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Question
The moment of inertia of a uniform semicircular wire of mass M and radius r about a line perpendicular to the plane of the wire through the centre is ___________ .
Options
\[M r^2\]
\[\frac{1}{2}M r^2\]
\[\frac{1}{4}M r^2\]
\[\frac{2}{5}M r^2\]
Solution
\[M r^2\]
Consider an element of length, dl = rdθ.
\[dm = \frac{M}{\pi r}dl = \frac{M}{\pi r}rd\theta\]
MOI of semicircular wire = \[\int_0^\pi r^2 dm\]
\[I = \int_0^\pi r^2 \frac{m}{\pi r}rd\theta\]
\[ \Rightarrow I = m r^2\]
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