A uniform circular disc of radius R and mass M has a circular hole of radius R/2 cut from it. The centre of the hole is at a distance R/2 from the centre O of the original disc. If the centre of mass of the remaining portion is at a distance R/N from O, find N.
Physics · Center of maas , Momentum and Collision · center of mass and question based on it
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Question

A uniform circular disc of radius R and mass M has a circular hole of radius R/2 cut from it. The centre of the hole is at a distance R/2 from the centre O of the original disc. If the centre of mass of the remaining portion is at a distance R/N from O, find N.

Physics · Center of maas , Momentum and Collision · center of mass and question based on it

G1 Physics · Accepted Answer

Take the centre O of the original disc as x = 0 and the centre of the removed circular part at x = +R/2. Since mass is proportional to area, the removed part has mass m = M(R/2)²/R² = M/4. Treat the removed part as negative mass. Then x(CM) = [M(0) − (M/4)(R/2)] / [M − M/4] = [−MR/8]/(3M/4) = −R/6. Therefore the remaining body's centre of mass is a distance R/6 from O, on the side opposite to the hole. Hence N = 6.