Linear relationships

Definition

Linear relationship

A relationship between two quantities in which equal changes in one produce equal changes in the other, giving a straight-line graph.

A relationship between force and extension is linear when equal increases in force always produce equal increases in extension.
A linear relationship gives a straight-line graph, and the gradient is the same everywhere along the line.
A spring loaded below its limit of proportionality has a linear force-extension relationship.
A linear relationship is also directly proportional only if the straight line passes through the origin.
1. A graph of extension against force for a spring is a straight line through the origin, so the relationship is linear and directly proportional.
2. A graph of total length against force for the same spring is still a straight line, but it crosses the length axis at the original length, so that relationship is linear but not directly proportional.

Non-linear relationships

Definition

Non-linear relationship

A relationship between two quantities in which equal changes in one produce unequal changes in the other, giving a curved graph.

A relationship between force and extension is non-linear when equal increases in force produce unequal increases in extension.
A non-linear relationship gives a curved graph, and its gradient changes from one part of the curve to another.
A rubber band has a non-linear relationship over most of its stretch.
1. At first it stretches easily, because the long, tangled chains of molecules in the rubber straighten out with little force.
2. Once the chains are nearly straight, the band becomes much harder to stretch, so each extra newton produces a smaller extension.
A strip of polythene and a strip of liquorice are also non-linear, stretching more and more for each extra newton as they start to thin.
A steel spring becomes non-linear when it is loaded beyond its limit of proportionality, because its coils begin to open out and each extra newton stretches it further than before.

Reading a force-extension graph

A spring loaded to large forces gives a graph with a straight section followed by a curved section.
The point where the straight line starts to curve is the limit of proportionality.
On a graph of extension against force, the line curves upwards beyond this point, showing that each newton now produces a larger extension.
On a graph of force against extension, the same behaviour makes the line curve towards the extension axis, so check which quantity is on which axis before describing the shape.
To find where a relationship stops being linear from a table of results, work out the extra extension for each equal step in force and look for the step where it changes.

The elastic limit

Definition

Elastic limit

The point beyond which an object no longer returns to its original shape when the deforming forces are removed.

Loading an object beyond its elastic limit causes plastic deformation, so the object is left with a permanent extension when the force is removed.
For many springs, the elastic limit lies close to the limit of proportionality, but the two are different points.
The limit of proportionality is found from the shape of the graph during loading.
The elastic limit can only be found by removing the force and checking whether the object returns to its original length.
A curved graph does not by itself show that an object has passed its elastic limit, because a rubber band behaves non-linearly yet still returns almost to its original length.

Example

Finding where a spring stops being linear

A spring is loaded in steps of $N1.0\ \text{N}$ , giving extensions of $cm1.5\ \text{cm}$ , $cm3.0\ \text{cm}$ , $cm4.5\ \text{cm}$ , $cm6.0\ \text{cm}$ , $cm8.2\ \text{cm}$ and $cm11.0\ \text{cm}$ .
The extra extension for each step is found by subtracting each extension from the next one.

1.5\ \text{cm},\quad 1.5\ \text{cm},\quad 1.5\ \text{cm},\quad 1.5\ \text{cm},\quad 2.2\ \text{cm},\quad 2.8\ \text{cm}

The first four steps are equal, so the relationship is linear up to $N4.0\ \text{N}$ .
The later steps grow larger, so the relationship becomes non-linear above $N4.0\ \text{N}$ , and the limit of proportionality is reached at about $N4.0\ \text{N}$ .

Example

Linear but not directly proportional

A spring has an original length of $cm20\ \text{cm}$ and extends by $cm5\ \text{cm}$ for each newton of force.
Its total length is $cm25\ \text{cm}$ at $N1\ \text{N}$ , $cm30\ \text{cm}$ at $N2\ \text{N}$ and $cm35\ \text{cm}$ at $N3\ \text{N}$ .
Each newton adds the same $cm5\ \text{cm}$ , so a graph of total length against force is a straight line and the relationship is linear.

2\times25\ \text{cm}=50\ \text{cm}\neq30\ \text{cm}

Doubling the force does not double the total length, so the total length is not directly proportional to the force, even though the extension is.

