Hooke’s Law Says that Extension is Proportional to Force

If a metal wire is supported at the top and then a weight attached to the bottom, it stretches. The weight pulls down with force F, producing an equal and opposite force at the support.

The material will only deform (stretch, bend, twist ect.) if there’s a pair of opposite forces acting on it.

1. Robert Hooke discovered in the 17th century that the extension of a stretched wire, △x, is proportional to the change in load or force, △F. This relationship is now called Hooke’s law.

2. It can be written as: △F = k△x . Where k is the stiffness of the object that is being stretched. k is called the force constant (or stiffness constant) and has units Nm¯¹.

Hooke’s Law Also Applies to Springs

A metal spring also changes length when you apply a pair of opposite forces.

1. The extension or compression of a spring is proportional to the force applied — so Hooke’s law applies.

2. For springs, k in the formula △F = k△x can also be called the spring stiffness or spring constant.

Hooke’s law also works for compressive forces as tensile forces. For spring, k has the same value whether the forces are tensile or compressive (isn’t true for all materials).

3. Hooke’s law doesn’t just apply to metal springs and wires — most other materials obey up to a point.

Hooke’s Law Stops Working when the Load is Great Enough

There’s a limit to the force you can apply for Hooke’s law to stay true.

1. The graph shows force against extension for a typical metal wire or spring.

2. The first part of the graph (up to point P) shows Hooke’s law being obeyed — there’s a straight-line relationship between force and extension.

3. When the force becomes great enough, the graph starts to curve. Metals generally obey Hooke’s law up to the limit of proportionality, P.

4. The point marked E on the graph is called the elastic limit. If you exceed the elastic limit. If you exceed the elastic limit, the material will be permanently stretched. When all the force is removed, the material will be longer that at the start.

5. Be careful — There are some materials, like rubber, that only obey Hooke’s law for really small extensions.

A Deformation can be Elastic or Plastic

A material will show elastic deformation up to its elastic limit, and plastic deformation beyond it. If a deformation is elastic, the material returns to its original shape once the forces are removed.

1. When the material is put under tension, the atoms of the material are pulled apart from one another.

2. Atoms can move slightly relative to their equilibrium positions, without changing position in the material.

3. Once the load is removed, the atoms return to their equilibrium distance apart.

If a deformation is plastic, the material is permanently stretched.

1. Some atoms in the material move position relative to one another.

2. When the load is removed, the atoms don’t return to their original positions.

Extension and compression are sometimes called tensile deformation and compressive deformation.

Investigating Extension

1. Set up the experiment shown in the diagram. Support the object being tested at the top (e.g. with a clamp) and measure its original (natural) length with a ruler.

2. Add weights one at a time to the bottom of the object.

3. After each weight is added, measure the new length of the object, then calculate the extension:

Extension = new length - original length

4. To measure an extension as accurately as possible, make sure you fix a ruler close to the extending object and take readings with your eye close to the ruler. It’s a good idea to use a ‘fiducial marker’ (a thin tag on the object that marks where you’re measuring) and a set square to ensure the ruler is vertical. Make sure the marker and ruler are both at eye level when you take readings.

5. Plot a graph of force (weight) against extension for your results. Where the line of best fit is straight, then the object obeys Hooke’s law and the gradient =k(as △F = k△x).

6. Make sure you carry out the experiment safely. You should be standing up so you can get out of the way quickly if the weights fall, and wearing safety goggles to protect your eyes in case the object snaps.

If the markings on your measuring equipment are quite far apart, you can often interpolate between them (e.g. if the length is halfway between the markings for 24mm and 25mm you could record it as 24.5 mm). But it’s better to use something with a finer scale if possible.