $\frac{9}{2}$

Answer 1

Answer: (B)

$ v=\omega(2 R)$
$ v=\omega_0 R: $ no slipping
$ \therefore \omega_0=2 \omega$

$ \overrightarrow{ L }= m \overrightarrow{ r } \times \overrightarrow{ v }_{ c }+ I _{ c } \omega_0$
$ = M 2 Rv +\frac{1}{2} MR ^2 \omega_0$
$ =4 MR ^2 \omega+\frac{1}{2} MR ^2(2 \omega)=5 MR ^2 \omega$
$ \therefore n =5$