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Question

A thin rod of length 4l, mass 4m is bent at the points shown in the figure. What is the moment of inertia of the rod about the axis passing point O and perpendicular to the plane of the paper?


A
Ml23
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B
10Ml23
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C
Ml212
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D
Ml224
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Solution

The correct option is B 10Ml23

Each rod is of mass M and length l. As we know, moment of inertia of a rod of mass M and length l about its center is
I=Ml212
For rod OB and OC, moment of inertia about point O, by parallel axis theorem.
IOB=IOC=Ml212+M(l2)2=Ml23

Distance between point O and center of rod AB, or point O and center of rod CD.
=l2+(l2)2=5l24
Hence, Moment of inertia of rod AB and CD about point O by parallel axis theorem:
IAB=ICD=Ml212+M(5l24)
IAB=ICD=4Ml23

Therefore, MOI of bent rod of length 4l and mass 4M about point O will be,
I=IAB+IOB+IOC+ICD
I=2(IAB+IOB)
I=2(Ml23+4Ml23)
I=10Ml23

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