A planar, irregular shaped object has a mass of 2 kg. Its moment of in

A planar, irregular shaped object has a mass of 2 kg. Its moment of inertia along the two orthogonal axes in the plane of the object are 3 kg m² and 4 kg m², respectively. The moment of inertia of the object along an axis perpendicular to the plane and passing through a point 0·5 m away from its centre of mass is represented by I. Which one among the following is the correct value of I ?

7·5 kg m²

7 kg m²

5 kg m²

1 kg m²

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Answer is Wrong!

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UPSC Geoscientist – 2024

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For a planar object, the moment of inertia about an axis perpendicular to the plane and passing through a point is related to the moments of inertia about two perpendicular axes in the plane by the perpendicular axis theorem and the parallel axis theorem.

By the perpendicular axis theorem, the moment of inertia about the center of mass (assuming the given orthogonal axes intersect at CM) and perpendicular to the plane is I_CM = Ix + Iy = 3 kg m² + 4 kg m² = 7 kg m². By the parallel axis theorem, the moment of inertia I about a parallel axis perpendicular to the plane and at a distance d = 0.5 m from CM is I = I_CM + m d².

Given mass m = 2 kg and distance d = 0.5 m. I = 7 kg m² + (2 kg)(0.5 m)² = 7 + 2(0.25) = 7 + 0.5 = 7.5 kg m².

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