## Hooke’s Law and Young’s Modulus

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## Hooke’s Law

About 350 years ago, a scientist named Robert Hooke discovered that **the extension of a material is proportional to the stretching force provided that the material is not permanently stretched**. This is now known as Hooke’s law.

According to Hooke’s law, doubling the force doubles the extension, trebling the force trebles the extension, and so on.

Mathematically, we can write this as:

- F ∝ e, which means Force is directly proportional to extension.

One important thing to note is that, Hooke’s law is true only if the elastic limit or limit of proportionality of the material is not exceeded.

The diagram above shows a graph of force versus extension of an object stretched beyond its elastic limit, E. From O to E, the object obeys Hooke’s law as shown by the graph that OE is a straight line passing through the origin O. If the force had been removed at E, the extension of the object would have return to 0. If the force is applied up to point P on the graph, the proportionality limit is exceeded and on removing the force some of the extension (OA) remains.

*If an experiment is performed to verify Hooke’s law, the conventional way of plotting the results would be to have the force on the horizontal axis and the extension on the vertical axis. The reason being that in the experiment we vary the force (which makes force the independent variable) to produce a resultant change in extension (which make extension the dependent variable). However, the graph above has extension on the horizontal axis and force on the vertical axis. This is usually done because the gradient of the straight section of this graph becomes an important quantity, known as the force constant of the spring.*

### Force constant and stiffness

With force on the vertical axis and extension on the horizontal axis, the behaviour of the material in the linear region of the graph can be expressed as F = kx, whereby k is the force constant of the material (also called the stiffness or the spring constant of the material).

The **force constant** is the force per unit extension.

The force constant k is the gradient of straight section of the graph. The SI unit for the force constant is newtons per metre or Nm^{−1}.

The stiffer spring the larger value for the force constant k.

## The Young modulus

**Young modulus** is the ratio of stress to strain.

Young modulus is a constant for a particular material and does not depend on its shape or size.

$latex \displaystyle Young’s\ modulus = \frac{stress}{strain}$

$latex \displaystyle E = \frac{\sigma}{\epsilon}$

whereby E is the Young modulus of the material, σ (sigma) is the stress and ε (epsilon) is the strain.

The SI unit of the Young modulus is the same as that for

stress, Nm^{−2} or Pa.

If we plot the stress-strain graph of a material, with stress on the vertical axis and strain on the horizontal axis, the gradient is the Young modulus of the material.

Here are the Young Modulus values of some materials:

Material | Young modulus in GPa |
---|---|

aluminium | 70 |

brass | 90–110 |

brick | 7–20 |

concrete | 40 |

copper | 130 |

glass | 70–80 |

wrought iron | 200 |

lead | 18 |

Perspex® | 3 |

polystyrene | 2.7–4.2 |

rubber | 0.01 |

steel | 210 |

tin | 50 |

wood | approx 10 |

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