Springs in parallel
Materials - OCR A-Level Physics
Springs in parallel
Springs arranged side by side, each carrying part of the load. The combined spring constant is stiffer than any one spring on its own.
$$k_{\text{total}} = k_1 + k_2 + \ldots$$
Springs in series
Springs joined end to end, each carrying the full load. The combined spring constant is less stiff than any one spring on its own.
$$\frac{1}{k_{\text{total}}} = \frac{1}{k_1} + \frac{1}{k_2} + \ldots$$
- Parallel: each spring shares the load, so the same force gives a smaller extension. Combined stiffness adds, just like capacitors in parallel.
- Series: each spring feels the full force, so each extends fully. The total extension is the sum, so the combination feels less stiff. Reciprocals add, just like resistors in parallel.
- For $n$ identical springs of constant $k$: parallel gives $k_{\text{total}} = nk$, series gives $k_{\text{total}} = k/n$.
- Worked example: two springs with $k_1 = 30 \text{ N m}^{-1}$ and $k_2 = 60 \text{ N m}^{-1}$ in parallel give $k_{\text{total}} = 90 \text{ N m}^{-1}$; in series they give $1/k_{\text{total}} = 1/30 + 1/60 = 3/60$, so $k_{\text{total}} = 20 \text{ N m}^{-1}$.
Common Mistake
MEDIUM
Wrong: Applying the resistor rules to springs (adding $k$ values in series, reciprocals in parallel).
Right: Springs combine the OPPOSITE way to resistors. Springs in parallel: $k$ values add directly (stiffer). Springs in series: use the reciprocal formula (less stiff).
Right: Springs combine the OPPOSITE way to resistors. Springs in parallel: $k$ values add directly (stiffer). Springs in series: use the reciprocal formula (less stiff).