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rajan jha Grade: 12
        A circular lamina of radius a and centre O has mass per unit area kx^2,where x  is 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.
6 years ago

Answers : (1)

vikas askiitian expert
510 Points
										

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

6 years ago
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