Elasticity and Hooke's Law

This lesson covers Hooke's law, elastic and plastic deformation, force-extension graphs, and the energy stored in a spring — as required by the Edexcel GCSE Physics specification (1PH0), Topic 1: Key Concepts of Physics. You need to understand the linear relationship between force and extension, interpret force-extension graphs, and calculate elastic potential energy (Higher tier).

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Hooke's Law

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

$F = kx$F=kx

Where:

• F = force applied (N)
• k = spring constant (N/m)
• x = extension (m)

Key Points

• Extension is the difference between the stretched length and the original (natural) length: x = stretched length − original length.
• The spring constant (k) is a measure of the stiffness of the spring. A higher spring constant means a stiffer spring — more force is needed to extend it by the same amount.
• Hooke's law applies to the linear region of a force-extension graph — where the line is straight and passes through the origin.
• Beyond the limit of proportionality, the extension is no longer directly proportional to the force.

Rearrangements

• k = F ÷ x (spring constant = force ÷ extension)
• x = F ÷ k (extension = force ÷ spring constant)

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Worked Examples

Example 1

A spring has a spring constant of 50 N/m. Calculate the extension when a force of 4 N is applied.

x = F ÷ k = 4 ÷ 50 = 0.08 m = 8 cm

Example 2

A spring extends by 0.12 m when a force of 6 N is applied. Calculate the spring constant.

k = F ÷ x = 6 ÷ 0.12 = 50 N/m

Example 3

A spring with k = 80 N/m has an original length of 15 cm. A force of 12 N is applied. What is the new length?

Extension: x = F ÷ k = 12 ÷ 80 = 0.15 m = 15 cm New length = original length + extension = 15 + 15 = 30 cm

Exam Tip: Always check whether the question asks for the extension or the new length. Extension = change in length. New length = original length + extension. This is a very common source of lost marks.

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The Limit of Proportionality

The limit of proportionality is the point beyond which Hooke's law no longer applies — the force and extension are no longer directly proportional.

• Below the limit of proportionality: the graph is a straight line through the origin, and F = kx applies.
• Above the limit of proportionality: the graph begins to curve, and the spring extends more per unit of additional force.

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Elastic Deformation vs Plastic Deformation

Feature	Elastic Deformation	Plastic Deformation
Definition	Object returns to its original shape when the force is removed	Object does not return to its original shape when the force is removed
Permanent change?	No	Yes
Energy	All stored elastic potential energy is recovered	Some energy is used to permanently rearrange atoms
Example	A spring stretched within its limit of proportionality	A spring stretched beyond its elastic limit

The Elastic Limit

The elastic limit is the maximum force that can be applied to a spring such that it still returns to its original length when the force is removed.

• Below the elastic limit: elastic deformation — the spring returns to its original length.
• Above the elastic limit: plastic deformation — the spring is permanently stretched.

Note: The elastic limit is at or just beyond the limit of proportionality.

Exam Tip: The limit of proportionality and the elastic limit are close together but are not necessarily the same point. The limit of proportionality is where the graph stops being straight. The elastic limit is where permanent deformation begins. At GCSE, they are often treated as the same point.

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Force-Extension Graphs

A force-extension graph plots force (y-axis) against extension (x-axis) for a material being stretched.

Interpreting the Graph

Feature	Meaning
Straight line through origin	Hooke's law is obeyed; F is directly proportional to x
Gradient of the straight section	Equals the spring constant (k)
Point where the line curves	The limit of proportionality has been exceeded
Area under the graph	Equals the elastic potential energy stored

Finding the Spring Constant from a Graph

The spring constant is the gradient of the straight-line section:

$k = \frac{\Delta F}{\Delta x} = \frac{\text{change in force}}{\text{change in extension}}$k=ΔxΔF​=change in extensionchange in force​

Worked Example

From a force-extension graph, the straight line passes through (0, 0) and (0.05 m, 10 N). Calculate the spring constant.

k = ΔF ÷ Δx = 10 ÷ 0.05 = 200 N/m

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Elastic Potential Energy (Higher Tier)

When a spring is stretched or compressed within its limit of proportionality, it stores elastic potential energy. This energy is given by:

$E_e = \frac{1}{2}kx^2$Ee​=21​kx2

Where:

• E_e = elastic potential energy (J)
• k = spring constant (N/m)
• x = extension (m)

This equation is equivalent to the area under the straight-line section of a force-extension graph (the area of a triangle = ½ × base × height = ½ × x × F = ½ × x × kx = ½kx²).

Worked Examples

Example 1: A spring with k = 40 N/m is extended by 0.3 m. Calculate the energy stored.

E_e = ½kx² = ½ × 40 × 0.3² = ½ × 40 × 0.09 = 1.8 J

Example 2: A spring stores 5 J of energy and has k = 250 N/m. What is the extension?

Rearrange: x² = 2E_e ÷ k = 2 × 5 ÷ 250 = 0.04 x = √0.04 = 0.2 m (20 cm)

Exam Tip: The equation E = ½kx² only applies when the spring is within the limit of proportionality (the linear region). If the spring has been stretched beyond this point, you cannot use this equation — you would need to calculate the area under the curved section of the graph instead.

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Core Practical: Investigating Hooke's Law

Aim

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

Equipment

• Spring, clamp stand, clamp, boss, ruler (mm scale), slotted masses (e.g. 100 g each), mass hanger, pointer attached to spring, safety goggles, cushion or sand tray (safety).

Method

1. Set up the spring on a clamp stand so it hangs vertically.
2. Attach a pointer to the bottom of the spring.
3. Measure the original length of the spring using the ruler (with no mass attached).
4. Add a mass (e.g. 100 g = 0.1 kg) to the mass hanger. Calculate the force: F = mg = 0.1 × 9.8 = 0.98 N.
5. Measure the new length of the spring. Calculate the extension = new length − original length.
6. Repeat, adding one mass at a time, up to 7 or 8 masses.
7. Record all results in a table.
8. Plot a force-extension graph (force on y-axis, extension on x-axis).

Expected Results

• The graph should be a straight line through the origin up to the limit of proportionality.
• The gradient of the straight section = spring constant (k).
• Beyond the limit of proportionality, the graph curves.

Safety Precautions