Young Modulus

Young Modulus

The spring constant k is useful for calculating the extension e of a rod, from the load F, using Hooke's law: F = ke.

However, if we change to a rod of different size and shape, then the spring constant could change, even if the rod is of the same material. For example, a longer rod would give more extension for the same load. It would be convenient if there is a number for each material that we can use, whatever the size and shape.

To see how this is possible, consider a rod of length L = 1 m, and cross-sectional area A = 1 cm². Suppose that a load F of 1 N produces an extension e of 1 mm.

Consider what happens if we have two of these rods joined side by side. Then a force of 1 N on each rod would produce the same extension of 1 mm. However, the total force is 2 N. It so happens that the cross-sectional area is also doubled to 2 cm². So there is in fact a force of 1 N, for 1 cm² of the area - the same as for the single rod.

This suggests that it may be useful to think of a force per unit area, or F/A. This F/A is called the stress. For the same stress, a rod would give the same extension, whether the rod is thick or thin. Since stress = F/A, F would of course be smaller for the thin rod.

Now, go back to the original rod of length 1 m, and cross-sectional area 1 cm². Join two of these rods end to end, and apply a load of 1 N. This load would produce the same 1 N tension in both rods. Each rod then extends by 1 mm. The total extension is therefore 2 mm.

This suggests that it may be useful to think in terms of the extension per unit length of the rod, or e/L. This e/L is called the strain. The same load F gives the same strain, no matter how long the rod is.

Combining these, we find that a rod of any length and any thickness gives the same strain, when under the same stress. The stress is proportional to the load F, and the strain is proprotional to the extension e. Since Hooke's law tells us that F and e a proportional, so stress and strain are also propostional.

Therefore, if we divide stress by strain, we get a number that is the same for different stresses. This number is called the Young modulus. Since stress and strain do not depend on the size and shape of the rod, the Young modulus depends only on the material.

The Young modulus is Y = stress / strain.

Substituting, Y = (F/A) / (e/L) = FL/Ae.

So if we know the load F, we can use this to find the extension e.

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