question_answer

Two springs of spring constants \[{{k}_{1}}\]and \[{{k}_{2}}\]are joined in series. The effective spring constant of the combination is given by:

A) \[\sqrt{{{k}_{1}}{{k}_{2}}}\]

B) \[({{k}_{1}}+{{k}_{2}})/2\]

C) \[{{k}_{1}}+{{k}_{2}}\]

D) \[{{k}_{1}}{{k}_{2}}/({{k}_{1}}+{{k}_{2}})\]  

Correct Answer: D

Solution :

 Let us consider two springs of spring constants \[{{k}_{1}}\]and \[{{k}_{2}}\]joined in series as shown in the figure. Under a force F, they will stretch by\[{{y}_{1}}\]and \[{{y}_{2}}.\] So \[y={{y}_{1}}+{{y}_{2}}\] or \[\frac{F}{k}=\frac{{{F}_{1}}}{{{k}_{1}}}+\frac{{{F}_{2}}}{{{k}_{2}}}\] but as springs are massless, so force on them must be the same, i.e., \[{{F}_{1}}={{F}_{2}}=F.\] So, \[\frac{1}{k}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}\]or \[k=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\] Aliter: In series combination, \[=\frac{1}{{{k}_{S}}}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}=\frac{{{k}_{2}}+{{k}_{1}}}{{{k}_{1}}{{k}_{2}}}\Rightarrow {{k}_{S}}=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]