Hooke's Law: Using the Equation (Worked Examples)

Explained

Using F equals k x

Hooke's law says the force applied to a spring is proportional to its extension, provided the limit of proportionality has not been passed.

In symbols, F equals k times x. F is the force in newtons, x is the extension in metres, and k is the spring constant in newtons per metre. A stiff spring has a large k, because it takes a large force to stretch it even a little.

Extension is a change

The x in the equation is the extension, not the length. Extension is the stretched length minus the original length.

A spring 8 cm long that becomes 14 cm long has an extension of 6 cm, not 14. Putting the full length into the equation is the commonest error in this topic, and the answer it gives is always too small a spring constant.

The three forms

To find the force, multiply the spring constant by the extension.

To find the spring constant, divide the force by the extension.

To find the extension, divide the force by the spring constant.

Substitute into F equals k times x first and solve for whichever letter is left, rather than trying to remember three versions.

A worked example

A spring extends by 0.04 m when a force of 6 N is applied. The spring constant is 6 divided by 0.04, which is 150 N/m.

What force would give an extension of 0.10 m? Using the same spring, 150 times 0.10 gives 15 N.

Check the proportion. The extension is two and a half times larger, and 15 N is two and a half times 6 N, so the answer is consistent.

Units, and the mass trap

Extension is usually given in centimetres or millimetres and must be converted to metres, dividing by 100 or 1000. A spring constant calculated with centimetres comes out a hundred times too small.

If the question gives a hanging mass rather than a force, the force is its weight, so multiply the mass in kilograms by g first. A 200 g mass is 0.2 kg, which weighs about 2 N, and that 2 N is what goes into the equation.

Those two conversions between them account for most wrong answers here, and both are easy to write out as separate lines.

Reading k off a graph

With force on the vertical axis and extension on the horizontal, the spring constant is the gradient of the straight section.

Take two points far apart on the line, find the change in force and the change in extension between them, and divide. Using two points close together magnifies any reading error.

If the axes are swapped, with extension vertical, the gradient is one over the spring constant instead. Check which quantity is on which axis before calling a gradient k.

Energy stored

Work is done stretching a spring, and that energy is stored in its elastic store. On a force against extension graph, the energy stored is the area under the line.

For a straight line that area is a triangle, so it is a half times the force times the extension. Doubling the extension therefore quadruples the energy stored, not doubles it, because both the force and the distance have doubled.