A) \[{{\left( \frac{M+m}{M} \right)}^{1/2}}\]
B) \[\frac{M}{M+m}\]
C) \[\frac{M+m}{M}\]
D) \[{{\left( \frac{M}{M+m} \right)}^{1/2}}\]
Correct Answer: A
Solution :
At mean position, initial velocity \[={{A}_{1}}{{\omega }_{1}}\] New velocity \[=\frac{M{{A}_{1}}{{\omega }_{1}}}{M+m}\] \[\Rightarrow \,{{A}_{2}}{{\omega }_{2}}=\frac{M{{A}_{1}}{{\omega }_{1}}}{M+m}\] \[\frac{{{A}_{2}}}{{{A}_{1}}}=\left( \frac{M}{M+m} \right)\frac{{{\omega }_{1}}}{{{\omega }_{2}}}\] \[{{\omega }_{1}}=\sqrt{\frac{k}{M}}\] \[{{\omega }_{2}}=\sqrt{\frac{k}{M+m}}\] \[\Rightarrow \,\,\,\frac{{{A}_{2}}}{{{A}_{1}}}\sqrt{\frac{M}{m+M}}\] or \[\frac{{{A}_{1}}}{{{A}_{2}}}\sqrt{\frac{m+M}{M}}\]
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