Discussion
The figure shows arrangements of springs. They have stiffness K1 and K2 as marked. Which of the following arrangements offers a stiffness \(= \frac{2K_{1}K_{2}}{K_{1} + 2K_{2}}\) ?
Answer: D
The arrangement is two \(K_{2}\) springs in parallel, in series with one \(K_{1}\) spring.
Springs in parallel add their stiffnesses, and springs in series add their reciprocals. Two \(K_{2}\) springs in parallel give \(2K_{2}\), which then acts in series with \(K_{1}\).
\[ \begin{aligned} \frac{1}{k} &= \frac{1}{2K_{2}} + \frac{1}{K_{1}} \\ k &= \frac{2K_{1}K_{2}}{K_{1} + 2K_{2}} \end{aligned} \]
The series chain in A gives \(\frac{K_{1}K_{2}}{K_{1} + 2K_{2}}\), half the required value. B gives \(\frac{2K_{1}K_{2}}{2K_{1} + K_{2}}\), and the three parallel springs in C give \(K_{1} + 2K_{2}\).
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