To compensate and keep angular momentum constant, the washer's angular velocity has to increase. The reverse happens when you loosen the string and allow the washer to move out farther.

Zero velocity vs. zero acceleration QUESTION: If the angular velocity of a link in a mechanism is zero, does this imply that the angular acceleration of that link is also zero? The short answer to ...

Nope: Clearly, this is not conserved. I could plot the angular velocity of each ball—but they would just have the same value and not add up to a constant.