A load–extension graph shows how a spring stretches
Definition
Extension
The increase in length of an object when a load is applied to it.
extension=stretched length−original length \text{extension} = \text{stretched length} - \text{original length} extension=stretched length−original length- A force can change an object's motion, but it can also change its size and shape, and stretching a spring is the standard way to study this.
- The load is the stretching force in newtons; hanging a 100 g100\ \text{g}100 g mass gives a load of about 0.98 N0.98\ \text{N}0.98 N, from the weight W=mgW = mgW=mg with g=9.8 N/kgg = 9.8\ \text{N/kg}g=9.8 N/kg (use 10 N/kg10\ \text{N/kg}10 N/kg if a question says so).
- By convention the graph is plotted with load on the vertical axis and extension on the horizontal axis.
- For a spring the graph has two clear parts.
- At first it is a straight line through the origin: extension is directly proportional to load, so doubling the load doubles the extension.
- Then the line curves away from the straight part, bending towards the extension axis: equal increases in load now give larger increases in extension, because the spring is stretching more easily.
- A steeper straight line means a stiffer spring, because more force is needed for each centimetre of stretch.
Practical
Investigation: stretching a spring
- Aim: to plot a load–extension graph for a spring and find how its extension depends on the load.
- Apparatus: eye protection, a stand and clamp, a spring, a hanger carrying 100 g100\ \text{g}100 g slotted masses, a pointer and a vertical metre rule.
- Method:
- Clamp the spring from a stand and fix a metre rule vertically beside it, with a pointer at the lower end of the spring.
- Record the unloaded reading of the pointer against the rule.
- Add masses in 100 g100\ \text{g}100 g steps (0.98 N0.98\ \text{N}0.98 N each at g=9.8 N/kgg = 9.8\ \text{N/kg}g=9.8 N/kg), recording the new reading each time.
- Work out the extension as the new reading minus the unloaded reading.
- Unload step by step, re-reading each length, then average the loading and unloading readings.
- Measure and plot:
- Record load in N\text{N}N and extension in cm\text{cm}cm in a table headed load / N and extension / cm.
- Plot load on the yyy-axis against extension on the xxx-axis and draw one thin best-fit line.
- Result: Extension is directly proportional to load, giving a straight line through the origin whose gradient is the spring constant k=F/xk = F/xk=F/x.
- Watch out:
- Read the pointer at eye level, square to the scale, to avoid parallax error.
- Keep the rule vertical and take at least six different loads.
- Do not overload the spring past the point where it stops returning to its original length, or the later readings are ruined.
- Safety: Wear eye protection and keep feet clear of where the masses would fall if the spring snapped.
The limit of proportionality is where the line stops being straight
Definition
Limit of proportionality
The point on a load–extension graph beyond which the extension is no longer directly proportional to the load, which is the point where the graph stops being a straight line.
- On a sketch, mark it where the line first begins to curve away from the straight part.
- Beyond this point each extra newton of load produces more extension than before, so the relationship k=F/xk = F/xk=F/x no longer holds.
The spring constant measures how stiff a spring is
Definition
Spring constant
The force per unit extension of a spring, that is, the force needed to stretch it by one unit of length.
k=Fx k = \frac{F}{x} k=xFwhere FFF is the load in N\text{N}N and xxx is the extension in m\text{m}m, giving kkk in N/m\text{N/m}N/m.
- Because load and extension are proportional below the limit of proportionality, kkk is the gradient of the load–extension graph over its straight section.
- A large kkk means a stiff spring that needs a big force for a small stretch.
- Watch the axes: if a graph plots extension against load instead, the gradient is 1/k1/k1/k, not kkk.
Common Mistake
- Extension is not length: always subtract the original length first.
- The limit of proportionality is not the elastic limit; only the limit of proportionality is needed here, and the two mark different points on the graph.
- Reserve the word proportional for a straight line through the origin; a straight line that misses the origin is linear, not proportional.
- The spring constant is a fixed property of a spring, so it does not increase as you add more load below the limit of proportionality.
Exam technique
- Describing the experiment is point-marked, so give the original length, a range of at least five or six loads, extension as new length minus original length, an eye-level reading to avoid parallax, and a plotted graph; "measure carefully" earns nothing.
- You have to recall the equation, so write k=F/xk = F/xk=F/x before substituting, convert cm to m, and give the unit with the final answer.
- If asked to mark the limit of proportionality, put a clear labelled point exactly where the line starts to curve, not at the end of the graph.
- Describe the shape in the right words: directly proportional for the straight part, and extension increases more for equal increases in load for the curved part.
Self review
- What is meant by the extension of a spring?
- Describe the shape of a load–extension graph for a spring.
- What does the limit of proportionality mark on the graph?
- Define the spring constant, and give its equation and unit.
- A 5.0 N5.0\ \text{N}5.0 N load extends a spring by 0.10 m0.10\ \text{m}0.10 m within the straight-line region. Find the spring constant.
