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The M.I. of a ring of mass radius R abou...

The M.I. of a ring of mass radius R about the axis passing through the centre of gravity and normal to its plane will be :

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Consider a thin elemental ring of thickness .dr. at a radial distance .r..
Mass of the disc = M
Mass per unit area of the disc `= (M)/(pi R^(2)) = sigma`
Area of the ring `= 2 pi r` dr
Mass of the ring `= sigma (3 pi r dr)`
Moment of Inertia of the ring = Mass `(r)^(2)`
Moment of inertia of disc `= int_(0)^(R) sigma (2 pi r dr) r^(2) = (sigma pi R^(4))/(2) = (MR^(2))/(2)`
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