Question

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)$

Answer 1

The Correct Option is D

To determine the effective spring constant when two springs are joined in series, we must understand how springs behave in such a configuration. The general formula for the effective spring constant $k_{\text{eff}}$ of two springs with spring constants $k_1$ and $k_2$ in series is given by:

\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2}

This can be re-arranged as follows to find $k_{\text{eff}}$:

\frac{1}{k_{\text{eff}}} = \frac{k_1 + k_2}{k_1 k_2}

By taking the reciprocal, we obtain:

k_{\text{eff}} = \frac{k_1 k_2}{k_1 + k_2}

Let's analyze why the other options do not align with this result:

• \sqrt{k_1 k_2} – This option does not consider the harmonic addition of the springs in series.
• \frac{k_1 + k_2}{2} – This formula would be more applicable for the equivalent spring constant of two springs in parallel based on similar reasoning, but averages aren't used for springs in series.
• k_1 + k_2 – This is incorrect for series. It applies when two springs are in parallel.

Therefore, the correct option is:

\frac{k_1 k_2}{k_1 + k_2}