Learning outcomes
- Use spring constant k = F/x.
- Interpret load–extension graphs and locate the limit of proportionality.
- Describe an experiment on spring extension.
- Explain qualitatively how inward force maintains circular motion.
8.1 Deformation and springs
A force can change an object’s size or shape. A spring extends when pulled and compresses when pushed. Extension x is the increase in length, not the final length. Within the proportional region, force is directly proportional to extension and k = F/x.
The spring constant k is measured in N m⁻¹ and indicates stiffness. A large k means a large force is required for a given extension. When using centimetres in raw data, convert extension to metres before calculating k in SI units.
8.2 Load–extension graph
Plot load on the vertical axis against extension on the horizontal axis. The initial straight line through the origin shows F ∝ x. Its gradient is the spring constant when F is vertical and x horizontal. The limit of proportionality is the point beyond which the graph is no longer straight.
The syllabus requires the limit of proportionality but not a detailed treatment of the elastic limit. Avoid claiming the spring will definitely fail immediately after the proportional limit; the graph only shows that the simple proportional relationship no longer applies.
8.3 Practical method and quality
Clamp the spring securely beside a vertical ruler. Measure its original length, add known loads in equal steps, wait for oscillations to stop and record the new length. Calculate extension for each load. Keep the ruler close to the spring and read at eye level. Do not exceed a safe load.
Repeat readings while unloading if appropriate. A pointer attached to the spring can reduce difficulty in judging the end position. Plot a best-fit line through the proportional region and identify any anomalous reading.
8.4 Circular motion
An object moving in a circle has changing velocity because its direction changes. A resultant force directed towards the centre continuously turns the velocity vector. If the inward force disappears, the object moves along the tangent at the release point, not radially outward.
Qualitatively, a larger inward force can maintain a higher speed for the same mass and radius, or a smaller radius for the same mass and speed. A larger mass requires a larger inward force to follow the same path at the same speed. The formula F = mv²/r is not required in syllabus 5054, but these relationships should be described.
Worked examples
Spring constantA 6.0 N load produces an extension of 3.0 cm = 0.030 m. k = 6.0/0.030 = 200 N m⁻¹.
Using a graphIf a force–extension graph is straight up to 5 N and then curves, 5 N marks the approximate limit of proportionality.
Circular pathFor the same mass and radius, increasing the speed requires a larger inward force. The force remains perpendicular to instantaneous motion.
Practical focus
InvestigationInvestigate extension against load. Use at least six load values, repeat readings and plot F against x. Discuss why zero extension should correspond to zero added load and why the graph may not pass exactly through the origin due to reading or preload effects.
Examination guidance
- Extension = stretched length − original length.
- For a force–extension graph, gradient is k only when force is on the y-axis and extension on the x-axis.
- The inward force changes direction, not necessarily speed.
Check your understanding
- A spring length changes from 12.0 cm to 15.5 cm. Extension?
- What does a high spring constant mean?
- What direction is the resultant force in circular motion?
Answers
- 3.5 cm.
- The spring is stiff; a large force is needed per unit extension.
- Towards the centre of the circle.
