Elastic Potential Energy and Springs

This lesson covers elastic potential energy and Hooke's law as required by the Edexcel GCSE Physics specification (1PH0), Topic 3: Conservation of Energy. You need to understand how energy is stored in stretched or compressed springs, use the relevant equations, and know the core practical for investigating Hooke's law.

---

Hooke's Law

When a spring is stretched or compressed by a force, it extends (or compresses) proportionally — as long as the limit of proportionality is not exceeded.

The Equation

$F = k x$F=kx

Where:

• F = force applied (newtons, N)
• k = spring constant (newtons per metre, N/m)
• x = extension (metres, m)

What Is the Spring Constant?

The spring constant (k) tells you how stiff a spring is:

• A large spring constant means the spring is stiff — it needs a large force to produce a small extension.
• A small spring constant means the spring is soft — a small force produces a large extension.
• The units of the spring constant are N/m (newtons per metre).

Exam Tip: Extension means the change in length, not the total length. Extension = stretched length − original length. A very common mistake is to use the total length instead of the extension.

---

The Limit of Proportionality

Hooke's law only applies up to the limit of proportionality. Beyond this point:

• The force and extension are no longer proportional.
• The spring may be permanently deformed — it will not return to its original length when the force is removed.
• This is shown on a force-extension graph as the point where the straight line begins to curve.

Region	Behaviour	Graph Shape
Before the limit of proportionality	Force is proportional to extension (Hooke's law applies)	Straight line through the origin
At the limit of proportionality	Extension starts to become non-proportional	The line begins to curve
Beyond the limit of proportionality	Permanent deformation occurs	Curved line

Exam Tip: Students often confuse the limit of proportionality with the elastic limit. The limit of proportionality is where the graph stops being a straight line. The elastic limit is where the spring stops returning to its original shape. At GCSE these are very close together but they are technically different points.

---

Finding the Spring Constant from a Graph

On a force-extension graph (force on the y-axis, extension on the x-axis), the spring constant is the gradient of the straight-line section.

$k = \frac{F}{x} = \frac{\Delta F}{\Delta x} = \text{gradient of the line}$k=xF​=ΔxΔF​=gradient of the line

Worked Example

Question: A spring extends by 0.04 m when a force of 8 N is applied. Calculate the spring constant.

Solution:

$k = \frac{F}{x} = \frac{8}{0.04} = 200 \text{ N/m}$k=xF​=0.048​=200 N/m

---

Elastic Potential Energy (Higher Tier)

When a spring is stretched or compressed, energy is stored in the elastic potential energy store. This energy can be calculated using:

The Equation

$E_e = \frac{1}{2} k x^2$Ee​=21​kx2

Where:

• E_e = elastic potential energy (joules, J)
• k = spring constant (newtons per metre, N/m)
• x = extension (metres, m)

Key Points

• This equation only applies when the spring has not been stretched beyond its limit of proportionality.
• Like kinetic energy, elastic PE depends on the square of the extension — double the extension and the energy stored increases by a factor of four.
• Elastic PE = area under a force-extension graph (which is a triangle for a spring obeying Hooke's law).

---

Area Under the Force-Extension Graph

For a spring obeying Hooke's law, the force-extension graph is a straight line through the origin. The energy stored is the area under this line, which forms a triangle:

$\text{Energy stored} = \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times F$Energy stored=Area of triangle=21​×base×height=21​×x×F

Since $F = kx$F=kx, this becomes:

$\text{Energy stored} = \frac{1}{2} \times x \times kx = \frac{1}{2} k x^2$Energy stored=21​×x×kx=21​kx2

This confirms that the area-under-the-graph method gives the same result as the elastic PE formula.

Exam Tip: You may be given a force-extension graph and asked to find the energy stored. Calculate the area of the triangle under the straight-line portion. If the graph curves, you would need to estimate the area (e.g., by counting squares) — but at GCSE the graph is usually straight.

---

Worked Examples — Elastic PE

Example 1: Calculating Elastic PE

Question: A spring with a spring constant of 250 N/m is stretched by 0.1 m. Calculate the elastic potential energy stored.

Solution:

$E_e = \frac{1}{2} k x^2 = \frac{1}{2} \times 250 \times 0.1^2 = \frac{1}{2} \times 250 \times 0.01 = 1.25 \text{ J}$Ee​=21​kx2=21​×250×0.12=21​×250×0.01=1.25 J

Example 2: Finding Extension from Elastic PE

Question: A spring (k = 400 N/m) stores 8 J of elastic potential energy. What is the extension?

Solution:

$x^2 = \frac{2 E_e}{k} = \frac{2 \times 8}{400} = 0.04$x2=k2Ee​​=4002×8​=0.04 $x = \sqrt{0.04} = 0.2 \text{ m}$x=0.04​=0.2 m

Example 3: Finding Spring Constant from Elastic PE

Question: A spring is extended by 0.05 m and stores 0.5 J. What is the spring constant?

Solution:

$k = \frac{2 E_e}{x^2} = \frac{2 \times 0.5}{0.05^2} = \frac{1.0}{0.0025} = 400 \text{ N/m}$k=x22Ee​​=0.0522×0.5​=0.00251.0​=400 N/m

---

Core Practical: Investigating Hooke's Law

Aim

To investigate the relationship between force and extension of a spring.

Equipment

• Retort stand, boss and clamp
• Spring
• Metre rule (or 30 cm ruler)
• Mass hanger and slotted masses (e.g., 100 g masses)
• Pointer (a piece of card or tape attached to the bottom of the spring)
• Set square (to read the ruler accurately at eye level)

Method

1. Set up the retort stand with the spring hanging from the clamp.
2. Place the metre rule vertically beside the spring.
3. Record the original length of the spring (with the mass hanger but no added masses).
4. Add a 1 N (100 g) mass to the hanger.
5. Record the new length of the spring. Calculate extension = new length − original length.
6. Repeat, adding masses one at a time (up to about 6–8 N), recording the extension each time.
7. Plot a graph of force (y-axis) against extension (x-axis).

Variables