Springs (3.4.1)

Forces don’t just make objects move; they can also change their size and shape. This effect is called deformation.

When an object deforms, it either stretches or compresses from its natural length.

Different materials respond differently to deformation: some deform elastically and return to their natural length, while others deform plastically and remain permanently deformed.

When two forces pull outwards on an object in opposite directions, they create tension, leading to tensile deformation. As a result, the object extends.

An example of this is a stretched spring

• When a load is applied to a spring, the force pulls down due to gravity.
• The spring stretches because the load applies a tensile force.
• The greater the load, the more the extension.

Other examples of tensile forces include:

• A stretched rubber band.
• A rope in a tug-of-war, where both teams pull in opposite directions.
• A metal wire in a suspension bridge holds up weight as it is stretched.

When two forces push inwards on an object from opposite directions, they create compression, leading to compressive deformation. Under this force, the object shortens or squashes.

An example of this is a compressed Spring:

• When a force is applied from above, the spring pushes back but shortens in response.
• This is an example of a compressive force at work.
• The greater the load, the more the spring will shorten.

Other examples of compressive forces:

• Standing on a sponge – your weight compresses it.
• A chair leg pressing on the floor – the force from the chair applies compression.
• A car tyre on the ground – the tyre compresses under the car’s weight.

The greater the force applied to a spring or elastic material, the greater the extension (if tensile) or compression (if compressive).

Remember, extension is the amount the length of the spring changes. It is not the final length of the spring.

There are several cases worth being familiar with where Hooke’s law is and is not obeyed:

A force-extension graph illustrates how a material deforms when a force is applied.

Different materials (e.g., brittle or ductile materials) produce unique graphs based on their mechanical properties.

These graphs help identify whether a material follows Hooke’s Law and determine its elastic and plastic behaviour.

A material obeys Hooke’s Law if the graph is a straight line through the origin.

This means that extension is directly proportional to force

The gradient of a force-extension graph can be used to determine the force constant (). The force constant measures a material’s stiffness — it tells us how much force is needed to produce a certain extension. It’s often called the spring constant when describing springs, but we generally use force constant for any material or object, not just springs.

The limit of proportionality represents the point where Hooke’s Law ceases to be valid—beyond this, extension is no longer proportional to force. Beyond this point, increasing force results in larger, non-uniform extensions

The elastic limit is the maximum extension a material can experience before returning to its natural length once the force is removed.

If stretched beyond the elastic limit, the material undergoes plastic deformation, meaning it will not return to its original length.

The elastic limit is always beyond the limit of proportionality on the graph.

As more force is applied beyond the limit of proportionality, the force-extension graph curves instead of remaining linear. The material exhibits non-linear behaviour and no longer obeys Hooke’s Law.

The force-extension relationship becomes non-linear, and the material may eventually fail or break if the force continues to increase.

It is important to note that force-extension graphs are commonly illustrated with force on the -axis and extension on the -axis!

Always check the axes before calculating the force constant

• If force () is on the y-axis and extension () is on the -axis, then the gradient gives the force constant
• If force is on the -axis and extension is on the y-axis, the force constant is =1/gradient instead.

The force-extension graph for an unknown-metal wire is shown below.

This graph shows that the wire:

• Follows Hooke’s Law in the initial elastic region, meaning force and extension are proportional.
• Elastic deformation occurs up to the elastic limit, where the material returns to its original shape upon unloading.
• Beyond the elastic limit, the wire undergoes plastic deformation, meaning it is permanently stretched.
• The unloading curve has the same gradient as the loading curve*, but the material now has a longer final length due to plastic deformation.

It is useful to know that the term curve is sometimes used broadly in physics to refer to any plotted relationship, even if it appears linear as a straight line.

The force-extension graph for a rubber band is shown below.

• The rubber band does not obey Hooke’s Law, as its force-extension graph is curved rather than linear.
• Elastic deformation only—it returns to its original shape after unloading.
• The loading and unloading curves do not overlap, forming a hysteresis loop.
• The hysteresis loop area represents the energy lost as thermal energy, meaning that not all the work done in stretching the material is recovered.
• The curve for contraction is always below the curve for stretching because some energy is dissipated.

The force-extension graph for a polyethene strip is shown below.

• Polyethene strips do not obey Hooke’s Law and behave very differently from metals or rubber.
• Plastic deformation occurs immediately—even a small force permanently stretches the material.
• This makes polymeric materials easy to reshape but difficult to return to their original form.

To validate Hooke’s Law, we may want to investigate the relationship between force and extension for:

A metal spring (expected to follow Hooke’s Law),
A rubber band (expected to form a hysteresis loop),
A polythene strip (expected to show immediate plastic deformation).

In this experiment, our variables are:

Independent variable: Force () (applied using weights).
Dependent variable: Extension ().
Control variables:

• Same measuring equipment.
• Same starting length for each material.
• Consistent increments of force are applied.

Equipment List

• Clamp stand, boss, and clamp.
• Metre ruler (resolution = ).
• Set square (to ensure accurate ruler alignment).
• Mass hanger and slotted masses.
• Fiducial marker (for measuring extension accurately).

Safety Precautions

• Risk of snapping materials → Wear eye protection to prevent injury.
• Falling weights hazard → Place a box or soft mat beneath to catch falling masses.
• Clamp stability → Ensure the stand is securely fixed to prevent tipping.

The method to validate Hooke’s law could follow the steps below.

1. Set up the apparatus
   • Secure a spring to a clamp stand and boss.
   • Attach a metre ruler vertically next to the spring using a set square to ensure it is straight.
   • Place a fiducial marker at the bottom of the spring for precise measurements.
2. Record initial length
   • Measure the spring’s original length with no load.
3. Apply force and record extension
   • Hang a mass hanger from the spring.
   • Measure and record the spring’s new length.
   • Calculate extension as: Extension=New length−Original length
   • Repeat by adding more masses and recording the new length each time.
4. Repeat this process for rubber bands and polythene strips, recording the extensions for each material.

Following the experiment to validate Hooke’s Law, to analyse the results, we should:

1. Plot a force-extension graph
2. Force () on the y-axis, Extension () on the -axis.
3. Draw a line of best fit for each material.
4. Determine the force constant for the metal spring
5. From Hooke’s Law:
6. The gradient of the straight-line section of the graph gives the force constant
7. If force is on the -axis, use =1/gradient

Identify key points on the graph:

• Elastic limit (where the graph curves for a metal wire).
• Hysteresis loop (for rubber).
• Permanent extension (for polythene).