A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string. The body is released from rest. Then the acceleration of the body is

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(1) (2mg)/(2m+M). Two equations, one for the falling body and one for the disc, joined by the string. Body: mg-T=ma. Disc: the string pulls tangentially at the rim, so TR=Iα with I=1/2MR². The string does not slip, so a=α R, giving TR=1/2MR²·a/R and T=1/2Ma. Substitute: mg-1/2Ma=ma. mg=a(m+M/2), so a=(mg)/(m+M/2)=(2mg)/(2m+M). Check the limits: with M=0 the body is in free fall, a=g; with M very large the disc barely turns and a→0.

A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string. The body is released from rest. Then the acceleration of the body is

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