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