A) 2/5
B) 4/5
C) 2/7
D) 3/7
Correct Answer: B
Solution :
Kinetic'energy of sphere \[{{K}_{{{r}_{0}}}}=\frac{1}{2}I{{\omega }^{2}}\] \[\therefore \]Moment of inertia of sphere, \[I=\frac{2}{5}M{{R}^{2}}\] \[\therefore \]Rotational kinetic energy of sphere \[{{K}_{{{r}_{0}}}}=\frac{1}{2}M{{R}^{2}}{{\omega }^{2}}\] Total energy of sphere \[{{K}_{{{r}_{0}}}}=\frac{1}{2}I{{\omega }^{2}}+\frac{1}{2}M{{v}^{2}}\] \[=\frac{1}{2}\times \frac{2}{5}M{{R}^{2}}{{\omega }^{2}}+\frac{1}{2}M{{R}^{2}}{{\omega }^{2}}\] \[=M{{R}^{2}}{{\omega }^{2}}\left( \frac{1}{5}+\frac{1}{2} \right)\] \[=\frac{7}{10}M{{R}^{2}}{{\omega }^{2}}\] Total energy of sphere\[{{K}_{{{t}_{0}}}}=\frac{7}{10}M{{R}^{2}}{{\omega }^{2}}\] \[\therefore \] \[\frac{{{K}_{{{r}_{0}}}}}{{{K}_{{{t}_{0}}}}}=\frac{1/5M{{R}^{2}}{{\omega }^{2}}}{7/10M{{R}^{2}}{{\omega }^{2}}}\] \[=\frac{2}{7}\]
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