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Question

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density of radius 2R as shown in the figure. The moments of inertia of this lamina about an axis passing through O and P are I0 and IP respectively. Both these axes are perpendicular to the plane of the lamina. The ratio IPIO is


A
378
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B
3713
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C
813
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D
14
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Solution

The correct option is B 3713
Assume the mass of bigger disc is M
Hence, mass of removed disc is m
Since disc has uniform mass density,
Mπ(2R)2=mπR2
M=4m ... (1)
MOI about point O for bigger disc =M(2R)22=2MR2
MOI about point O for removed disc =m(R)22+mR2
=3mR22=3MR28 (M=4m)
Hence I0=2MR2[3MR28]
(-ve because we removed that portion)
I0=13MR28 ... (2)

MOI about point P for bigger disc =M(2R)22+M(2R)2
=6MR2
Distance between point P and center of removed disc is
=(2R)2+(R)2=5R
MOI about point P for removed disc
=mR22+m(5R)2
=11mR22 (m=M4)
=11MR28
Hence IP=6MR2[11MR28] {-ve because we removed that portion}
IP=37MR28
IPIO=37MR2813MR28=3713

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