A) \[1:\sqrt{{}}2:\frac{1}{\sqrt{{}}2}\]
B) \[1:\sqrt{{}}2:\sqrt{{}}2\]
C) \[\sqrt{{}}2:1:\frac{1}{\sqrt{{}}2}\]
D) \[\sqrt{{}}2:1:2\]
Correct Answer: A
Solution :
: For a spring, \[T=2\pi \sqrt{\frac{m}{k}}\] In case (i), \[T=2\pi \sqrt{\frac{m}{k}}\] In case (ii), two springs are in series. \[\therefore \] \[{{k}_{s}}=\frac{k\times k}{k+k}=\frac{k\times k}{2k}=\frac{k}{2}\] \[\therefore \] \[{{T}_{s}}=2\pi \sqrt{\frac{2m}{k}}=\sqrt{2}T\] In case (iii), the two springs are in parallel. \[\therefore \] \[{{k}_{p}}=k+k+2k\] \[\therefore \] \[{{T}_{p}}=2\pi \sqrt{\frac{m}{2k}}=\frac{T}{\sqrt{2}}\] \[\therefore \] \[T:{{T}_{s}}:{{T}_{p}}=1:\sqrt{2}:\frac{1}{\sqrt{2}}\]
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