A) \[\frac{12}{25}\]
B) \[\frac{7}{5}\]
C) \[\frac{24}{25}\]
D) \[\frac{5}{7}\]
Correct Answer: A
Solution :
Under the action of first force, \[{{\operatorname{F}}_{1}}=\,\,m\omega _{1}^{2}y\] Under the action of second force, \[{{\operatorname{F}}_{2}}=\,\,m\varepsilon _{2}^{2}y\] Under the action of resultant force, \[{{F}_{1}}+{{F}_{2}}\,\,=\,\,m{{\omega }^{2}}y\] \[\Rightarrow \,\,\,\operatorname{m}{{\omega }^{2}}y=\,\,m\omega _{1}^{2}y\,\,+\,\,m\omega _{2}^{2}y\] \[\Rightarrow \,\,\,{{\omega }^{2}}=\omega _{1}^{2}+\omega _{2}^{2}\] \[\Rightarrow \,\,\,{{\left( \frac{2\pi }{T} \right)}^{2}}\,\,={{\left( \frac{2\pi }{{{T}_{1}}} \right)}^{2}}+{{\left( \frac{2\pi }{{{T}_{2}}} \right)}^{2}}\] \[\Rightarrow \,\,\,T=\sqrt{\frac{T_{1}^{2}T_{2}^{2}}{T_{1}^{2}+T_{2}^{2}}}\,\,=\,\,\sqrt{\frac{{{\left( \frac{4}{5} \right)}^{2}}.{{\left( \frac{3}{5} \right)}^{2}}}{{{\left( \frac{4}{5} \right)}^{2}}+{{\left( \frac{3}{5} \right)}^{2}}}}=\frac{12}{25}\]
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