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Question

Two semi-circular discs of mass density 1 kg/m2 and 2 kg/m2, radius r=1 m each are joined to form a complete disc. Find the moment of inertia (MI) of complete disc about an axis passing through it's centre.

A
π kg-m2
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B
2π kg-m2
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C
3π4 kg-m2
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D
π2 kg-m2
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Solution

The correct option is C 3π4 kg-m2
Let O be the centre of complete disc, axis of rotation is as shown in figure:

Let the mass density of semi-circular discs be σ1 & σ2
σ1=1 kg/m2, σ2=2 kg/m2.
mass=σ×area
Mass of the complete disk
m=(m1+m2)=σ1(πR22)+σ2(πR22)
m=1×(π(1)22)+2×(π×(1)22)
m=3π2 kg
The MI of complete disc about an axis passing through it's centre and perpendicular to the plane:
I=mr22
I=(3π2×122)=3π4 kg-m2
I=3π4 kg-m2

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