Spring Stiffness in Series and Parallel Connections

This problem explores the concept of equivalent stiffness for closed coil springs when arranged in two common configurations: series and parallel. Understanding how spring stiffness changes with these arrangements is fundamental in mechanics and engineering.

We are given two closed coil springs with stiffnesses $\text{s}_1 = \text{s}$ and $\text{s}_2 = 2\text{s}$. We need to find the ratio of their equivalent stiffness when connected in series to when connected in parallel.

Series Connection of Springs

When springs are connected in series, they are linked end-to-end, and the total displacement is the sum of the displacements of individual springs. The force applied is the same across all springs. For springs in series, the reciprocal of the equivalent stiffness is the sum of the reciprocals of the individual stiffnesses. This is analogous to resistors in parallel in electrical circuits.

The formula for equivalent stiffness ($\text{S}_{\text{series}}$) for two springs $\text{s}_1$ and $\text{s}_2$ in series is:

$$ \frac{1}{\text{S}_{\text{series}}} = \frac{1}{\text{s}_1} + \frac{1}{\text{s}_2} $$

Substituting the given stiffness values $\text{s}_1 = \text{s}$ and $\text{s}_2 = 2\text{s}$:

$$ \frac{1}{\text{S}_{\text{series}}} = \frac{1}{\text{s}} + \frac{1}{2\text{s}} $$

To combine the terms on the right side, we find a common denominator:

$$ \frac{1}{\text{S}_{\text{series}}} = \frac{2}{2\text{s}} + \frac{1}{2\text{s}} $$ $$ \frac{1}{\text{S}_{\text{series}}} = \frac{2 + 1}{2\text{s}} $$ $$ \frac{1}{\text{S}_{\text{series}}} = \frac{3}{2\text{s}} $$

Inverting both sides to find $\text{S}_{\text{series}}$:

$$ \text{S}_{\text{series}} = \frac{2\text{s}}{3} $$

Parallel Connection of Springs

When springs are connected in parallel, they are arranged side-by-side, and the force applied is distributed among them, while the displacement is the same for all springs. For springs in parallel, the equivalent stiffness is simply the sum of the individual stiffnesses. This is analogous to resistors in series in electrical circuits.

The formula for equivalent stiffness ($\text{S}_{\text{parallel}}$) for two springs $\text{s}_1$ and $\text{s}_2$ in parallel is:

$$ \text{S}_{\text{parallel}} = \text{s}_1 + \text{s}_2 $$

Substituting the given stiffness values $\text{s}_1 = \text{s}$ and $\text{s}_2 = 2\text{s}$:

$$ \text{S}_{\text{parallel}} = \text{s} + 2\text{s} $$

Adding the terms:

$$ \text{S}_{\text{parallel}} = 3\text{s} $$

Ratio of Stiffnesses

Finally, we need to find the ratio of the stiffness of springs connected in series to parallel. This means we need to calculate $\frac{\text{S}_{\text{series}}}{\text{S}_{\text{parallel}}}$.

Using the equivalent stiffness values we calculated:

$$ \text{Ratio} = \frac{\text{S}_{\text{series}}}{\text{S}_{\text{parallel}}} = \frac{\frac{2\text{s}}{3}}{3\text{s}} $$

To simplify the fraction, we can multiply the numerator by the reciprocal of the denominator:

$$ \text{Ratio} = \frac{2\text{s}}{3} \times \frac{1}{3\text{s}} $$ $$ \text{Ratio} = \frac{2\text{s}}{9\text{s}} $$

The term $\text{s}$ cancels out from the numerator and denominator:

$$ \text{Ratio} = \frac{2}{9} $$

Therefore, the ratio of the stiffness of springs connected in series to parallel is $\frac{2}{9}$.

The final answer is 2/9.