Suppose we have an extensible massless string satisfying Hooke's Law, tied to a wall at one end and going over a 'fricional pulley'. Now if we tie the other end with a mass 'M', the total length of the string changes by ΔL and the pulley rotates by angle θ. I came across that ΔL=Rθ, which I couldn't understand. I know that the point A (on the string) will move anticlockwise w.r.t. the pulley and point B clockwise. Can anyone find how ΔL and θ are related?
We know that
$s = r\theta$
for small angular displacements, where $s$ is the arc length, $\theta $ is the angle and $r$ is the radius of the circle.
When pulley rotated due to friction, we assume no relative motion occured between pulley and wire. So the arc is equal to the change in length of wire.