Question

From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

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Solution

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Let mass per unit area of the original disc=
Thus mass of original disc=
Radius of smaller disc=.
Thus mass of the smaller disc=
After the smaller disc has been cut from the original, the remaining portion is considered to be a system of two masses. The two masses are:
M(concentrated at O), and -M(=M/4) concentrated at O'
(The negative sign indicates that this portion has been removed from the original disc.)
Let x be the distance through which the centre of mass of the remaining portion shifts from point O.
The relation between the centres of masses of two masses is given as:
(The negative sign indicates that the centre of mass gets shifted toward the left of point O) 

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