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A lamina is made by removing a small di...

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing though O and P is `I_O and I_P` respectively. Both these axes are perpendiucalr to the plane of the lamina. The ratio `I_P/I_O` ot the nearest integer is

Text Solution

Verified by Experts

The correct Answer is:
C
Let `sigma` be the surface mass density. Then
`I_O = (1)/(2)sigma(2R)^2]xx(2R)^2 - `
`[(1)/(2)(sigmapiR^2)^2 + sigma(pi R^2)xxR^2]`
`=(13)/(2) pi sigma^4`
`I_P = 8pi sigma R^4 + sigma pi(2R)^2xx(2R)^2`
`[(1)/(2) sigma(pi R^2)R^2+sigma(piR^2)(sqrt((2R)^2+R^2))^2]`
`=24pi sigma R^4 - 5.5 sigma R^4 =18.5 pi sigma R^4`
`:. (I_P)/(I_O) = (18.5pi sigmaR^4)/(13)/(2) pi sigmaR^4 = (37)/(13) ~~3`
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Knowledge Check

  • A square lamina is as shown in figure. The moment of inertia of the frame about the three axes as shown in figure I_(1),I_(2) and I_(3) respectively. Select the correct alternative.

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