Решение

Consider the instantaneous axis of rotation passing through the point $O_1$, then the speed of the spool axis is

$$\fbox{$v_\text{reel} = \omega R$}$$

To find the speed of the load, consider the speed point $O$ as a superposition of the velocities of translational motion with the speed of the wheel center and the speed of rotation point $O$ relative to the wheel center.

$$\vec{v_O} = \vec{v}_{in} + \vec{v}_{out}$$

$$v_O = v_{out} – v_{in} = \omega R - \omega r = \omega\cdot (R - r)$$

Any point of the thread, due to its inextensibility, has the same speed, therefore, the speed of the load

$$\fbox{$v_\text{rope} = v_O = \omega\cdot (R - r)$}$$