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Forces 4: 4.3 An experiment to investigate the force applied to a spring and resulting extension - experimental procedure, method of processing results, Hooke's Law graphs and calculations

Sub-index of physics notes on FORCES section 4 Elastic potential energy

4.3a An experiment to investigate the force applied to a spring and resulting extension

How can we investigate the relationship between a spring's extension and weights added to it?

What is the relationship between the extension of a spring when increasing weights are connected to it?

If weights are attached to a firmly fixed suspended spring, the spring will elongate depending on the value of the weight attached. The greater the weight, the greater the spring is extended. The extra length the spring attains is called the spring extension and this phenomena can be systematically investigated using the simple apparatus described below.

When a weight is added and the spring is static, the weight of the mass (and the spring itself) is counterbalanced by the force of tension in the spring.

A metre rule is fixed in a vertical position using a stand and several clamps. Preferably with the linear scale pointing downwards!

The metre rule scale can be read in mm or cm.

From the top of the 1 m ruler a spring is suspended with a hook-base is added to which extra mass can be added e.g. in 50g increments (0.5 N force increase increment).

Fix a pointer onto the hook to which the weights will be attached. If you can't fix a pointer on, just use the base of the weight hook and sight it horizontally onto the scale.

You take the initial reading with no extra weight on (other than the weight of the spring itself plus hook) and take the initial reading on the scale. This first reading with no extra load applied is the crucial starting point for all the successive measurements.

You then add an extra mass and take the scale reading in mm/cm. From each successive reading you must subtract the initial reading to obtain the true extension of the spring (I haven't actually shown this in the table of results below).

Record a minimum of five observations carefully in a prepared table and convert the mm/cm scale readings to the extension in m.

Below is a typical table of results, already corrected by subtracting the 'initial' reading.

50 g load = 0.05 kg ~ force/weight of 0.5 N

For simplicity I've taken gravity as 10 N/kg, therefore every 50 g mass added equals an incremental weight increase of 0.5 N.

 From the data table you plot a graph of total force (= tension) versus the total extension in the spring length (graph sketched on the left).

 (the tension in the spring equals the force created downwards by the weight of mass).

 Draw the line of best fit from the 0,0 graph origin.

 If the spring is truly elastic a linear graph is obtained.

 This means a simple linear equation describes the behaviour of the spring under these conditions.

The experiment is a simple proof and demonstration that the extension of a spring or any elastic material is directly proportional to the force applied (the load or weight in newtons).

This relationship is expressed with the simple equation:

force = a spring constant x extension

F = ke

where F = the applied force in newtons (N), e = the spring extension in metres (m)

and k is the spring (elastic) constant in N/m.

The stiffer the spring (or any material being stretched) the greater the spring constant.

and you would see a steeper gradient of the graph line, or a smaller gradient for a weaker spring,

the point illustrated by the 'theoretical' purple lines on the above graph.

This linear equation relationship between force applied and the extension (or compression) of an elastic material is also known as Hooke's Law of proportionality.

This can be stated as:

The extension of a spring or wire (or any elastic object) is proportional to the load (force applied)


If the deformation of a material is proportional to the force applied, the material is truly elastic and is said to obey Hooke's Law (a law of proportionality).

From the graph you can calculate the spring constant e.g. rearranging the equation (Hooke's Law equation)

k = F/e = gradient of the graph = 3.0/0/0.06 = 50 N/m

Five important points to note:

This equation works for compression where e is the difference between the full length and compressed length

The spring constant varies with the material of the spring, the size and number of coils of the spring.

The stronger/stiffer the spring the greater the value of the spring constant k.

This spring system is the basis for simple instruments used to measure the weight of an object like a fish you have caught!

A force meter is used for experiments in a laboratory - school, college, university or in the engineering industry where it is used e.g. to test the strength of materials.

Above is an illustration of a simple instrument for weighing objects.

It is essentially a 'force meter' calibrated to read in g and kg.

(so it takes into account gravity at the Earth's surface, but it would be any good on the Moon or Mars with their different strength of gravitational fields)

Prior to taking a reading the pointer should be adjusted to read zero.

You place the object on the hook which stretches the spring and read off its weight on the calibrated scale.


Extending the investigation and alternative graphs

Different groups in a class can look at different springs, or if time permits each group of students can look at several springs. The class results can be pooled and graphs drawn.

Instead of plotting force versus extension (which  I prefer) you can plot extension versus force.

Since F = ke, e = F / k, so the gradient will be 1 / k, the reciprocal of the spring constant.

In the 'idealised' right-hand graph of spring extension versus force, springs A and C results did not go beyond the limit of proportionality.

However, the results for spring B showed a deviation from linearity and the graph curves upwards from point L, the limit of proportionality.

4.3b What happens if you keep on increasing the force applied to an elastic material?

In the above experiment, if you add even more weights to the spring then the resulting graph of results may not be linear for the higher weight readings.

This is because the spring is overstretched beyond its elastic limit (the limit of proportionality).

Beyond point L Hooke's Law is no longer obeyed.

In other words the non-linear section of the graph is beyond L, the limit of proportionality - the spring stretches more than you expect and the graph begins to curve over.

From zero force to L Hooke's Law is obeyed - the linear section of elasticity.

After that, between point L and point D, the stretching is greater than expected - non-linear graph, but the spring will return to its original length - the spring is still behaving elastically, but only for a relatively small further increase in the applied force.

Just because an object behaves elastically, it doesn't mean that Hooke's Law is obeyed.

An elastic band is 'elastic' but it doesn't obey Hooke's Law!

Eventually at point D, called the elastic limit, the force is too great for spring will not return to its original length - permanent deformation beyond the limit of elasticity.

This happens with a repeatedly stretched elastic band - eventually it breaks!

From point D onwards the spring behaves with plastic deformation.

On the right-hand graph, an alternative representation of the graphical data, I've indicated the permanent extension showing the spring will NOT return to its original length.


Sub-index of physics notes: FORCES 4. Elastic potential energy

Keywords, phrases and learning objectives for elastic potential energy

Be able to describe an experiment to investigate the force applied to a spring and resulting extension.

Explain  the experimental procedure to validate whether a stretched material obeys Hooke's Law - processing data with graphs and calculations.

Know what happens if the material (e.g. a spring) is stretched beyond the elastic limit - interpret a graph to aid your explanation.


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Sub-index of physics notes: FORCES 4. Elastic potential energy