A) \[\frac{a+b}{2}\]
B) \[\frac{a+b}{3}\]
C) \[\sqrt{\frac{{{a}^{2}}+{{b}^{2}}+ab}{2}}\]
D) \[\sqrt{\frac{{{a}^{2}}+{{b}^{2}}+ab}{3}}\]
Correct Answer: B
Solution :
\[dI=(dm){{r}^{2}}\] \[=(\sigma dA){{r}^{2}}\] \[=\left( \frac{{{\sigma }_{0}}}{r}2\pi rdr \right){{r}^{2}}\] \[=({{\sigma }_{0}}2\pi ){{r}^{2}}dr\] \[I=\int_{{}}^{{}}{dI}=\int\limits_{a}^{b}{{{\sigma }_{0}}2\pi {{r}^{2}}dr}\] \[={{\sigma }_{0}}2\pi \left( \frac{{{b}^{3}}-{{a}^{3}}}{3} \right)\] \[m=\int_{{}}^{{}}{dm}=\int_{{}}^{{}}{\sigma dA}\] \[={{\sigma }_{0}}2\pi \int\limits_{a}^{b}{dr}\]\[m={{\sigma }_{0}}2\pi \,(b-a)\] Radius of gyration \[k=\sqrt{\frac{I}{m}}=\sqrt{\frac{({{b}^{3}}-{{a}^{3}})}{3(b-a)}}\] \[=\sqrt{\left( \frac{{{a}^{3}}+{{b}^{3}}+ab}{3} \right)}\]You need to login to perform this action.
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