Practical

Investigating the elastic limit

Aim: to find the load at which a spring and a strip of polythene stop returning to their original length, and to compare this with the shape of the force-extension graph.
Apparatus: a clamp stand with two bosses and clamps, a G-clamp, an old steel spring that can be damaged, a strip of polythene about $cm2\ \text{cm}$ wide cut from a carrier bag, two small wooden blocks, a bulldog clip, a $g100\ \text{g}$ mass hanger with $g100\ \text{g}$ slotted masses, a metre rule, a set square, a pointer, a marker pen, a tray of sand and eye protection.
Method, the spring:
- Hang the spring from the clamp beside a vertical metre rule, fix the pointer to its lower end and record the unloaded pointer position at eye level.
- Hang the mass hanger on the spring, which applies $N1\ \text{N}$ , and record the loaded pointer position once the spring is still.
- Remove the hanger completely and record the unloaded pointer position again.
- If the pointer returns to its original position, reload the spring with $g100\ \text{g}$ more than last time and repeat the load, measure, unload, measure cycle.
- Continue until the unloaded reading no longer returns to the original position, which shows a permanent extension has appeared.
- Use the loaded readings to plot extension against force and mark where the line starts to curve.
Method, the polythene strip:
- Grip the top of the strip between the two wooden blocks in the clamp and fix the bulldog clip to its lower end to hold the mass hanger.
- Draw two thin lines across the strip $cm10.0\ \text{cm}$ apart with the marker pen, and measure the distance between them.
- Repeat the load, measure, unload, measure cycle, measuring the distance between the two lines each time.
Results: the spring returns to its original position for the smaller loads and then keeps a permanent extension above a certain load, while the polythene strip becomes permanently longer after quite small loads and starts to narrow before it tears.
Conclusion: the elastic limit lies between the last load that gave full recovery and the first load that left a permanent extension, and for the spring this was close to the load at which its graph began to curve.
Watch out: unload completely every time, leave the same settling time before each reading, and use a spring that is not needed again because it will be permanently stretched.
Safety: wear eye protection, place the tray of sand under the masses to catch them, keep your feet clear, and stop adding masses if the polythene starts to tear.

Common Mistake

Do not treat linear and directly proportional as the same thing, because a straight line that misses the origin is linear but not directly proportional.
Do not mix up the limit of proportionality with the elastic limit, because the first concerns the shape of the loading graph and the second concerns whether the object returns to its original shape.
Do not call a curve non-linear without saying how the extension changes, such as each extra newton giving a larger extension.

Exam technique

Comparing linear and non-linear behaviour

Describe a linear relationship as a straight line, and add through the origin only if the line really passes through it.
Describe a non-linear relationship as a curve and explain it as unequal extensions for equal increases in force.
Quote values from the graph or table, such as the force where the line starts to curve, instead of describing the shape only in general terms.
When asked how to show that the elastic limit has been passed, say that the force is removed and the length is measured again to check for a permanent extension.

Self review

Describe the difference between a linear and a non-linear force-extension relationship.
Explain why a graph of total length against force for a spring is linear but not directly proportional.
State what happens to the force-extension graph for a spring beyond the limit of proportionality.
Describe how you would show that a spring has been loaded beyond its elastic limit.
Name a material that behaves non-linearly over most of its stretch.

Linear relationships

Definition

Linear relationship

A relationship between two quantities in which equal changes in one produce equal changes in the other, giving a straight-line graph.

A relationship between force and extension is linear when equal increases in force always produce equal increases in extension.
A linear relationship gives a straight-line graph, and the gradient is the same everywhere along the line.
A spring loaded below its limit of proportionality has a linear force-extension relationship.
A linear relationship is also directly proportional only if the straight line passes through the origin.
1. A graph of extension against force for a spring is a straight line through the origin, so the relationship is linear and directly proportional.
2. A graph of total length against force for the same spring is still a straight line, but it crosses the length axis at the original length, so that relationship is linear but not directly proportional.

Non-linear relationships

Definition

Non-linear relationship

A relationship between two quantities in which equal changes in one produce unequal changes in the other, giving a curved graph.

