Hooke's Law (Cambridge (CIE) IGCSE Physics) : Revision Note

Hooke's law

Extended tier only

  • 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

The extension of a spring due to an applied load

Load extension and force, downloadable AS & A Level Physics revision notes

Hooke's Law states that a force applied to a spring will cause it to extend by an amount proportional to the force

  • Hooke's law can be described by the following equation:

F space equals space k x

  • Where:

    • F = force applied, measured in newtons (N)

    • k = the spring constant, measured in newtons per metre (N/m)

    • x = extension of spring, measured in metres (m)

  • The force applied to the spring is sometimes referred to as the load

Spring constant

  • The spring constant is defined as:

The force per unit extension

  • Therefore, the units are newtons per metre (N/m)

  • The spring constant is a measure of how stiff the spring is

    • Stiff springs have a high spring constant

    • Stretchy springs have a low spring constant

  • The spring constant can be applied to objects other than springs

  • The Hooke's law equation can be used to calculate the spring constant of a material

k space equals fraction numerator space F over denominator x end fraction

The force-extension graph

  • Hooke’s law is a linear relationship 

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

  • Any material beyond its limit of proportionality will have a non-linear relationship between force and extension

Force-extension graph for a spring

limit-of-proportionality-graph

Hooke's Law is associated with the linear region of a force-extension graph. Beyond the limit of proportionality, Hooke's law no longer applies

Important features of the force-extension graph 

  • The linear portion of the graph

    • This represents the load or force under which the spring obeys Hooke's law

    • Force and extension are directly proportional

    • The gradient of the linear portion is equal to the spring constant for a force-extension graph

    • The gradient of the linear portion is equal to 1 over k for an extension-force graph

  • The limit of proportionality

    • This is the point at which the graph begins to curve

    • Beyond this point, force and extension are no longer proportional

  • The curved portion of the graph

    • This is where the material does not obey Hooke's law

    • Force and extension are not proportional

Worked Example

The figure below shows the forces acting on a child who is balancing on a pogo stick. The child and pogo stick are not moving.

Hookes Law Worked Example, downloadable IGCSE & GCSE Physics revision notes

The spring constant of the spring on the pogo stick is 4900 N/m. The weight of the child causes the spring to compress elastically from a length of 40 cm to a new length of 33 cm.

Calculate the weight of the child.

Answer:

Step 1: List the known quantities

  • Spring constant, k = 4900 N/m

  • Original length = 40 cm

  • Final length = 33 cm

Step 2: Write the relevant equation

F space equals space k x

Step 3: Calculate the compression, x

x space equals space final space length space minus space original space length

x space equals space 33 space minus space 40 space equals space minus 7 space cm

  • A negative extension represents a compression of 7 cm

Step 4: Convert any units

  • Since the spring constant is given in N/m, x must be in metres (m)

x space equals space 7 over 100 space equals space minus 0.07 space straight m

Step 5: Substitute the values into the Hooke's Law equation

F space equals space 4900 space cross times space minus 0.07

F space equals space minus 343 space straight N

  • The minus sign simply indicates the direction of the force, downwards in this case

  • The child's weight is 343 N

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