A) \[\frac{8}{3}{{\pi }^{2}}\]
B) \[\frac{5}{3}{{\pi }^{2}}\]
C) \[\frac{3}{2}{{\pi }^{2}}\]
D) \[\frac{2}{3}{{\pi }^{2}}\]
Correct Answer: D
Solution :
: Let M be the mass of the rod and L be its length. Then \[I=\frac{1}{12}M{{L}^{2}}\] When the same rod is bent into the ring of radius R, then \[2\pi R=L\] or \[R=\frac{L}{2\pi }\] ?...(i) Its moment of inertia about its diameter is \[I=\frac{1}{2}M{{R}^{2}}=\frac{1}{2}M{{\left( \frac{L}{2\pi } \right)}^{2}}\] (using (i)) \[=\frac{1}{8{{\pi }^{2}}}M{{L}^{2}}\] \[\therefore \] \[\frac{I}{I}=\frac{\frac{1}{12}M{{L}^{2}}}{\frac{1}{8{{\pi }^{2}}}M{{L}^{2}}}=\frac{8{{\pi }^{2}}}{12}=\frac{2}{3}{{\pi }^{2}}\]
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