Question

6) The figure below shows a uniform metal plate P of radius 2R from which a disk of radius R has been stamped out (removed) in an assembly line. Using the xy coordinate system shown, locate the center of mass "CM" of the remaining plate.

Answer 1

@saifoo.khan

Answer 2

The area of the removed portion is one fourth the area of the bigger disc of radius 2R.
Suppose the complete disc had mass 4M. Then the smaller cut out disc has mass M.
Now ,the given remaining disc can be thought of as a disc of mass 4M with cm at origin and a smaller disc of mass -M with cm at distance -R. The significance of negative mass is only mathematical simplicity.

Now ,we have two masses - one of mass -M at distance -R , one of mass 4M at 0 (origin).

The cm of these 2 masses will be given by\[x = [4M * 0 +(-M)*(-R)] \div (4M-M)\]

This gives x= +R/3 which means cm lies R/3 to the right of the origin as it should.