Elastic Potential Energy and Dissipation

This lesson covers elastic potential energy — the energy stored when an object is stretched or compressed — and explores the concept of energy dissipation in more detail. Both topics are part of the AQA GCSE Physics specification (Section 4.1) and are commonly examined.

Elastic Potential Energy

Elastic potential energy is the energy stored in an object when it is stretched or compressed. This applies to springs, rubber bands, trampolines, and any other elastic object.

When a spring is stretched or compressed, work is done on it. This work is stored as elastic potential energy. When the force is released, the stored energy is transferred to other stores (such as kinetic energy).

The Elastic Potential Energy Equation

E_e = 0.5 x k x e^2

Where:

E_e = elastic potential energy in joules (J)
k = spring constant in newtons per metre (N/m)
e = extension (or compression) in metres (m)

Quantity	Symbol	Unit
Elastic potential energy	E_e	joules (J)
Spring constant	k	newtons per metre (N/m)
Extension	e	metres (m)

Exam Tip: This equation only applies when the spring has not been stretched beyond its limit of proportionality. Beyond this point, the spring deforms permanently and the equation is no longer valid. AQA will always specify that the spring obeys Hooke's law if they want you to use this equation.

Hooke's Law

Hooke's law states that the extension of an elastic object is directly proportional to the force applied, provided the limit of proportionality is not exceeded.

F = k x e

Where:

F = force applied in newtons (N)
k = spring constant in newtons per metre (N/m)
e = extension in metres (m)

The spring constant (k) tells you how stiff a spring is. A high value of k means a stiff spring that requires more force to stretch it by a given amount.

Spring Constant	Meaning
High k value	Stiff spring — hard to stretch
Low k value	Soft spring — easy to stretch

Force-Extension Graphs

A force-extension graph for a spring obeying Hooke's law is a straight line through the origin.

The gradient of the straight section equals the spring constant, k.
The limit of proportionality is the point where the line starts to curve. Beyond this, the spring no longer obeys Hooke's law.
The elastic limit is the point beyond which the spring will not return to its original length when the force is removed.

graph LR
    subgraph "Force-Extension Graph"
        A["Origin (0,0)"] -->|"Straight line: Hooke’s law obeyed"| B["Limit of proportionality"]
        B -->|"Curve: Hooke’s law NOT obeyed"| C["Spring deforms permanently"]
    end

Exam Tip: Be careful to distinguish between the limit of proportionality and the elastic limit. The limit of proportionality is where the graph stops being a straight line. The elastic limit is where the spring is permanently deformed. These are close together but NOT the same point.

Worked Examples — Elastic Potential Energy

Example 1: A spring with spring constant 40 N/m is extended by 0.15 m. Calculate the elastic potential energy stored.

E_e = 0.5 x k x e^2 E_e = 0.5 x 40 x 0.15^2 E_e = 0.5 x 40 x 0.0225 E_e = 0.45 J

Example 2: A spring stores 2 J of elastic potential energy when extended by 0.1 m. Calculate the spring constant.

E_e = 0.5 x k x e^2 2 = 0.5 x k x 0.1^2 2 = 0.5 x k x 0.01 2 = 0.005 x k k = 400 N/m

Example 3: A spring with spring constant 80 N/m stores 10 J of elastic potential energy. Calculate the extension.

E_e = 0.5 x k x e^2 10 = 0.5 x 80 x e^2 10 = 40 x e^2 e^2 = 0.25 e = 0.5 m

Energy Transfers Involving Springs

When a stretched spring is released, the elastic potential energy is transferred to other energy stores.

Scenario	Energy Transfer
Spring launches a ball upwards	Elastic potential --> Kinetic --> Gravitational potential
Spring on a toy car released on a flat surface	Elastic potential --> Kinetic (+ internal energy due to friction)
Bungee cord at maximum stretch	Gravitational potential --> Kinetic --> Elastic potential

graph TD
    A["Elastic potential energy store (spring compressed)"] -->|"Spring released"| B["Kinetic energy store (ball moves)"]
    B -->|"Ball rises"| C["Gravitational potential energy store"]
    B -->|"Friction / air resistance"| D["Internal energy of surroundings (dissipated)"]
    style D fill:#ffcccc,stroke:#cc0000

Exam Tip: In a spring-launched ball question, if you are told to ignore air resistance, then elastic potential energy = gravitational potential energy at the highest point. Set E_e = E_p and solve.

Energy Dissipation

In every real energy transfer, some energy is dissipated. This means it is transferred to the surroundings in a way that is not useful, usually increasing the internal (thermal) energy of the surroundings.

Why Is Dissipated Energy Wasted?

Dissipated energy has not disappeared — it has spread out into the environment. It is stored in the kinetic energy of millions of particles in the surroundings. Because it is spread so thinly across so many particles, it is very difficult to collect and use again.

Examples of Energy Dissipation

Device / System	Useful Energy Transfer	Wasted (Dissipated) Energy
Light bulb	Electrical to light (radiation)	Internal (thermal) energy of bulb and surroundings
Car engine	Chemical to kinetic	Internal (thermal) energy of engine, exhaust, friction in moving parts; sound
Electrical appliance	Electrical to various useful stores	Internal (thermal) energy of wires and components
Bouncing ball	GPE to KE and back	Internal (thermal) energy each time the ball hits the ground (ball bounces lower each time)

Reducing Energy Dissipation

We can reduce unwanted energy transfers by:

Elastic Potential Energy and Dissipation

Elastic Potential Energy and Dissipation