Spring constant measures stiffness
Definition
Spring constant
The force required per unit extension of a spring within its linear region.
- For a spring in its linear elastic region, force and extension are related by F=kxF=kxF=kx.
- In this equation, FFF is the force exerted on the spring in newtons, kkk is the spring constant in newtons per metre, and xxx is the extension in metres.
- Extension is the change in length, so x=stretched length−original lengthx=\text{stretched length}-\text{original length}x=stretched length−original length.
- A larger value of kkk means a stiffer spring because more force is required for the same extension.
Rearrange and convert before substituting
- Use k=Fxk=\dfrac{F}{x}k=xF to calculate spring constant and x=Fkx=\dfrac{F}{k}x=kF to calculate extension.
- Convert centimetres or millimetres to metres before using F=kxF=kxF=kx: 1 cm=0.01 m1\,\text{cm}=0.01\,\text{m}1cm=0.01m and 1 mm=0.001 m1\,\text{mm}=0.001\,\text{m}1mm=0.001m.
- The unit follows from k=F/xk=F/xk=F/x, giving N m−1\text{N m}^{-1}N m−1 or N/m\text{N}/\text{m}N/m.
Example
Finding the spring constant
- A force of 6.0 N6.0\,\text{N}6.0N produces an extension of 4.0 cm=0.040 m4.0\,\text{cm}=0.040\,\text{m}4.0cm=0.040m.
- k=Fx=6.00.040=150 N m−1k=\dfrac{F}{x}=\dfrac{6.0}{0.040}=150\,\text{N m}^{-1}k=xF=0.0406.0=150N m−1.
Direct proportion has a precise meaning
- While kkk is constant, doubling the force doubles the extension, so F∝xF\propto xF∝x.
- A force against extension graph is a straight line through the origin in this region, and its gradient equals kkk.
Common Mistake
Do not substitute total spring length for xxx; calculate the extension from the change in length.
Self review
- What does the spring constant measure?
- How is extension calculated from two length measurements?
- What equation links force, spring constant and extension?
- What does the gradient of a force against extension graph represent?
