Hooke's Law

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Use Hooke's law as a model relating the force in an elastic string or spring to the extension or compression, and understand the term modulus of elasticity
  2. Use the formula for the elastic potential energy stored in a string or spring
  3. Solve problems involving forces due to elastic strings or springs, including those where considerations of work and energy are needed (e.g. particles moving horizontally, vertically, on inclined planes, or in conical pendulum setups)

What is Hooke's Law?

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Hooke's Law describes what happens when you stretch or compress an elastic object (like a spring or elastic string). It tells us that the force needed to stretch or compress the object is directly proportional to how much you stretch or compress it.

Think of it like this: if you pull on a spring and it stretches by 2 cm, and then you pull twice as hard, it will stretch by 4 cm. The force and the extension increase together in the same ratio.

The Basic Formula

For an elastic string or spring, Hooke's Law is written as:

T = (λ/l) × x

Where:

  • T = tension (the pulling force) in the string or spring, measured in Newtons (N)
  • λ (lambda) = modulus of elasticity (explained below), measured in Newtons (N)
  • l = natural length of the string or spring (its length when nothing is pulling it), measured in metres (m)
  • x = extension (how much longer the string is than its natural length), measured in metres (m)

Important: This formula only works when the string or spring is being stretched. If there's no extension (x = 0), then there's no tension (T = 0).

Understanding Modulus of Elasticity

The modulus of elasticity (λ) is a measure of how stiff the string or spring is. Think of it as the "stiffness constant."

  • A large λ means the string is very stiff and hard to stretch (you need a big force to stretch it even a little bit)
  • A small λ means the string is stretchy and easy to extend (a small force creates a large extension)

The modulus of elasticity is a property of the particular string or spring you're using – it doesn't change unless you change to a different string.

Extension vs Compression

  • Extension means the string or spring is being stretched and is longer than its natural length
  • Compression means the spring is being squashed and is shorter than its natural length

Note: Elastic strings can only be stretched (they go slack if you try to compress them), but springs can be both stretched AND compressed. When a spring is compressed by a distance x, we use the same formula with x representing the compression.

Alternative Form of Hooke's Law

Sometimes you'll see Hooke's Law written as:

T = k × x

Where k = λ/l is called the spring constant or stiffness. This is exactly the same law, just written in a different way.

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