Force and extension

Definition

Extension

The increase in the length of an object when it is stretched, equal to its stretched length minus its original length.

Definition

Directly proportional

A relationship between two quantities in which multiplying one by a factor multiplies the other by the same factor.

When a force stretches a spring, the extension of the spring depends on the size of the force.
A larger stretching force pulls the coils further apart, so it produces a larger extension.
For a spring, equal increases in force produce equal increases in extension, provided the spring is not stretched too far.
1. If adding $N1\ \text{N}$ stretches a spring by $cm2\ \text{cm}$ , adding another $N1\ \text{N}$ stretches it by a further $cm2\ \text{cm}$ .
This means the extension of a spring is directly proportional to the force applied to it.
1. Doubling the force doubles the extension.
2. Halving the force halves the extension.
3. Applying no force gives no extension.
The relationship is written as $extension∝force\text{extension}\propto\text{force}$ , where the symbol $∝\propto$ means "is directly proportional to".

Hooke's law

Definition

Hooke's law

The extension of a spring is directly proportional to the force applied to it, provided the limit of proportionality is not exceeded.

Definition

Limit of proportionality

The point beyond which the extension of an object is no longer directly proportional to the force applied to it.

The direct proportion between force and extension for a spring is called Hooke's law, after the scientist Robert Hooke, who described it in 1678.
Hooke's law holds only up to a point called the limit of proportionality.
Below the limit of proportionality, the spring stretches by the same amount for every extra newton of force.
Beyond the limit of proportionality, the extension is no longer directly proportional to the force.
Many objects obey Hooke's law for small forces, including steel springs and metal wires.
The spring inside a newton meter obeys Hooke's law over its whole working range, which is why the markings on its scale are evenly spaced.

Force-extension graphs

The relationship between force and extension is shown by plotting a force-extension graph.
In an investigation, force is the quantity you change, so it is often plotted on the horizontal axis with extension on the vertical axis.
Some graphs have force on the vertical axis and extension on the horizontal axis instead, so always read the axis labels first.
For a spring obeying Hooke's law, the graph is a straight line through the origin.
1. The line is straight because equal steps in force give equal steps in extension.
2. The line passes through the origin because zero force gives zero extension.
On a graph of extension against force, a steeper line belongs to a spring that stretches further for each newton.
Values can be read from the straight part of the graph to find the extension for any force within the range plotted.
1. If the line shows an extension of $cm6.0\ \text{cm}$ at $N3.0\ \text{N}$ , the extension at $N1.5\ \text{N}$ is $cm3.0\ \text{cm}$ .

Other simple systems

Other simple systems can be loaded in the same way as a spring, with the extension measured for each force.
A metal wire behaves like a spring for small forces, because its extension is directly proportional to the force, although the extensions are tiny and need very careful measurement.
A rubber band and a strip of polythene also extend more as the force increases, but their extension is not directly proportional to the force, so their graphs are not straight lines through the origin.
Two identical springs hung one below the other each carry the full load, so each stretches by the same amount and the total extension is double that of one spring.
Two identical springs hung side by side share the load equally, so each carries half the force and the extension is half that of one spring.

Example

Checking results against Hooke's law

A spring has an original length of $cm12.0\ \text{cm}$ , a length of $cm15.0\ \text{cm}$ with a $N2.0\ \text{N}$ load and a length of $cm18.0\ \text{cm}$ with a $N4.0\ \text{N}$ load.

\text{extension at } 2.0\ \text{N}=15.0\ \text{cm}-12.0\ \text{cm}=3.0\ \text{cm}

\text{extension at } 4.0\ \text{N}=18.0\ \text{cm}-12.0\ \text{cm}=6.0\ \text{cm}

Doubling the force from $N2.0\ \text{N}$ to $N4.0\ \text{N}$ doubles the extension from $cm3.0\ \text{cm}$ to $cm6.0\ \text{cm}$ .
The extension is directly proportional to the force, so these results support Hooke's law.
The stretched lengths of $cm15.0\ \text{cm}$ and $cm18.0\ \text{cm}$ do not double, which is why the extension must be worked out before testing for proportionality.

