The moment of inertia of the two uniform joint rods about point P as shown in figure, where mass of each rod is $M$ , axis is perpendicular to the plane, is $\frac{x}{y} M ℓ^{2}$ . Find $(x + y)$ . ( $x$ & $y$ are least possible integer)

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I 1 ​ = 3 1 ​ M ℓ 2 r 2 = ℓ 2 + ( 2 ℓ ​ ) 2 = ℓ 2 + 4 ℓ 2 ​ = 4 5 ℓ 2 ​ I 2 ​ = 12 1 ​ M ℓ 2 + M r 2 = 12 1 ​ M ℓ 2 + 4 5 ​ M ℓ 2 = 12 1 ​ M ℓ 2 + 12 15 ​ M ℓ 2 = 12 16 ​ M ℓ 2 = 3 4 ​ M ℓ 2 I = I 1 ​ + I 2 ​ = 3 1 ​ M ℓ 2 + 3 4 ​ M ℓ 2 = 3 5 ​ M ℓ 2 Thus, y x ​ = 3 5 ​ . If x + y = 8 , then x = 5 and y = 3 .

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