Two springs, P and Q, each obey Hooke's law over the range of forces considered. Spring P has spring constant 200 N m^-1 and spring Q has spring constant 300 N m^-1. (a) The springs are joined end-to-end, so that the same force acts through both springs and the total extension i…

AS Level Physics 9702: Springs in Series and in Parallel

Two springs, P and Q, each obey Hooke's law over the range of forces considered. Spring P has spring constant $200\text{ N m}^{-1}$ and spring Q has spring constant $300\text{ N m}^{-1}$ .

(a) The springs are joined end-to-end, so that the same force acts through both springs and the total extension is the sum of the individual extensions (a series combination). Determine the effective spring constant of this series combination. [2]

(b) The same two springs are instead arranged side-by-side between two rigid plates, so that both springs are forced to have the same extension when a load is applied (a parallel combination). Calculate the effective spring constant of this parallel combination. [2]

(c) A load of $12\text{ N}$ is hung from the parallel combination described in (b). Calculate the extension produced. [2]

Part (a): Springs in series

For springs in series, the same force $F$ acts through each spring, and the total extension is the sum of the individual extensions. This leads to the reciprocal combination rule: $\frac{1}{k_{\text{series}}} = \frac{1}{k_P} + \frac{1}{k_Q}$

Substituting the values: $\frac{1}{k_{\text{series}}} = \frac{1}{200} + \frac{1}{300} = \frac{3}{600} + \frac{2}{600} = \frac{5}{600} = \frac{1}{120}$

Taking the reciprocal: $k_{\text{series}} = 120\text{ N m}^{-1}$

Part (b): Springs in parallel

For springs in parallel, both springs share the same extension, and the total force supported is the sum of the forces in each spring. This leads to a direct sum of spring constants: $k_{\text{parallel}} = k_P + k_Q$

Substituting the values: $k_{\text{parallel}} = 200 + 300 = 500\text{ N m}^{-1}$

Part (c): Extension of the parallel combination

Using Hooke’s law with the combined spring constant from (b): $F = k_{\text{parallel}}\,x \quad\Rightarrow\quad x = \frac{F}{k_{\text{parallel}}}$

Substituting $F = 12\text{ N}$ and $k_{\text{parallel}} = 500\text{ N m}^{-1}$ : $x = \frac{12}{500} = 0.024\text{ m}$

Converting to centimetres for clarity: $x = 0.024\text{ m} = 2.4\text{ cm}$

Final answers

(a) Series effective spring constant $= \boxed{120}\text{ N m}^{-1}$
(b) Parallel effective spring constant $= \boxed{500}\text{ N m}^{-1}$
(c) Extension of parallel combination $= \boxed{0.024}\text{ m}$ (i.e. $2.4\text{ cm}$ )