A) \[{{T}_{1}}+{{T}_{2}}\]
B) \[\frac{{{T}_{1}}{{T}_{2}}}{{{T}_{1}}+{{T}_{2}}}\]
C) \[\sqrt{T_{1}^{2}+T_{2}^{2}}\]
D) \[\frac{{{T}_{1}}+{{T}_{2}}}{2}\]
Correct Answer: C
Solution :
Let spring constants of two springs are\[{{k}_{1}}\]and \[{{k}_{2}}\]respectively, then \[{{T}_{1}}=2\pi \sqrt{\frac{m}{{{k}_{1}}}}\] and \[{{T}_{2}}=2\pi \sqrt{\frac{m}{{{k}_{2}}}}\] \[\Rightarrow \] \[{{k}_{1}}=\frac{4{{\pi }^{2}}m}{T_{1}^{2}}\] and \[{{k}_{2}}=\frac{4{{\pi }^{2}}m}{T_{2}^{2}}\] When two springs are connected in series, then \[T=2\pi \sqrt{\frac{m}{{{k}_{eq}}}}\] where \[{{k}_{eq}}=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\] \[\Rightarrow \] \[T=\sqrt{T_{1}^{2}+T_{2}^{2}}\]
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