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Find the moment of inertia of a lamina bounded by concentric circles of radii a and b about a diameter.
Let the mass of the lamina be m.
Therefore the density of the lamina § = m/¶(b2-a2).
Consider an element of thickness dr at adistance r from the centre.
dm = §(2¶rdr)
Therefore moment of inertia about the centre O
I = dmr2
= §(2¶rdr)r2
= 2§¶r3dr
= 2§¶(b4-a4)/4
= 2m¶(b4-a4)/(4¶(b2-a2))
= m(a2+b2)/2
Applying the perpendicular axis theoram
Iz = Ix + Iy
Io = Iab + Icd
By symmetry Iab = Icd
Iab = Icd = Io/2
= m(a2+b2)/4
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