Deformation
When a force acts on an object, the object may change shape by bending, stretching or compressing – or a combination of all three shape changes. However, to change the shape of a stationary object there must be more than one force acting to do the following.
Bend an object’s ends past each other – eg when an archer pulls an arrow back against a bow.
Pull an object’s ends apart – eg when a rubber band is stretched.
Push an object’s ends together – eg when an empty drinks can is squashed.
A change in shape is called distortion:
- elastic distortion is reversed when the force is removed
- inelastic distortion is not fully reversed when the force is removed – there is a permanent change in shape
A rubber band undergoes elastic distortion when stretched a little. A metal drinks can undergoes inelastic distortion when it is squashed.
Hooke’s law
Extension and compression
Extension happens when an object increases in length and compression happens when it decreases in length. The extension of an elastic object, such as a spring, is described by Hooke’s law:
Limit of proportionality
Spring constant is a measure of the stiffness of a spring up to its limit of proportionality. The limit of proportionality refers to the point beyond which Hooke’s law is no longer true when stretching a material
The higher the spring constant, the stiffer the spring. The spring constant is different for different elastic objects. For a given spring and other elastic objects, the extension is directly proportional to the force applied. For example, if the force is doubled, the extension doubles. This works until the limit of proportionality is exceeded.
The elastic limit of a material is the furthest point it can be stretched or deformed while being able to return to its previous shape. When an elastic object is stretched beyond its elastic limit, the object does not return to its original length or shape when the force is removed. Once a material has gone past its elastic limit, its deformation is said to be inelastic.
In this instance, the relationship between force and extension changes from being linear, or directly proportional, to being non-linear.
Non-linear extension occurs more in some materials than others. Materials like clay or putty usually show non-linear extension.
Force-extension graphs
Linear extension and elastic distortion can be seen below the limit of proportionality.
Non-linear extension and inelastic distortion can be seen above the limit of proportionality. The limit of proportionality is also described as the elastic limit. The gradient of a force-extension graph before the limit of proportionality is equal to the spring constant
Energy stored in a spring
Work is done when a spring is extended or compressed. Elastic potential energy is stored in the spring. Provided inelastic distortion has not happened, the work done is equal to the elastic potential energy stored.
The elastic potential energy stored can be calculated using the equation:
elastic potential energy = 0.5 × spring constant × (extension)2
Key fact
This equation also works for the reduction in length when a spring is compressed.
Example
A spring has a spring constant, k, of 3 N/m. It is stretched until it is extended by 50 cm. Calculate the elastic potential energy stored by the spring, assuming it is not stretched beyond the limit of proportionality.
First convert centimetres to metres:
50 cm = 50 ÷ 100 = 0.5 m
Then calculate using the values in the question
Required practical – investigating force and extension with a spring
Investigate the relationship between force, extension and work done extending a spring
There are different ways to investigate the relationship between force and extension for a spring. In this required practical activity, it is important to:
- make and record length accurately
- measure and observe the effect of force on the extension of springs
- collect the data required to plot a force-extension graph
Aim of the experiment
To investigate the relationships between force and extension for a spring, and the work done in extending the spring.
Method
- Secure a clamp stand to the bench using a G-clamp or a large mass on the base.
- Use bosses to attach two clamps to the clamp stand.
- Attach the spring to the top clamp and a ruler to the bottom clamp.
- Adjust the ruler so that it is vertical and with its zero level with the top of the spring.
- Measure and record the unloaded length of the spring.
- Hang a 100 g slotted mass carrier – weight 0.98 newtons (N) – from the spring. Measure and record the new length of the spring.
- Add a 100 g slotted mass to the carrier. Measure and record the new length of the spring.
- Repeat step 7 until you have added a total of 1,000 g.
Results
Record your results in a suitable table.
| Force (N) | 0 (unloaded) |
|---|---|
| Length (mm) | 22 |
| Extension (mm) | 0 |
| Force (N) | 0.98 |
|---|---|
| Length (mm) | 52 |
| Extension (mm) | 30 |
| Force (N) | 1.96 |
|---|---|
| Length (mm) | 83 |
| Extension (mm) | 61 |
Analysis
- For each result, calculate the extension:
- extension = length – unloaded length
- Plot a line graph with extension on the vertical axis, and force on the horizontal axis. Draw a suitable line or curve of best fit.
- Identify the range of force over which the extension of the spring is directly proportional to the weight hanging from it.
- For the region where extension is proportional to force, find the gradient of the line. The spring constant, k, is the reciprocal of this gradient.
- Work done = force × distance moved. Here, the work done in extending the spring is given by the area under the line on the graph.
- The energy transferred to a spring’s elastic store is given by the equation: Compare the area under the line, from the origin up to a point, with the calculation of the energy stored in the spring for that extension.
Evaluation
It is important to keep the ruler vertical. Suggest another way to improve the accuracy of the length measurements.
Hazards and control measures
| Hazard | Equipment falling off table |
|---|---|
| Consequence | Heavy objects falling on feet – bruise/fracture |
| Control measures | Use a G-clamp to secure the stand |
| Hazard | Sharp end of spring recoiling if the spring breaks |
|---|---|
| Consequence | Damage to eyes and cuts to skin |
| Control measures | Wear eye protection and support and gently lower masses whilst loading the spring |
| Hazard | Masses falling to floor if the spring fails |
|---|---|
| Consequence | Heavy objects falling on feet – bruise/fracture |
| Control measures | Gently lower load onto spring and step back |
