In this article, I will be looking at the tensile strength, stiffness, Hooke’s law, how to use Hooke’s law, Young Modulus (E) and example questions. The theory for working out tensile strength and stiffness is relatively simple to use. However, before anything, let’s start with the definitions of strength and stiffness.
Strength – Force needed to cause fracture or failure. It is measured in Newtons (N).
Stiffness – Force needed to cause unit change in length. It is measured in Newtons per metre (N/m)
Hooke’s law is k = f / x. Rearrangements include f = kx and x = f / k
k is the spring constant and is measured in N/m.
x is the extension and it is measured in metres.
f is the force and is measured in Newtons.
This is why k is measured in newtons/ metre as k = f / x is the same as k = Newtons / metres.
k is a sample dependent. This means that the sample of the same material will have a new value of k.
We need a quantity which is substance dependent only (does not depend on the initial dimensions of the material).
Using Hooke’s Law
A 2.2m length of steel wire stretches 0.4mm when loaded with 27N. Calculate k for this length.
First of all, lets make clear the 2.2m length is not relevant. We do not need the length of wire when using k = f / x.
- k = f / x.
- k = 0.4mm (which is the same as 4 x 10 to the power of -4 metres) / 27
- k = 67500 Nm which means it takes 67500N for the steel wire to stretch a metre.
For force, we use stress:
Stress = force / cross-sectional area = f / A
For extension, we use strain:
Strain = extension / orginial length = x / l
We can replace k with Young Modulus:
E or YM = Stress / Strain = (f/A) / (x/l) = fl / Ax
f the load in Newtons.
A is the cross-sectional area of the cable or bar and is measured in metres squared.
x is the extension and is measured in metres.
l is the original length and is measured
Young Modulus is measured Newtons per metres squared.