Hooke's Law & Springs

Explained

Hooke's law and how springs stretch

Hang a mass on a spring and it stretches. Hang twice the mass and it stretches twice as far. That proportionality is Hooke's law, and it holds only up to a point, which is why the graph is the most examined part of this topic.

Extension is not length

Extension means the increase in length compared with the original length, not the length itself. A spring 12 cm long that started at 10 cm has an extension of 2 cm.

Forgetting to subtract the original length is the most common error in the calculation, and it is worth checking before anything else if an answer looks strange.

The spring constant

The spring constant, k, is the force needed per unit extension, so k equals F divided by x. A large spring constant means a stiff spring that needs a large force for a small extension.

Watch the units. If force is in newtons and extension in metres, k comes out in newtons per metre. Extensions are often measured in centimetres, so a conversion is usually needed and is a frequent source of answers that are a hundred times too big or too small.

Reading the force extension graph

Plot force against extension and the line is straight through the origin while Hooke's law holds. The gradient of that straight section is the spring constant.

Beyond a certain force the line curves. That point is the limit of proportionality. Past it, extension is no longer proportional to force, and the spring may not return to its original length when the force is removed.

A question showing a graph that curves at the top is almost always asking you to identify that limit, or to explain that Hooke's law no longer applies beyond it.

The required practical

Measure the natural length of the spring first. Add masses one at a time, recording the new length each time, and calculate extension by subtracting the natural length. Convert masses to weights using W equals m times g, because the stretching force is the weight, not the mass.

Plot force on the vertical axis and extension on the horizontal axis, then take the gradient of the straight portion. Two controls matter: read the ruler at eye level to avoid parallax error, and do not exceed the limit of proportionality, because a permanently stretched spring makes every later reading wrong.