Graph shape distinguishes linear behaviour
Definition
Limit of proportionality
The point beyond which force and extension are no longer directly proportional.
- A linear force-extension relationship produces a straight line, while a non-linear relationship produces a curve.
- Direct proportionality requires a straight line through the origin, not merely any straight line.
- Below the limit of proportionality, F=kxF=kxF=kx and the gradient of a force against extension graph is the constant kkk.
- Beyond this limit, extension no longer increases in direct proportion to force and the gradient changes.
Loading and unloading reveal permanent change
- If unloading returns the graph to the origin, the object has recovered its original length.
- If the unloading path finishes at a non-zero extension, the object has undergone inelastic distortion.
- Rubber bands commonly give curved force-extension graphs even when they return to their original length, so non-linear does not automatically mean inelastic.
Common Mistake
Do not confuse the limit of proportionality with the point where permanent deformation begins; they are not necessarily the same point.
Practical
Investigating a spring
- Apparatus: clamp stand, boss and clamp, spring, mass hanger and slotted masses, metre rule, pointer, set square and eye protection.
- Set-up: clamp the stand securely, hang the spring beside a vertical metre rule and attach a horizontal pointer to the lower end of the spring.
- Method:
- record the unstretched spring length L0L_0L0 at eye level, using a set square to align the pointer with the scale.
- add masses in equal steps, allow oscillations to stop, record the new length LLL and calculate extension using x=L−L0x=L-L_0x=L−L0.
- calculate the applied force from F=mgF=mgF=mg, using the total hanging mass in kilograms and g=10 N kg−1g=10\,\text{N kg}^{-1}g=10N kg−1.
- take at least six force-extension pairs without exceeding the spring's safe load, then remove the masses in the same steps to obtain unloading readings.
- repeat each reading, identify anomalous values and calculate a mean extension for each force.
- plot FFF on the vertical axis against xxx on the horizontal axis, draw a best-fit line or curve and use a large gradient triangle in the linear region to find k=ΔF/Δxk=\Delta F/\Delta xk=ΔF/Δx.
- determine the area beneath the force-extension graph; use E=12kx2E=\dfrac{1}{2}kx^2E=21kx2 only for the straight-line region.
- Variables: change the applied force, measure extension, and keep the same spring, starting length, ruler position and temperature.
- Errors: avoid parallax by reading at eye level, clamp the ruler, use a pointer and wait for the spring to stop moving.
- Safety: wear eye protection, keep feet clear of falling masses, secure the stand and do not overload the spring.
Example
Using experimental data
- A best-fit line rises from (0.010 m,2.0 N)(0.010\,\text{m},2.0\,\text{N})(0.010m,2.0N) to (0.040 m,8.0 N)(0.040\,\text{m},8.0\,\text{N})(0.040m,8.0N).
- k=8.0−2.00.040−0.010=200 N m−1k=\dfrac{8.0-2.0}{0.040-0.010}=200\,\text{N m}^{-1}k=0.040−0.0108.0−2.0=200N m−1.
- At x=0.040 mx=0.040\,\text{m}x=0.040m, E=12×200×(0.040)2=0.16 JE=\dfrac{1}{2}\times200\times(0.040)^2=0.16\,\text{J}E=21×200×(0.040)2=0.16J.
Self review
- How does a linear relationship differ from a directly proportional relationship?
- What is the limit of proportionality?
- How is extension obtained in the spring practical?
- How is spring constant found from the graph?
- How is work done found from a non-linear graph?
