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   \[{{k}_{1}}=k+k=2k\] Hence, time period                           \[{{T}_{1}}=2\pi \sqrt{\frac{M}{{{k}_{1}}}}=2\pi \sqrt{\frac{M}{2k}}\]      ?.(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 \[\frac{{{T}_{1}}}{{{T}_{2}}}=\sqrt{\frac{M/2k}{2M/k}}=\sqrt{\frac{1}{4}}=\frac{1}{2}=0.5\]