A lamina is made by removing a small disc of diameter 2 R from a bigger disc of uniform mass density and radius 2 R , as shown in the figure. The moment of inertia of this lamina about axes perpendicular to the plane of the lamina and passi

Question

A lamina is made by removing a small disc of diameter 2 R from a bigger disc of uniform mass density and radius 2 R , as shown in the figure. The moment of inertia of this lamina about axes perpendicular to the plane of the lamina and passing through the points O and P is I O and I P ​ respectively. If the ratio I P I O = m n , where m and n are the smallest integers, then what is the value of m + n ?

Quantrex Academy · Accepted Answer

Correct answer: 50

I P = 4 m R 2 2 + m 4 R 2 - m 4 R 2 2 + m 4 5 R 2 I P = mR 2 2 + 4 - 1 8 + 5 4 I P = mR 2 6 - 1 1 8 = 3 7 8 mR 2 &#8230; 1 I O = 4 mR 2 2 - m 4 R 2 2 + m 4 R 2 I O = mR 2 2 - 1 8 + 1 4 = mR 2 2 - 3 8 = mR 2 13 8 &#8230; 2 So I P I O = 3 7 / 8 1 3 / 8 = 37 13 &#8658; m + n = 37 + 13 = 50

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