A) \[\frac{{{I}_{1}}\omega _{1}^{2}}{12}\]
B) \[\frac{3}{8}{{I}_{1}}\omega _{1}^{2}\]
C) \[\frac{{{I}_{1}}\omega _{1}^{2}}{6}\]
D) \[\frac{{{I}_{1}}\omega _{1}^{2}}{24}\]
Correct Answer: D
Solution :
\[{{E}_{i}}=\frac{1}{2}{{I}_{1}}\times \omega _{1}^{2}+\frac{1}{2}\frac{{{I}_{1}}}{2}\times \frac{\omega _{1}^{2}}{4}\] \[=\frac{{{I}_{1}}\omega _{1}^{2}}{2}\left( \frac{9}{8} \right)=\frac{9}{16}{{I}_{1}}\omega _{1}^{2}\] \[{{I}_{1}}{{\omega }_{1}}+\frac{{{I}_{1}}\omega _{1}^{{}}}{4}=\frac{3{{I}_{1}}}{2}\omega \] \[\frac{5}{4}{{I}_{1}}\omega _{1}^{{}}=\frac{3{{I}_{1}}}{2}\omega \] \[\omega =\frac{5}{6}{{\omega }_{1}}\] \[{{E}_{f}}=\frac{1}{2}\times \frac{3{{I}_{1}}}{2}\times \frac{25}{36}\omega _{1}^{2}\]\[=\frac{25}{48}{{I}_{1}}\omega _{1}^{2}\] \[\Rightarrow \]\[{{E}_{f}}-{{E}_{i}}={{I}_{1}}\omega _{1}^{2}\left( \frac{25}{48}-\frac{9}{16} \right)=\frac{-2}{48}{{I}_{1}}\omega _{1}^{2}\] \[=\frac{-{{I}_{1}}\omega _{1}^{2}}{24}\]You need to login to perform this action.
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