Example

Predicting an extension

A spring obeying Hooke's law extends by $cm2.5\ \text{cm}$ when a force of $N0.50\ \text{N}$ is applied.
The force is increased to $N3.0\ \text{N}$ , which is still below the limit of proportionality.

\text{scale factor}=\frac{3.0\ \text{N}}{0.50\ \text{N}}=6

\text{new extension}=6\times2.5\ \text{cm}=15\ \text{cm}

The force is six times larger, so the extension is also six times larger.

Practical

Investigating the effect of forces on a spring

Aim: to investigate how the extension of a spring depends on the force applied to it, and to test whether the spring obeys Hooke's law.
Apparatus: a clamp stand with two bosses and clamps, a G-clamp, a steel spring, a $g100\ \text{g}$ mass hanger with eight $g100\ \text{g}$ slotted masses, a metre rule, a set square, a pin or strip of tape as a pointer, and eye protection.
Variables: the force on the spring is the independent variable, the extension is the dependent variable, and the spring, the pointer position and the zero reading are kept the same throughout.
Method:
- Fix the clamp stand to the bench with the G-clamp and hang the spring from the upper clamp.
- Clamp the metre rule vertically beside the spring, close enough that the pointer almost touches the scale.
- Fix the pointer horizontally to the bottom of the spring and record its position with no load, using the set square and keeping your eye level with the pointer.
- Hang the $g100\ \text{g}$ mass hanger on the spring, which applies a force of $N1\ \text{N}$ , wait until the spring is completely still, and record the pointer position.
- Add $g100\ \text{g}$ masses one at a time, so the force rises in $N1\ \text{N}$ steps up to $N8\ \text{N}$ , recording the pointer position after each addition.
- Subtract the unloaded reading from every loaded reading to find the extension at each force.
- Repeat the whole set of readings twice more and calculate a mean extension for each force.
- Plot a graph of extension on the vertical axis against force on the horizontal axis, and draw a line of best fit.
Results: a typical steel spring extends by about $cm2\ \text{cm}$ for every $N1\ \text{N}$ , so its points lie on a straight line that passes through the origin.
Conclusion: the straight line through the origin shows that the extension is directly proportional to the force, so the spring obeys Hooke's law over this range of forces.
Watch out: plot the extension and not the stretched length, read the scale at the same point on the pointer every time, and treat any points that curve away at the largest forces as evidence that the limit of proportionality has been passed.
Safety: wear eye protection in case the spring snaps or flies off, keep your feet clear of the area under the masses, and do not load the spring beyond the masses provided.

Common Mistake

Do not plot the stretched length instead of the extension, because a graph of length against force does not pass through the origin.
Do not say that force and extension are proportional for any amount of stretching, because Hooke's law applies only up to the limit of proportionality.
Do not call a straight line that misses the origin directly proportional.

Exam technique

Describing force-extension relationships

Use the words directly proportional and add up to the limit of proportionality, because writing only that the extension increases as the force increases describes a trend and scores less.
Back up a claim of proportionality with data from the question, showing that doubling the force doubles the extension.
Describe the graph as a straight line through the origin, naming both features.
In a practical question, name force as the independent variable and extension as the dependent variable, and say how the extension is calculated.

Self review

State Hooke's law.
Describe the shape of the force-extension graph for a spring that obeys Hooke's law.
Calculate the extension of a spring with a $N5.0\ \text{N}$ load if it extends by $cm4.0\ \text{cm}$ with a $N2.0\ \text{N}$ load and obeys Hooke's law.
Explain why a graph of stretched length against force does not pass through the origin.
Describe what happens to the total extension when two identical springs are hung one below the other.

Force and extension

Definition

Extension

The increase in the length of an object when it is stretched, equal to its stretched length minus its original length.

Definition

Directly proportional

A relationship between two quantities in which multiplying one by a factor multiplies the other by the same factor.

When a force stretches a spring, the extension of the spring depends on the size of the force.
A larger stretching force pulls the coils further apart, so it produces a larger extension.
For a spring, equal increases in force produce equal increases in extension, provided the spring is not stretched too far.
1. If adding $N1\ \text{N}$ stretches a spring by $cm2\ \text{cm}$ , adding another $N1\ \text{N}$ stretches it by a further $cm2\ \text{cm}$ .
This means the extension of a spring is directly proportional to the force applied to it.
1. Doubling the force doubles the extension.
2. Halving the force halves the extension.
3. Applying no force gives no extension.
The relationship is written as $extension∝force\text{extension}\propto\text{force}$ , where the symbol $∝\propto$ means "is directly proportional to".

