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Young’s Modulus
Introduction
Born some seventy years after Robert Hooke’s death, Thomas Young refined Hooke’s elasticity law by identifying the modulus described in this chapter that is independent of the dimensions of the material, thus aiding engineers in their calculations. Young made extensive contributions to many other areas of science – in particular, optics, in which the double-slit experiment bears his name.
The problem with a stiffness constant, which is measured in the Hooke's Law experiment, is that its value will depend on the dimensions of the material used. If we want to be able to compare materials, we need a measure that is independent of the materials’ dimensions. This requires us to use stress and strain rather than force and extension.
We need to define these terms accurately, which we will do in the Physics section of this chapter. They are known formally as tensile stress and tensile strain (tensile just means something to do with stretching).
The objective
To measure Young’s modulus for a thin wire.
The apparatus
·A ruler;
·A protractor;
·A thin steel wire (a top E guitar string works well);
·A set of weights;
·A weight support;
·Two G-clamps;
·An axle with pulley and indicator;
·Axle supporting block;
·A wood block.
The variables
The independent variable is the weight from which the tensile stress is calculated, and the dependent variable is the angle from which the strain is calculated.
The Physics
Tensile stress is defined as the force applied divided by the cross-sectional area. This can be expressed as:
Stress = F/A
Where F is the force applied to stretch the wire and A is the cross-sectional area of the wire. The units are Nm−2, which is the same as what we measure pressure with, i.e., Pascals or Pa.
Tensile strain is defined as the change in length as a proportion of the original length. This can be expressed as:
Strain = e/l
Where e is the amount that the wire has stretched and l is its original length. This is a ratio of two numbers with the same units. So the units are mm−1, which cancel, so there are no units at all. Strain is simply a numeric ratio.
Young’s modulus is defined as the stress divided by the strain. That is:
E = tensile stress/tensile strain
Substituting from the previous two definitions, this gives us:
E = Fl/Ae (units Pa)
If we take a wire and stretch it within its elastic limit, it will return to its original length when we stop applying a stretching force. However, if we were to carry on applying more and more force, there comes a point – the elastic limit – that when exceeded no longer allows the wire to go back to its original length. Any deformation beyond this elastic limit is called a plastic deformation. We are not concerned with such deformations here.
The method
We add just enough weight to the cradle to make the wire just tense, i.e. no kinks. Note the position of the pointer against the protractor and the weight. Now add weights and for every different weight note the pointer position.
The Video
Watch a video of the Young's Modulus e-practical here.
This shows how to use it and how to collect the data.
The E-Practical
Perform the experiment yourself, collect your own data, make mistakes and be able to correct them. The e-practical requires that your browser can run WebGL 2 (usually found on Windows browsers, safari on iOS, and various Mobile browsers, test with https://get.webgl.org/webgl2/). This link is for students and evaluation only, schools should purchase a site licence.
In the e-practical, we are assuming that the cradle is supplying sufficient force to minimally extend the wire. We are taking this as the zero point for the force and the extension. It is at this cradle-weight that we are measuring the angle from.
The weights in the cradle can be altered by placing the mouse cursor over the cradle and clicking.
The Results
YCreate a table with headings for the mass, the angle of deflection (call this theta) in degrees, the same angle in radians, the amount of stretch (call this delta x), the force, the stress, and the strain. Your table should look something like this:
You may find it convenient to do this using a spreadsheet program such as Microsoft Excel.
The stress and strain can be calculated from the formulae given in the Physics section.
To calculate the cross-sectional area of the wire, use the formula for the area of a circle:
A = пr2
Once your table is complete, plot a graph with stress on the Y-axis and strain on the X-axis for all the readings that you have taken. You should get a straight line where the gradient equals the value for Young’s modulus.
Further Discussion
What do you think the area under the graph might represent? Think about what it is the product of.
What are the units on the two axes?
What do you get if you multiply these units?
What is force times a distance?
Is this something that you have met before?
Where else do you meet this formula?
What is measured by this formula?
Where s is the distance, g is the acceleration (the value you are looking for), and t is the time of descent.
If you are using the e-practical, you can repeat this experiment for the Moon and Mars.
This works out at 2.59722E-06 radians. Given that the radius of the Moon’s orbit is approximately 384,400,000 m, the distance moved by the Moon in 1 second is:
D = 2.59722E-06 × 384,400,000 m = 998.37 m
From the earlier diagram:
R – h = Rcos(ɵ)
Therefore, h, the amount that the Moon has dropped towards us, is given by:
h = R – Rcos(ɵ)
Which works out at around 1.3 mm.
This section is adapted from material developed by Dr Robert Lucas and is related to the book High School and Undergraduate Physics Practicals, published by CRC Press.