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