Hooke's Law (AQA GCSE Combined Science: Synergy: Physical Sciences): Revision Note

What is Hooke's Law

• The relationship between the extension of an elastic object and the applied force is defined by Hooke's Law

• Hooke's Law states that:

The extension of an elastic object is directly proportional to the force applied, up to the limit of proportionality

• Directly proportional means that as the force is increased, the extension increases 

  • If the force is doubled, then the extension will double

  • If the force is halved, then the extension will also halve

• The limit of proportionality is the point beyond which the relationship between force and extension is no longer directly proportional 

  • This limit varies according to the material

Using Hooke's Law

Hooke's Law Equation

• Hooke's Law is defined by the equation:

F = k × e

• Where:

  • F = force in newtons (N)

  • k = spring constant in newtons per metres (N/m)

  • e = extension in metres (m)

• The symbol e can represent either the extension or compression of an elastic object

• The spring constant represents how stiff a spring is

  • The higher the spring constant, the higher the stiffness

• The extension of an object can be calculated by:

final length – original length

• The extension of the spring can be measured by marking the position of bottom of the unstretched spring

• When the spring is stretched the final length must be measured from the bottom of the spring

• The Hooke's Law equation can be rearranged with the help of the following equation triangle:

Linear & Non-Linear Extension

• Hooke’s law is the linear relationship between force and extension

  • This is represented by a straight line on a force-extension graph

• Materials that do not obey Hooke's law, i.e force and extension are no longer directly proportional; they have a non-linear relationship

  • This is represented by a curve on a force-extension graph

Calculating Spring Constant

• The spring constant can be calculated by rearranging the Hooke's law equation for k:

k =

• Where:

  • k = spring constant in newtons per metres (N/m)

  • F = force in newtons (N)

  • e = extension in metres (m)

• This equation shows that the spring constant is equal to the force per unit extension needed to extend the spring, assuming that its limit of proportionality is not reached

• The stiffer the spring, the greater the spring constant and vice versa

  • This means that more force is required per metre of extension compared to a less stiff spring

• The spring constant is also used in the equation for elastic potential energy

Interpreting Graphs of Force v Extension

• The relationship between force and extension is shown on a force-extension graph

• If the force-extension graph is a straight line, then the material obeys Hooke's law

  • Sometimes, this may only be a small region of the graph, up to the material's limit of proportionality

• The symbol Δ means the 'change in' a variable

  • For example, ΔF and Δe are the 'change in' force and extension respectively

  • This is the same as rise ÷ run for calculating the gradient

• The '∝' symbol means 'proportional to'

  • i.e. F ∝ e means the 'the force is proportional to the extension'

• If the force is on the y axis and the extension on the x axis, the spring constant is the gradient of the straight line (Hooke's law) region of the graph

  • If the graph has a steep straight line, this means the material has a large spring constant

  • If the graph has a shallow straight line, this means the material has a small spring constant

• If the force is on the x axis and the extension on the y axis, the spring constant is 1 ÷ gradient of the straight line (Hooke's law) region of the graph

  • If the graph has a steep straight line, this means the material has a small spring constant

  • If the graph has a shallow straight line, this means the material has a large spring constant