Torque applied: \( \tau = F \cdot R \).
Moment of Inertia: \( I = \frac{1}{2}mR^2 \).
Angular acceleration: \( \alpha = \frac{\tau}{I} = \frac{FR}{\frac{1}{2}mR^2} = \frac{2F}{mR} \).

Starting from rest, the angle turned in time \( t \) is: \[ \theta = \frac{1}{2} \alpha t^2 = \frac{1}{2} \left( \frac{2F}{mR} \right) t^2 = \frac{Ft^2}{mR} \]

The length of the cord unwound is related to the angle by \( l = R\theta \): \[ l = R \cdot \left( \frac{Ft^2}{mR} \right) = \frac{Ft^2}{m} \]

Notice that the radius \( R \) cancels out! Since \( F \), \( t \), and \( m \) are equal for both discs, the length unwound is the same.