question_answer

Two springs having force constants k each are arranged in parallel and in series. A mass M is attached to two arrangements separately. If time period in first case is \[{{T}_{1}}\] and in second case is \[{{T}_{2}}\], then ratio \[\frac{{{T}_{1}}}{{{T}_{2}}}\] is :

A)  1.5              

B)  3.2

C)  0.5               

D)  2.1

Correct Answer: C

Solution :

Key Idea: In first set up, the springs are joined in parallel and in second, the springs are joined in series. When springs are connected in parallel, the effective spring constant is \[{{k}_{1}}=k+k=2\,k\] Hence, time period \[{{T}_{1}}=2\,\pi \sqrt{\frac{M}{{{k}_{1}}}}=2\pi \sqrt{\frac{M}{2\,k}}\] ... (i) When springs are connected in series, the effective spring constant is \[\frac{1}{{{k}_{2}}}=\frac{1}{k}+\frac{1}{k}=\frac{2}{k}\] \[\Rightarrow \] \[{{k}_{2}}\frac{k}{2}\] Therefore, time period, \[{{T}_{2}}=2\pi \sqrt{\frac{M}{{{k}_{2}}}}\]      \[=2\pi \sqrt{\frac{M}{k/2}}\] \[{{T}_{2}}=2\pi \sqrt{\frac{2M}{k}}\] ... (ii) Dividing Eq. (i) by Eq. (ii), we have \[\frac{{{T}_{1}}}{{{T}_{2}}}=\sqrt{\frac{M/2k}{2M/k}}=\sqrt{\frac{1}{4}}=\frac{1}{2}=0.5\]