Teaching guide for the module

Hooke's Law: Springs in series and parallel: Teacher's Guide

Teaching guide for the Hooke's Law simulator: explain in class Hooke's Law ( $F = k \cdot x$ ), stored elastic energy and: most importantly, the behaviour of systems of springs in series and in parallel, with the powerful (inverted) analogy with electrical resistors. Designed for physics and electrical-engineering teachers.

Module: Hooke's Law · Three tabs: Single · Series · Parallel

Physical phenomenon

Hooke's Law, formulated by Robert Hooke in 1660, describes the elastic behaviour of an ideal spring: the force needed to stretch (or compress) a spring by an amount $x$ is proportional to that displacement.

$F = k \cdot x$

The coefficient $k$ , called the spring constant or stiffness, is measured in $N/m$ and is a property of the spring (geometry, material, number of coils). High $k$ means a stiff spring, low $k$ a soft one.

The energy stored in the deformation, called elastic potential energy, is:

$U = \frac{1}{2} \cdot k \cdot x^{2}$

On the $F$ - $x$ graph it is represented by the area under the line $F = k x$ , the highlighted triangle in the simulator.

When several springs are combined, the equivalent stiffness depends on the configuration:

In series (chained): $\frac{1}{k _{e q}} = \frac{1}{k _{1}} + \frac{1}{k _{2}}$ : the system is softer than the smallest individual spring.
In parallel (side by side): $k_{e q} = k_{1} + k_{2}$ : the system is stiffer than the largest individual spring.

This behaviour is exactly inverted with respect to electrical resistors (series additive, parallel reciprocal): one of the most powerful conceptual bridges between physics and electrical engineering.

Key concepts

Hooke's Law: direct proportionality $F = k x$ within the elastic limit (beyond which the spring deforms plastically and the law no longer holds).
Spring constant $k$ : slope of the $F$ - $x$ line. Steeper = stiffer.
Elastic energy $U = \frac{1}{2} k x^{2}$ : quadratic dependence on displacement: doubling $x$ quadruples the energy.
Restoring force: the spring always exerts a force opposite to the deformation: $F_{rest} = - k x$ .
Series composition: same force, different elongations; equivalent stiffness less than each.
Parallel composition: same elongation, different forces; equivalent stiffness greater than each.
Inverted analogy with resistors: springs in series ↔ resistors in parallel; springs in parallel ↔ resistors in series.

How to use it in the classroom

Opening, Single tab. Start with a single spring. Move the force slider $F$ and watch the spring stretch linearly; the red dot on the $F$ - $x$ graph slides along the line of slope $k$ . Have students compute $x = F / k$ mentally with simple values (e.g. $F = 10 N$ , $k = 200 N/m \Rightarrow x = 5 cm$ ) before reading the KPI. Then vary $k$ at fixed $F$ : the slope of the line changes, the elongation shrinks as stiffness grows.

Development: elastic energy. Still on the Single tab, highlight the triangular area under the line on the graph: it is $U = \frac{1}{2} F x = \frac{1}{2} k x^{2}$ . Have them double $F$ and observe the energy KPI: it grows by a factor of four, not two. This is the moment to fix the quadratic dependence and introduce the idea of "stored work".

Conceptual core, Series tab. Switch to the series tab with $k_{1} = 100 N/m$ and $k_{2} = 400 N/m$ . Highlight that the same force $F$ passes through both springs (action-reaction at the intermediate node), but the elongations $x_{1} = F / k_{1}$ and $x_{2} = F / k_{2}$ are different: the softer spring stretches more. On the $F$ - $x$ graph the solid line ( $k_{e q}$ ) lies below the two dashed ones: the combined system is softer than either component.

Conceptual core, Parallel tab. Same pair $k_{1}$ and $k_{2}$ in parallel configuration. Now it is the elongation that is common (the load bar translates rigidly), while the force splits: $F_{1} = k_{1} x$ and $F_{2} = k_{2} x$ . The $k_{e q}$ line now lies above both dashed ones: the system is stiffer than the stiffest of the two components.

Closing: analogy with resistors. Sketch on the board a parallel: two resistors $R_{1}$ , $R_{2}$ and two springs $k_{1}$ , $k_{2}$ . Show that the maths is identical but the configurations are swapped: springs in series behave like resistors in parallel, and vice versa. A precious moment for students in electrical/electronic curricula.

Real-world examples

Car suspension. The springs at each wheel work in parallel with respect to the chassis: effective stiffness is the sum. Replacing one spring with a stiffer one only changes ride height on that side.
Pocket-spring mattress. Dozens of independent springs in parallel: each responds only to the local load, without "propagating" deformation to neighbours.
Trampoline. The perimeter springs all work in parallel: only this way you reach a high effective $k_{e q}$ while keeping each individual spring softer and safer.
Train wagon chain with spring buffers. Series system: a single traction force propagated from wagon to wagon, different elongations on each buffer.
Spring scale (dynamometer). A single calibrated spring: force read directly from elongation, $F = k x$ in action.
Analogue mechanical speedometer. A torsion spring opposes the rotating magnet; the equilibrium angle is proportional to speed.

Classroom discussion questions

Two identical springs with $k = 200 N/m$ are connected first in series, then in parallel. What are the two $k_{e q}$ ? Which configuration is "softer"?
A spring with $k = 100 N/m$ is stretched by $10 cm$ . How much energy does it store? And if I stretch it by $20 cm$ ?
Why are car suspensions in parallel and not in series? What would happen if they were in series?
In a series configuration, which spring stores more energy: the stiffer or the softer one? (Hint: $U_{i} = \frac{1}{2} k_{i} x_{i}^{2} = \frac{F ^{2}}{2 k _{i}}$ .)
Comparison with resistors: two $100 Ω$ resistors in parallel give $50 Ω$ . Two $100 N/m$ springs in parallel give $200 N/m$ . Why is the analogy inverted?

Related modules

Forces & Vectors: elastic force is a vector, opposite to the displacement; in static-equilibrium systems it must be summed with all the other applied forces.
Ohm's Law & Power Management: the formal analogy with series/parallel resistors is the most direct "bridge" between mechanics and electrical engineering: same maths, swapped configurations.