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:                                                                                                  [AIPMT (S) 2004]

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 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 same i.e., \[{{F}_{1}}={{F}_{2}}=F\]

So,       \[\frac{1}{k}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}\] or \[k=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]