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 bobin, find the velocity `v_(0)` of the centre of the bobbin at the instant when the thread forms an angle `alpha` with the vertical: (`R` and `r` are outer and inner radii off the babbin)

Options

A. `(vR)/(Rsinalpha-r)`
B. `(vR)/(Rsin alpha+r)`
C. `(2vR)/(Rsinalpha+r)`
D. `v/(Rsinalpha+r)`

Answer

A

Explanation

When the thread is pulled, the bobbin rolls to the right. Resultant velocity of point `B` along the thread is `upsilon=upsilon_(0) sin alpha-omegar`, where `upsilon_(0) sin alpha` is the and `omegar` linear velocity due to rotation. As the bobbin rolls without slipping, `upsilon_(0)=omegaR`. solving the obtained equations, we get `upsilon_(0)=(upsilonR)/(R sin alpha-r)`