Hooke's law

Definition

Hooke's law

The extension of a spring is directly proportional to the force applied to it, provided the limit of proportionality is not exceeded.

Definition

Limit of proportionality

The point beyond which the extension of an object is no longer directly proportional to the force applied to it.

The direct proportion between force and extension for a spring is called Hooke's law, after the scientist Robert Hooke, who described it in 1678.
Hooke's law holds only up to a point called the limit of proportionality.
Below the limit of proportionality, the spring stretches by the same amount for every extra newton of force.
Beyond the limit of proportionality, the extension is no longer directly proportional to the force.
Many objects obey Hooke's law for small forces, including steel springs and metal wires.
The spring inside a newton meter obeys Hooke's law over its whole working range, which is why the markings on its scale are evenly spaced.

Force-extension graphs

The relationship between force and extension is shown by plotting a force-extension graph.
In an investigation, force is the quantity you change, so it is often plotted on the horizontal axis with extension on the vertical axis.
Some graphs have force on the vertical axis and extension on the horizontal axis instead, so always read the axis labels first.
For a spring obeying Hooke's law, the graph is a straight line through the origin.
1. The line is straight because equal steps in force give equal steps in extension.
2. The line passes through the origin because zero force gives zero extension.
On a graph of extension against force, a steeper line belongs to a spring that stretches further for each newton.
Values can be read from the straight part of the graph to find the extension for any force within the range plotted.
1. If the line shows an extension of $cm6.0\ \text{cm}$ at $N3.0\ \text{N}$ , the extension at $N1.5\ \text{N}$ is $cm3.0\ \text{cm}$ .

Other simple systems

Other simple systems can be loaded in the same way as a spring, with the extension measured for each force.
A metal wire behaves like a spring for small forces, because its extension is directly proportional to the force, although the extensions are tiny and need very careful measurement.
A rubber band and a strip of polythene also extend more as the force increases, but their extension is not directly proportional to the force, so their graphs are not straight lines through the origin.
Two identical springs hung one below the other each carry the full load, so each stretches by the same amount and the total extension is double that of one spring.
Two identical springs hung side by side share the load equally, so each carries half the force and the extension is half that of one spring.

Example

Checking results against Hooke's law

A spring has an original length of $cm12.0\ \text{cm}$ , a length of $cm15.0\ \text{cm}$ with a $N2.0\ \text{N}$ load and a length of $cm18.0\ \text{cm}$ with a $N4.0\ \text{N}$ load.

\text{extension at } 2.0\ \text{N}=15.0\ \text{cm}-12.0\ \text{cm}=3.0\ \text{cm}

\text{extension at } 4.0\ \text{N}=18.0\ \text{cm}-12.0\ \text{cm}=6.0\ \text{cm}

Doubling the force from $N2.0\ \text{N}$ to $N4.0\ \text{N}$ doubles the extension from $cm3.0\ \text{cm}$ to $cm6.0\ \text{cm}$ .
The extension is directly proportional to the force, so these results support Hooke's law.
The stretched lengths of $cm15.0\ \text{cm}$ and $cm18.0\ \text{cm}$ do not double, which is why the extension must be worked out before testing for proportionality.

Example

Predicting an extension

A spring obeying Hooke's law extends by $cm2.5\ \text{cm}$ when a force of $N0.50\ \text{N}$ is applied.
The force is increased to $N3.0\ \text{N}$ , which is still below the limit of proportionality.

\text{scale factor}=\frac{3.0\ \text{N}}{0.50\ \text{N}}=6

\text{new extension}=6\times2.5\ \text{cm}=15\ \text{cm}

The force is six times larger, so the extension is also six times larger.

