Section 1 of 6The Law
Many materials behave like ideal springs, as long as they are not stretched too far.
Statement
- Up to the limit of proportionality, the extension of a material is directly proportional to the force applied to it.
Equation
- F = kx, where F is the force in newtons, x is the extension in metres and k is the spring constant in N m⁻¹.
Spring constant
- The constant k measures stiffnessa large k means a large force is needed for each metre of extension.
Extension
- Extension is the new length minus the original length, not the length of the spring itself.
Compression
- The same law holds when a spring is compressed, with x as the decrease in length.
Section 2 of 6Testing the Law
A simple experiment checks whether a spring obeys the law.
1
Hang a spring from a clamp beside a vertical metre rule and record its unloaded length with a set-square to avoid parallax.
2
Add a 1.0 N weight and record the new length, then calculate the extension by subtraction.
3
Continue adding equal weights, recording the length at each step, until several readings beyond the straight region are taken.
4
Plot force on the vertical axis against extension on the horizontal axis and draw the best-fit line through the early points.
5
The gradient of the straight section is the spring constant, and the point where the line curves is the limit of proportionality.
Section 3 of 6Reading the Graph
Reading the Graph
Straight region
- Force and extension are proportional here, so the graph is a straight line through the origin.
Gradient
- The slope of force against extension equals k; if the axes are swapped the slope is 1/k instead.
Limit of proportionality
- Beyond this point the line curves, and the material no longer obeys Hooke’s law even if it still returns to its shape.
Stiffness
- A steeper line on the force-extension graph represents a stiffer spring with a greater spring constant.
Section 4 of 6Limits and Uses
Springs
- Car suspensions and weighing scales rely on a predictable proportional response between load and extension.
Permanent change
- Overstretching a spring causes permanent deformation, so its natural length and k both change.
Wires
- A metal wire follows Hooke’s law for small loads, but rubber and polymers often do not, even for small extensions.
Section 5 of 6Worked Example
Question
- A spring extends by 0.080 m when a 12 N force is applied and obeys Hooke’s law.
- Find (a) its spring constant and (b) the extension produced by 20 N.
- (a) k = F/x = 12 / 0.080 = 150 N m⁻¹.
- (b) x = F/k = 20 / 150 = 0.133 m.
Answer
- (a) k = 150 N m⁻¹;
- (b) extension = 0.13 m, provided 20 N is below the limit of proportionality.
Section 6 of 6Exam Checklist & Tips
3.7.2 – Key Points for the Exam
- Hooke’s law Extension is directly proportional to applied force up to the limit of proportionality.
- Spring constant The force needed per unit extension, k = F/x, measured in N m⁻¹.
- Extension The increase in length of a stretched object.
- Limit of proportionality The point beyond which force and extension are no longer directly proportional.
Common Mistakes
- Using the total length of the spring instead of the extension in F = kx.
- Forgetting to convert centimetres or millimetres to metres, giving k wrong by a factor of 100 or 1000.
- Taking the gradient as k when the graph has extension on the vertical axis.
- Assuming a material still obeys Hooke’s law beyond the limit of proportionality.