Introduction to Classical Mechanics · Problem 10.11
Accelerating frames of reference — Morin
Physics — Mechanics · rotating-frame-kinematics
Problem
Unwinding string A wheel with radius R is placed flat on a table. A massless string with one end attached to the rim of the wheel is wrapped clockwise around the wheel a large number of times. When the string is wrapped completely around the wheel, a point mass m is attached to the free end and glued to the wheel. The wheel is then made to rotate with constant angular speed ω. At some point, the glue on the mass breaks, and the mass and string gradually unwind (with the speed of the wheel kept constant at ω by a motor, if necessary). Show that the length of the unwound string increases at the constant rate Rω, for both the clockwise and counterclockwise directions for ω. (The latter is the tricky one.)