Forces and elasticity – GCSE Combined Science Physics

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Think about the spring inside a pen. You press the button and the spring squashes. Let go and it springs back. Now think about a trampoline. You push the springs down, and they push you back up into the air.

Springs are everywhere. They are in mattresses, car suspension, bathroom scales and the clips on clothes pegs. A spring is useful because it stores energy when it is stretched or squashed, and then gives it back.

Now compare a spring with a lump of plasticine. Squash the plasticine and it stays squashed. Squash the spring and it springs back to its old shape. That difference is the first big idea in this lesson.

The second big idea is a pattern. Hang one weight on a spring and it stretches a little. Hang two weights and it stretches twice as much. One short equation captures that pattern, and it is the one you will use most.

By the end you will be able to do three things. You will tell elastic from inelastic and work out an extension. You will use force = spring constant × extension and read a force-extension graph. You will explain what the limit of proportionality means.

One more idea comes at the end. It is the energy stored in a stretched spring, and it uses a square, so take care with it.

Stretching and squashing

To stretch, squash or bend something you need more than one force. Hold a spring by one end and it just hangs there, but fix one end to a stand and pull the other, and the stand pushes back. Those two forces stretch the spring, and squashing needs two forces that push towards each other.

The extension is how much longer a spring gets: extension = new length − natural length. The natural length is the length with no force on it. A spring that is 10 cm long and stretches to 14 cm has an extension of 4 cm.

Changing the shape of an object is called distorting it. If the object goes back to its original shape when the force is removed, the distortion is elastic. A spring and an elastic band are elastic objects.

If the object stays changed when the force is removed, the distortion is inelastic. Plasticine is inelastic. So is a paper clip that you bend too far, because it stays bent.

Be careful with the word extension. It is not the whole length of the stretched spring. It is only the extra length, and in your sums you will almost always use the extension.

Measuring extension

You can test a spring in class. Clamp the spring to a stand next to a ruler and measure its natural length first. Always wear safety glasses in case a spring snaps, and read the ruler at eye level.

Hang a 100 g mass on the spring. A 100 g mass is 0.1 kg, so its weight is 0.1 × 10 = 1 N. Take g = 10 N/kg. Read the new length from the ruler and write it in a table.

Add another 100 g mass and repeat. Each time, work out the extension by taking away the natural length. Each mass adds 1 N to the force.

The table shows a spring with a natural length of 10 cm. Each extra newton makes it 2 cm longer. So 1 N gives 2 cm, 2 N gives 4 cm and 3 N gives 6 cm.

Watch your units. The spring constant is measured in N/m, so convert the extension from centimetres to metres by dividing by 100, and an extension of 4 cm is 0.04 m. Forgetting this is one of the most common ways to lose marks.

The spring constant

Look at the table again. Double the force and the extension doubles, and treble the force and the extension trebles. We say the force and the extension are directly proportional.

This pattern is called Hooke's law: force = spring constant × extension, or F = k × e in symbols. The force is in N, the extension is in m and the spring constant k is in N/m. Use it for any spring that is stretched by a force.

The spring constant tells you how stiff the spring is. A stiff spring has a large k, so it takes a big force to stretch it by 1 m. A soft spring has a small k.

To find the force, multiply: F = k × e. A spring with k = 80 N/m stretched by 0.1 m needs 80 × 0.1 = 8 N. To find the extension, rearrange to e = F ÷ k. To find the spring constant, use k = F ÷ e.

Before you start, check that the extension is in metres, and if it is in cm, change it first. Then write the equation, substitute the numbers and give the unit. The unit of k is newtons per metre, N/m, which is just force ÷ extension.

Force-extension graphs

A graph makes the pattern easy to see. Put the extension along the bottom and the force up the side, then plot each pair of readings from your table as a point. Use a ruler to draw the best straight line through the points.

For a spring that obeys Hooke's law, the points make a straight line through the origin, which is the point (0, 0). This shows that force is directly proportional to extension. No force means no extension, and double the force means double the extension.

The steepness of the line is its gradient. gradient = change in force ÷ change in extension. On a force-extension graph, the gradient is the spring constant k.

In this graph the line rises from 0 N to 8 N while the extension goes from 0 to 0.2 m. The gradient is 8 ÷ 0.2 = 40 N/m. So this spring has a spring constant of 40 N/m.

A steeper line means a stiffer spring and a shallower line means a softer one, so two springs on the same axes can be compared at a glance. Check which way round the axes are. If they are swapped, the gradient becomes extension ÷ force, which is not k.

The limit of proportionality

Springs do not follow Hooke's law forever. If you keep adding weights, the spring eventually stops behaving in the neat way, and the point where this happens is called the limit of proportionality. Think of it as the edge of the rule, beyond which the pattern no longer holds.

Up to the limit, the force-extension line is straight. Beyond it the line bends and gets flatter. The extension grows faster than the force, so the extension is no longer proportional to the force.

Past the limit, F = k × e no longer works. Do not use it for those readings. It only works for the straight part of the graph.

Stretch a spring too far and it may be damaged. It may not go back to its original length when the weights are removed. It has then been permanently stretched, which is inelastic distortion.

In an exam, spot the limit by looking for where the straight line starts to curve. Say that the force is no longer proportional to the extension. Do not say the spring breaks, as that is a different point.

A straight line through the origin on a force-extension graph shows a linear relationship: the force is directly proportional to the extension. Beyond the limit of proportionality the line curves, so the relationship is non-linear and doubling the force no longer doubles the extension. Only the straight part obeys F = k × e.

Energy stored in a spring

When you stretch a spring, you do work on it, and the work is transferred to the elastic potential store of the spring. When you let go, the spring gives that energy back, which is how a pull-back toy car gets its movement.

A catapult is a good picture. Pull back the elastic and you store energy. Let go and the energy is transferred to the kinetic store of the stone.

For a spring that obeys Hooke's law, energy stored = 0.5 × spring constant × extension². In symbols, Ee = ½ × k × e². The energy is in J, the spring constant is in N/m and the extension is in m.

Notice that the extension is squared. Square it first, then multiply by ½ and by k. A spring with k = 100 N/m stretched by 0.1 m stores 0.5 × 100 × 0.1² = 0.5 × 100 × 0.01 = 0.5 J.

Because of the square, doubling the extension makes the energy four times bigger. In the bars, k = 50 N/m. An extension of 0.1 m stores 0.25 J, 0.2 m stores 1 J and 0.4 m stores 4 J. A common slip is to forget to square the extension.

Recap

A spring needs two forces to stretch it. The extension is the new length minus the natural length. Elastic distortion goes back to the original shape, and inelastic distortion does not. A paper clip bent too far is a good example of the second kind.

force = spring constant × extension, F = k × e. The force is in N, the extension is in m and k is in N/m. A stiff spring has a large k, and a soft spring has a small one.

On a force-extension graph, a straight line through the origin shows direct proportion, and its gradient is k. Beyond the limit of proportionality the line curves, and F = k × e no longer works.

Energy stored = ½ × k × e². Doubling the extension stores four times as much energy.

How this is asked in an exam: a spring calculation is usually two marks, one for the substitution and one for the answer with its unit. Convert centimetres to metres first. Graph questions ask you to find k from the gradient or to say what a curve shows. When you describe a curve, say that force is no longer proportional to extension.

A quick self-test before you move on. Cover the page and say the equation, the unit of the spring constant and what the limit of proportionality means. If you can also change 5 cm into metres, you are ready for the practice questions.