Practical

Investigating the effect of forces on a spring

Aim: to investigate how the extension of a spring depends on the force applied to it, and to test whether the spring obeys Hooke's law.
Apparatus: a clamp stand with two bosses and clamps, a G-clamp, a steel spring, a $g100\ \text{g}$ mass hanger with eight $g100\ \text{g}$ slotted masses, a metre rule, a set square, a pin or strip of tape as a pointer, and eye protection.
Variables: the force on the spring is the independent variable, the extension is the dependent variable, and the spring, the pointer position and the zero reading are kept the same throughout.
Method:
- Fix the clamp stand to the bench with the G-clamp and hang the spring from the upper clamp.
- Clamp the metre rule vertically beside the spring, close enough that the pointer almost touches the scale.
- Fix the pointer horizontally to the bottom of the spring and record its position with no load, using the set square and keeping your eye level with the pointer.
- Hang the $g100\ \text{g}$ mass hanger on the spring, which applies a force of $N1\ \text{N}$ , wait until the spring is completely still, and record the pointer position.
- Add $g100\ \text{g}$ masses one at a time, so the force rises in $N1\ \text{N}$ steps up to $N8\ \text{N}$ , recording the pointer position after each addition.
- Subtract the unloaded reading from every loaded reading to find the extension at each force.
- Repeat the whole set of readings twice more and calculate a mean extension for each force.
- Plot a graph of extension on the vertical axis against force on the horizontal axis, and draw a line of best fit.
Results: a typical steel spring extends by about $cm2\ \text{cm}$ for every $N1\ \text{N}$ , so its points lie on a straight line that passes through the origin.
Conclusion: the straight line through the origin shows that the extension is directly proportional to the force, so the spring obeys Hooke's law over this range of forces.
Watch out: plot the extension and not the stretched length, read the scale at the same point on the pointer every time, and treat any points that curve away at the largest forces as evidence that the limit of proportionality has been passed.
Safety: wear eye protection in case the spring snaps or flies off, keep your feet clear of the area under the masses, and do not load the spring beyond the masses provided.

Common Mistake

Do not plot the stretched length instead of the extension, because a graph of length against force does not pass through the origin.
Do not say that force and extension are proportional for any amount of stretching, because Hooke's law applies only up to the limit of proportionality.
Do not call a straight line that misses the origin directly proportional.

Exam technique

Describing force-extension relationships

Use the words directly proportional and add up to the limit of proportionality, because writing only that the extension increases as the force increases describes a trend and scores less.
Back up a claim of proportionality with data from the question, showing that doubling the force doubles the extension.
Describe the graph as a straight line through the origin, naming both features.
In a practical question, name force as the independent variable and extension as the dependent variable, and say how the extension is calculated.

Self review

State Hooke's law.
Describe the shape of the force-extension graph for a spring that obeys Hooke's law.
Calculate the extension of a spring with a $N5.0\ \text{N}$ load if it extends by $cm4.0\ \text{cm}$ with a $N2.0\ \text{N}$ load and obeys Hooke's law.
Explain why a graph of stretched length against force does not pass through the origin.
Describe what happens to the total extension when two identical springs are hung one below the other.

2.3.3 Force and extension of a spring

Force and extension

Hooke's law

Force-extension graphs

Other simple systems

Checking results against Hooke's law

Predicting an extension

Investigating the effect of forces on a spring

Describing force-extension relationships

Review 2.3.3 Force and extension of a spring by teaching Genie

2.3.3 Force and extension of a spring

Force and extension

Hooke's law

Force-extension graphs

Other simple systems

Checking results against Hooke's law

Predicting an extension

Investigating the effect of forces on a spring

Describing force-extension relationships

Review 2.3.3 Force and extension of a spring by teaching Genie

2.3.3 Force and extension of a spring

Force and extension

Hooke's law

Force-extension graphs

Other simple systems

Checking results against Hooke's law

Predicting an extension

Investigating the effect of forces on a spring

Describing force-extension relationships

Review 2.3.3 Force and extension of a spring by teaching Genie

2.3.3 Force and extension of a spring Revision Guide

2.3.3 Force and extension of a spring

Force and extension

Hooke's law

Force-extension graphs

Other simple systems

Checking results against Hooke's law

Predicting an extension

Investigating the effect of forces on a spring

Describing force-extension relationships

Review 2.3.3 Force and extension of a spring by teaching Genie

2.3.3 Force and extension of a spring Revision Guide