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Calculate the moment of inertia of a solid sphere about its diameter.

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Moment of Inertia of a Solid Sphere about its Diameter
According to the figure a sphere of mass M and radius R is shown, whose density is p. We have to calculate the moment of inertia of the sphere about the diameter XX’.

We can assume the sphere to be made up of many discs whose surfaces are parallel to YY’ and the center is on XX’ axis. One of these discs has a center at O’ and radius y; and the distance of the O’ circle from center O is x; the width of this disc is dx.

Fig: Moment of Inertia of a solid sphere about its diameter

Density of the sphere (ρ) = \(\frac{M}{\frac{4}{2} \pi R^{3}}\) ……………. (1)
Volume of the disc = π y2dx
and the mass of the disc = π y2dx ρ …………….. (2)
Therefore, the moment of inertia of the sphere about the axis XX’ perpendicular to the surface (plane) and passing through the center is;

The moment of inertia of the total sphere about the XX’ axis will be equal to the sum of the moment of inertia of all the discs between x = -R and x = +R.

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