A relationship between force and extension is non-linear when equal increases in force produce unequal increases in extension.
A non-linear relationship gives a curved graph, and its gradient changes from one part of the curve to another.
A rubber band has a non-linear relationship over most of its stretch.
1. At first it stretches easily, because the long, tangled chains of molecules in the rubber straighten out with little force.
2. Once the chains are nearly straight, the band becomes much harder to stretch, so each extra newton produces a smaller extension.
A strip of polythene and a strip of liquorice are also non-linear, stretching more and more for each extra newton as they start to thin.
A steel spring becomes non-linear when it is loaded beyond its limit of proportionality, because its coils begin to open out and each extra newton stretches it further than before.

Reading a force-extension graph

A spring loaded to large forces gives a graph with a straight section followed by a curved section.
The point where the straight line starts to curve is the limit of proportionality.
On a graph of extension against force, the line curves upwards beyond this point, showing that each newton now produces a larger extension.
On a graph of force against extension, the same behaviour makes the line curve towards the extension axis, so check which quantity is on which axis before describing the shape.
To find where a relationship stops being linear from a table of results, work out the extra extension for each equal step in force and look for the step where it changes.

The elastic limit

Definition

Elastic limit

The point beyond which an object no longer returns to its original shape when the deforming forces are removed.

Loading an object beyond its elastic limit causes plastic deformation, so the object is left with a permanent extension when the force is removed.
For many springs, the elastic limit lies close to the limit of proportionality, but the two are different points.
The limit of proportionality is found from the shape of the graph during loading.
The elastic limit can only be found by removing the force and checking whether the object returns to its original length.
A curved graph does not by itself show that an object has passed its elastic limit, because a rubber band behaves non-linearly yet still returns almost to its original length.

Example

Finding where a spring stops being linear

A spring is loaded in steps of $N1.0\ \text{N}$ , giving extensions of $cm1.5\ \text{cm}$ , $cm3.0\ \text{cm}$ , $cm4.5\ \text{cm}$ , $cm6.0\ \text{cm}$ , $cm8.2\ \text{cm}$ and $cm11.0\ \text{cm}$ .
The extra extension for each step is found by subtracting each extension from the next one.

1.5\ \text{cm},\quad 1.5\ \text{cm},\quad 1.5\ \text{cm},\quad 1.5\ \text{cm},\quad 2.2\ \text{cm},\quad 2.8\ \text{cm}

The first four steps are equal, so the relationship is linear up to $N4.0\ \text{N}$ .
The later steps grow larger, so the relationship becomes non-linear above $N4.0\ \text{N}$ , and the limit of proportionality is reached at about $N4.0\ \text{N}$ .

Example

Linear but not directly proportional

A spring has an original length of $cm20\ \text{cm}$ and extends by $cm5\ \text{cm}$ for each newton of force.
Its total length is $cm25\ \text{cm}$ at $N1\ \text{N}$ , $cm30\ \text{cm}$ at $N2\ \text{N}$ and $cm35\ \text{cm}$ at $N3\ \text{N}$ .
Each newton adds the same $cm5\ \text{cm}$ , so a graph of total length against force is a straight line and the relationship is linear.

2\times25\ \text{cm}=50\ \text{cm}\neq30\ \text{cm}

Doubling the force does not double the total length, so the total length is not directly proportional to the force, even though the extension is.

