The free end of a thread wound on a bobbin is passed round a nail A hammered into the wall. The thread is pulled at a constant velocity 'v’ Assuming pure rolling of bobbin, find the velocity v₀ of the centre of the bobbin at the instant when the thread forms an angle $α$ with the vertical:(R and r are outer and inner radii off the babbin)

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$\frac{v R}{R \sin α - r}$

$\frac{v R}{R \sin α + r}$

$\frac{v}{R \sin α + r}$

$\frac{2 v R}{R \sin α + r}$

answer is A.

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Detailed Solution

When the thread is pulled, the bobbin rolls to the right. Resultant velocity of point B along the thread is $v = v_{0} \sin α - ω r$ , where $v_{0} \sin α$ is the component of translation velocity along the thread and $ω r$ linear velocity due to rotation. As the bobbin rolls without slipping, $v_{0} = ω R$ . Solving the obtained equations, we get $v_{0} = \frac{v R}{R \sin α - r}$ .