A) \[\left( \frac{M+m}{M} \right){{\omega }_{1}}\]
B) \[\left( \frac{M+m}{m} \right){{\omega }_{1}}\]
C) \[\left( \frac{M}{M+4m} \right){{\omega }_{1}}\]
D) \[\left( \frac{M}{M+2m} \right){{\omega }_{1}}\]
Correct Answer: C
Solution :
| [c] Conservation of angular momentum gives |
| \[\frac{1}{2}M{{R}^{2}}{{\omega }_{1}}=\left( \frac{1}{2}M{{R}^{2}}+2M{{R}^{2}} \right){{\omega }_{2}}\] |
| \[(\because {{l}_{1}}{{\omega }_{1}}={{l}_{2}}{{\omega }_{2}}\] and \[{{l}_{1}}=\frac{1}{2}m{{R}^{2}}\]s |
| \[{{l}_{2}}=\frac{1}{2}m{{R}^{2}}+2m{{R}^{2}})\] |
| \[\Rightarrow \,\,\frac{1}{2}M{{R}^{2}}{{\omega }_{1}}=\frac{1}{2}{{R}^{2}}(M+4m){{\omega }_{2}}\] |
| \[\therefore \] \[{{\omega }_{2}}=\left( \frac{M}{M+4m} \right){{\omega }_{1}}\] |
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