Practical

Investigating the elastic limit

Aim: to find the load at which a spring and a strip of polythene stop returning to their original length, and to compare this with the shape of the force-extension graph.
Apparatus: a clamp stand with two bosses and clamps, a G-clamp, an old steel spring that can be damaged, a strip of polythene about $cm2\ \text{cm}$ wide cut from a carrier bag, two small wooden blocks, a bulldog clip, a $g100\ \text{g}$ mass hanger with $g100\ \text{g}$ slotted masses, a metre rule, a set square, a pointer, a marker pen, a tray of sand and eye protection.
Method, the spring:
- Hang the spring from the clamp beside a vertical metre rule, fix the pointer to its lower end and record the unloaded pointer position at eye level.
- Hang the mass hanger on the spring, which applies $N1\ \text{N}$ , and record the loaded pointer position once the spring is still.
- Remove the hanger completely and record the unloaded pointer position again.
- If the pointer returns to its original position, reload the spring with $g100\ \text{g}$ more than last time and repeat the load, measure, unload, measure cycle.
- Continue until the unloaded reading no longer returns to the original position, which shows a permanent extension has appeared.
- Use the loaded readings to plot extension against force and mark where the line starts to curve.
Method, the polythene strip:
- Grip the top of the strip between the two wooden blocks in the clamp and fix the bulldog clip to its lower end to hold the mass hanger.
- Draw two thin lines across the strip $cm10.0\ \text{cm}$ apart with the marker pen, and measure the distance between them.
- Repeat the load, measure, unload, measure cycle, measuring the distance between the two lines each time.
Results: the spring returns to its original position for the smaller loads and then keeps a permanent extension above a certain load, while the polythene strip becomes permanently longer after quite small loads and starts to narrow before it tears.
Conclusion: the elastic limit lies between the last load that gave full recovery and the first load that left a permanent extension, and for the spring this was close to the load at which its graph began to curve.
Watch out: unload completely every time, leave the same settling time before each reading, and use a spring that is not needed again because it will be permanently stretched.
Safety: wear eye protection, place the tray of sand under the masses to catch them, keep your feet clear, and stop adding masses if the polythene starts to tear.

Common Mistake

Do not treat linear and directly proportional as the same thing, because a straight line that misses the origin is linear but not directly proportional.
Do not mix up the limit of proportionality with the elastic limit, because the first concerns the shape of the loading graph and the second concerns whether the object returns to its original shape.
Do not call a curve non-linear without saying how the extension changes, such as each extra newton giving a larger extension.

Exam technique

Comparing linear and non-linear behaviour

Describe a linear relationship as a straight line, and add through the origin only if the line really passes through it.
Describe a non-linear relationship as a curve and explain it as unequal extensions for equal increases in force.
Quote values from the graph or table, such as the force where the line starts to curve, instead of describing the shape only in general terms.
When asked how to show that the elastic limit has been passed, say that the force is removed and the length is measured again to check for a permanent extension.

Self review

Describe the difference between a linear and a non-linear force-extension relationship.
Explain why a graph of total length against force for a spring is linear but not directly proportional.
State what happens to the force-extension graph for a spring beyond the limit of proportionality.
Describe how you would show that a spring has been loaded beyond its elastic limit.
Name a material that behaves non-linearly over most of its stretch.

2.3.4 Linear and non-linear force-extension behaviour

Linear relationships

Non-linear relationships

Reading a force-extension graph

The elastic limit

Finding where a spring stops being linear

Linear but not directly proportional

Investigating the elastic limit

Comparing linear and non-linear behaviour

Review 2.3.4 Linear and non-linear force-extension behaviour by teaching Genie

2.3.4 Linear and non-linear force-extension behaviour

Linear relationships

Non-linear relationships

Reading a force-extension graph

The elastic limit

Finding where a spring stops being linear

Linear but not directly proportional

Investigating the elastic limit

Comparing linear and non-linear behaviour

Review 2.3.4 Linear and non-linear force-extension behaviour by teaching Genie

2.3.4 Linear and non-linear force-extension behaviour

Linear relationships

Non-linear relationships

Reading a force-extension graph

The elastic limit

Finding where a spring stops being linear

Linear but not directly proportional

Investigating the elastic limit

Comparing linear and non-linear behaviour

Review 2.3.4 Linear and non-linear force-extension behaviour by teaching Genie

2.3.4 Linear and non-linear force-extension behaviour Revision Guide

2.3.4 Linear and non-linear force-extension behaviour

Linear relationships

Non-linear relationships

Reading a force-extension graph

The elastic limit

Finding where a spring stops being linear

Linear but not directly proportional

Investigating the elastic limit

Comparing linear and non-linear behaviour

Review 2.3.4 Linear and non-linear force-extension behaviour by teaching Genie

2.3.4 Linear and non-linear force-extension behaviour Revision Guide