Find the ratio RiRo for which the centroid of the shaded reg

Find the ratio Ri/Ro for which the centroid of the shaded region is located at point C, the highest point on the inner radius.

Solution

The shaded portion is obtained by removing a smaller semi-circle of radius R_i from a bigger semi-circle of radius R_o.

The centroid distance from the centre point at the baseline of semicircle is given by: Y= 4R/3

Thus, centroid of bigger semicircle, Y_o= 4R_o/3 and that of smaller semicircle, Y_i = 4R_i /3

Therefore centroid of shaded region, Y = 4(R_o- R_i) /3

Now, to have centroid as C, Y should be equal to R_i

R_i =  4(R_o- R_i) /3

So, by simplyfying and cross multiplication: R_o/R_i = (3/4 + 1)