Question

The free end of a thread wound on a bobbin of inner radius $r$ and outer radius $R$ is passed round a nail $A$ hammered into a wall as shown in the figure. The thread is pulled at a constant velocity $v$. Find the velocity $v_0$ of the bobbin instant when the thread from an angle $\alpha$ with the vertical, assuming that the bobbin rolls over the horizontal surface without slipping.

Answer 1

For pure rolling, $v_0=R\omega$------------1

Speed of point in contact with string should be $v$.

$\Rightarrow v=r\omega -v_0\cos (90-\alpha )=r\omega -v_0\sin \alpha$-----------------------2

Using Equation 1 in Equation 2,

$v=\frac{r}{R}\cdot v_0-v_0\sin \alpha$

$\therefore v_0=\frac{v}{\frac{r}{R}-\sin \alpha }$