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Find the moment of inertia of a lamina bounded by concentric circles of radii a and b about a diameter.

Find the moment of  inertia of a lamina bounded by concentric circles of radii a and b about a diameter.

Grade:Upto college level

1 Answers

ashish kumar
13 Points
12 years ago

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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