A pulley is made using a thin rim and two rods of length equal to the diameter of the rim, the two rods lying along perpendicular diameters. The rim and each rod have a mass M. Two blocks of mass M and m are attached to the two ends of a light string passing over the pulley, which is hinged to rota…

Question

A pulley is made using a thin rim and two rods of length equal to the diameter of the rim, the two rods lying along perpendicular diameters. The rim and each rod have a mass M. Two blocks of mass M and m are attached to the two ends of a light string passing over the pulley, which is hinged to rotate freely in a vertical plane about its centre. The magnitude of the acceleration experienced by the blocks is . (Assume no slipping of the string on the pulley.)

Accepted Answer

(2) ((M-m)g)/([(8/3)M+m]). Build the pulley up piece by piece, about the axis through its centre and perpendicular to its plane. The thin rim has all its mass at radius R: Iᵣᵢₘ=MR². Each rod is a diameter, so its length is 2R, and it turns about an axis through its centre perpendicular to its length: I_rod=(M(2R)²)/(12)=(MR²)/3. There are two rods, so I=MR²+2·(MR²)/3=5/3MR². For a string that does not slip, the standard result is a=((M-m)g)/(M+m+I/(R²)). I/(R²)=5/3M, so the denominator is M+m+5/3M=8/3M+m. a=((M-m)g)/((8/3)M+m).

Step-by-step solution

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