A) \[\frac{65\pi a{{R}^{4}}}{2592}\]
B) \[\frac{25\pi a{{R}^{4}}}{72}\]
C) \[\frac{65\pi {{a}^{2}}{{R}^{3}}}{2938}\]
D) \[\frac{81\pi {{a}^{2}}{{R}^{4}}}{144}\]
Correct Answer: A
Solution :
[a] Given; \[J=a{{r}^{2}}\] \[i=\int_{1}^{2}{J\times 2\pi rdr}=\int_{R/3}^{R/2}{a{{r}^{2}}\times 2\pi rdr}\] \[=2\pi a\int_{R/3}^{R/2}{{{r}^{3}}dr}=2\pi a\left| \frac{{{r}^{4}}}{4} \right|_{R/3}^{R/2}\] \[=\frac{\pi a}{2}\left[ {{\left( \frac{R}{2} \right)}^{2}}-{{\left( \frac{R}{3} \right)}^{4}} \right]\] \[=\frac{\pi a{{R}^{4}}}{2}\times \frac{65}{81\times 16}=\frac{65\pi a{{R}^{4}}}{2592}\]
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