A circular lamina of radius a and centre O has mass per unit area kx^2,where xis the distance from O and k is a constant.If the mass of the lamina is M,find hn terms of M and a, the moment of inertia of the lamina about an axis thrgh O and perpendicular to the lamina.

consider a circular lamina placed in xy plane whose center lies ot origin...

mass per unit area (d) is given =kx^2

consider a small element of lamina ...let it be a ring of radius x and thickness dx which is co axial to the lamina...

length of this element =dl=2pix

area of this element is 2pixdx...............1

mass of dis element is dm=(area).(mass per unit area)=2pixdx(d)

                               dm=2pikx^3dx............2

moment of inertia of this element about a axis passing through its center and perpendicular

to plane of  element is given by dI=dmx^2

                                              dI =2pikx^5dx.........3

total moment of inertia is equla to integral dI limit from 0 to a

I=2pik (x^6/6) lim 0 to a

I = (pika^6)/3 ..............4

we have eq 2 as dm=2pikx^3dx

 if we integrate this eq then it will give the total mass of lamina (M)

 M=integral(2pikx^3)dx  limit 0 to a

 M=(pieka^4)/2...............5

solving 4 and 5 

     I=2/3 ma^2     ans