A) \[\frac{12}{25}\,s\]
B) \[\frac{24}{25}\,s\]
C) \[\frac{35}{24}\,s\]
D) \[\frac{15}{12}\,s\]
Correct Answer: A
Solution :
Under the influence of one force \[F=m\omega _{1}^{2}y\] and under the action of another force, \[{{F}_{2}}=m\omega _{2}^{2}y\] Under the action of both the forces \[F={{F}_{1}}+{{F}_{2}}\] \[\Rightarrow \] \[m{{\omega }^{2}}y=m\omega _{1}^{2}y+m\omega _{2}^{2}y\] \[\Rightarrow \] \[{{\omega }^{2}}=\omega _{1}^{2}+\omega _{2}^{2}\] \[\Rightarrow \]\[{{\left[ \frac{2\pi }{T} \right]}^{2}}={{\left[ \frac{2\pi }{{{T}_{1}}} \right]}^{2}}+{{\left[ \frac{2\pi }{{{T}_{2}}} \right]}^{2}}\] \[\Rightarrow \]\[T=\sqrt{\frac{T_{1}^{2}\times T_{2}^{2}}{T_{1}^{2}+T_{2}^{2}}}\] \[=\sqrt{\frac{{{\left( \frac{4}{5} \right)}^{2}}\left( \frac{3}{5} \right)}{{{\left( \frac{4}{5} \right)}^{2}}+{{\left( \frac{3}{5} \right)}^{2}}}}\]\[=\frac{12}{25}s\]
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