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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?enter image description here

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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.

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  • $\begingroup$ I think you didn't understand my question. The string can expand so the motion of string and pulley has to be relative. And I think there will be a single point on the string that will not move relative to the pulley. $\endgroup$ – Kartik Kataria Oct 18 '18 at 13:53
  • $\begingroup$ there shoulnt be relative motion as the string has friction and friction is the force which rotates the pulley. When the mass was attached, the wire was elongated and that rotated the pulley. $\endgroup$ – Mechanic7 Oct 18 '18 at 13:58
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    $\begingroup$ Ok, even if that's true, Rθ should correspond to the change in length of wire from wall to point A not the total length. $\endgroup$ – Kartik Kataria Oct 18 '18 at 14:06

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