For three mutually perpendicular axes through the same point, the sum of the three moments of inertia is 2Σmᵢrᵢ². This follows by adding the coordinate-axis expressions. For a thin spherical shell, symmetry gives the moment of inertia about any diameter as 2MR²/3.
Physics · Rotational Motion · moment of inertia and kinetic energy in pure rotation
For three mutually perpendicular axes through the same point, the sum of the three moments of inertia is 2Σmᵢrᵢ². This follows by adding the coordinate-axis expressions. For a thin spherical shell, symmetry gives the moment of inertia about any diameter as 2MR²/3.
Sub-question 1
Consider a solid cube of mass M and side L. What is the value of Σmᵢrᵢ² when the three mutually perpendicular axes intersect at the centre of the cube?
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Sub-question 2
Find the moment of inertia of ring of mass m and radius R about an axis passing through its centre and making an angle 45° with its plane:
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