1. Thread Kinematics The thread is pulled up with velocity \( u \). The speed along the thread direction (at angle \( \theta \)) is: \[ v_{thread} = u \cos \theta \]

2. Bobbin Kinematics The bobbin rolls without slipping on the outer radius \( R \). \[ v = \omega R \implies \omega = \frac{v}{R} \]

3. Unwinding at the Tangent Point The thread leaves the inner spool tangentially.
From geometry, the radius to the tangent point makes an angle \( \theta \) with the vertical.
The velocity of the thread at this contact point is the sum of: 1. The component of the center’s horizontal velocity \( v \) along the thread. Since the thread is perpendicular to the radius (which is at angle \( \theta \) to the vertical), the thread makes an angle \( \theta \) with the horizontal. \[ v_{projected} = v \cos \theta \] 2. The tangential rotational velocity \( \omega r \), which is directed along the thread.
Total thread speed: \[ v_{thread} = v \cos \theta + \omega r \]

4. Solving Equate the expressions for thread speed: \[ u \cos \theta = v \cos \theta + \left(\frac{v}{R}\right) r \] Divide by \( \cos \theta \) (assuming \( \theta \neq 90^\circ \)): \[ u = v + v \left(\frac{r}{R \cos \theta}\right) \] \[ u = v \left( 1 + \frac{r}{R \cos \theta} \right) \] Given \( R = \eta r \), so \( r/R = 1/\eta \): \[ u = v \left( 1 + \frac{1}{\eta \cos \theta} \right) = v \left( \frac{\eta \cos \theta + 1}{\eta \cos \theta} \right) \] Solving for \( v \): \[ v = \frac{\eta u \cos \theta}{\eta \cos \theta + 1} \]

5. Velocity of Approach The total velocity of approach is \( 2v \): \[ V_{approach} = \frac{2 \eta u \cos \theta}{\eta \cos \theta + 1} \]