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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 perpendicular to the plane of the lamina. The ratio `I_P/I_O` to the nearest integer is

A

`13//37`

B

`37//13`

C

`73//31`

D

`8//13`

Text Solution

Verified by Experts

The correct Answer is:
B
Let M be mass of the whold disc.
then, the mass of the removed disc `=(M)/(pi(2R)^(2)) pi R^(2) = M/4`
So, moment of inertia of the remaining disc about an axis passing through O
`I_(O) = 1/2M(2R)^(2) -[1/2(M/4)R^(2)+M/4R^(2)]`
`2MR^(2) -[(MR^(2) + 2MR^(2))/(8)] = MR^(2) [2-3/8] = 13/8MR^(2)`
Moment of the inertia of the remaining disc about an axis passing throuth P is
`I_(P) =1/2M(2R)^(2) + M(2R)^(2)]-[1/2(M/4)R^(2)+M/4(sqrt(5R)^(2))`
`=2MR^(2) + 4MR^(2)-[(MR^(2))/8 + (5MR^(2))/(4)]`
`=6MR^(2) -11/8MR^(2) = 37/8 MR^(2)`
`therefore I_(P)/I_(O) = 37/8 xx 8/13 = 37/13`
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