A) \[\frac{3}{2}\]
B) \[\frac{1}{2}\]
C) \[\frac{3}{5}\]
D) \[\frac{8}{5}\]
Correct Answer: D
Solution :
\[M\int\limits_{0}^{R}{{{\rho }_{0}}r(2\pi rdr)}=\frac{{{\rho }_{0}}\times 2\pi \times {{R}^{3}}}{3}\] \[\underset{(MOI\,about\,COM)}{\mathop{{{I}_{0}}}}\,=\int\limits_{0}^{R}{{{\rho }_{0}}r(2\pi rdr)}\times {{r}^{2}}=\frac{{{\rho }_{0}}\times 2\pi {{R}^{5}}}{5}\] by parallel axis theorem\[I={{I}_{0}}+M{{R}^{2}}\] \[=\frac{{{\rho }_{0}}\times 2\pi {{R}^{5}}}{5}+\frac{{{\rho }_{0}}\times 2\pi {{R}^{3}}}{3}\times {{R}^{2}}={{\rho }_{0}}2\pi {{R}^{5}}\times \frac{8}{15}\] \[=M{{R}^{2}}\times \frac{8}{5}\]
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