Force & Extension
- Forces on materials can cause them to compress (get shorter) or extend (get longer)
- The amount of compression or extension depends on:
- The stiffness of the object, represented by the spring constant, k
- The force applied to the object
Force and Extension of an Object
The amount of extension is directly proportional to the magnitude of the force applied
- The force applied to the material is directly proportional to the amount it is extended
- The more force applied then the greater the extension
- It is given by the equation:
F = kx
- Where:
- F = force applied to the object (N)
- k = spring constant (N/m)
- x = extension (or compression) of the object under the force applied (m)
- The extension of the material can be calculated by:
extension = final length − initial length
- This directly proportional relationship is true up to the limit of proportionality
- The limit of proportionality is the point beyond which the relationship between force and extension is no longer directly proportional
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- This limit varies according to the material
- Very stiff springs require a lot of force to extend or compress
- These springs have very high spring constants
The Limit of Proportionality
The spring on the right has been stretched beyond the limit of proportionality
- A graph of this force-extension relationship until the limit of proportionality shows a straight line through the origin
- The spring constant k is the gradient of this graph
A Force-Extension Graph
The graph shows that force and extension are directly proportional to each other
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.
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 = kx
Step 3: Calculate the extension, x
x = final length – original length = 40 – 33 = 7 cm
Step 4: Convert any units
- Since the spring constant is given in N/m, x must be in metres (m)
7 cm = 0.07 m
Step 5: Substitute the values into the force-extension equation
F = 4900 × 0.07 = 343 N
Weight of child = 340 N (2 s.f.)
Examiner Tip
Look out for unit conversions! Unless the spring constant is given in N/cm, make sure the extension is converted into metres (÷ 100) before substituting values into the equation. Remember that weight is a force given in Newton's (N).
If you want to understand more about the directly proportional relationship of force and extension and the properties of the graph then use the GCSE maths revision notes.
Higher-tier students can rearrange the force-extension equation using this formula triangle. Other students will only be required to calculate force.
Force-Extension Formula Triangle
The formula triangle for Force = spring constant × extension
