Spring into Action

A spring keeps it simple, up to a point

Hang a weight on a spring and it stretches. Hang double the weight and it stretches twice as far. While a spring is not pulled too hard, its extension is PROPORTIONAL to the force stretching it, and that one fact makes the whole topic tidy. ⭐⭐ Because extension is proportional to force, a graph of force against extension is a straight line through the origin. The rule in words is force equals the spring constant times the extension, where the spring constant is the stiffness: a stiff spring needs a big force for a small stretch. ⭐ The steepness of that straight line IS the spring constant. A steep line means a stiff spring; a shallow line means an easily stretched one. Reading the gradient reads the stiffness straight off the graph. ⭐⭐ But the simplicity has a boundary called the limit of proportionality. Push past it and the line stops being straight, because the spring no longer stretches in step with the force. Stretch it far beyond that and the change becomes inelastic: it does not spring back to its first shape when you let go. And stretching stores energy. The work you do pulling a spring is stored in its elastic store, ready to be released, which is how a catapult or a clockwork toy works. ⚠️ Carry one question the whole way through: is it still in the straight-line region, and what does force equals spring constant times extension give me?

Straight line, the limit, and the store

The first column is how a spring behaves below the limit, the second is what changes at and beyond it, and the third is the energy it holds.

Tap the two elastic cases

Tap the TWO cases that show ELASTIC deformation, where the object returns to its shape.

• A spring stretched gently and released, springing back to its length.
• A rubber band pulled a little, then let go, returning to shape.
• A paperclip bent right open and staying bent.
• A lump of dough squashed flat and staying flat.

Past the straight line

On a force-extension graph for a spring, what happens at the limit of proportionality?

• The line stops being straight, because extension no longer keeps step with force
• The line suddenly becomes perfectly vertical
• The line starts passing through the origin
• Nothing changes; it stays straight for ever

The language of springs

Six terms this topic turns on. The first two describe the stretch, the rest the rule and its edges.

Match each to what it means

Two true below the limit

Select the TWO statements that are TRUE while a spring is stretched BELOW its limit of proportionality.

• The extension is proportional to the force
• The force-extension graph is a straight line through the origin
• The graph is already a curve
• The spring is permanently stretched out of shape

Answering a spring question

Spring questions reward using the rule cleanly and respecting its limit. ⭐⭐ USE THE RULE THE RIGHT WAY. Force equals the spring constant times the extension. To find a force, multiply the spring constant by the extension; to find the spring constant, divide the force by the extension. Rearranging it is where the marks are. ⭐ CHECK YOU ARE BELOW THE LIMIT. The proportional rule only holds up to the limit of proportionality. If a question describes a spring stretched far beyond that, do not use the straight-line rule, because the extension is no longer proportional to the force. ⭐⭐ READ THE GRADIENT AS STIFFNESS. On a force-extension graph, the steeper the straight part, the stiffer the spring, so the larger the spring constant. Two springs can be compared just by which line is steeper. KNOW ELASTIC FROM INELASTIC. Elastic deformation springs back; inelastic deformation leaves the object changed for good. A spring stretched gently is elastic; stretched far past its limit, it can be left inelastic. ⚠️ A units habit. Keep the force in newtons, the extension in metres and the spring constant in newtons per metre, and convert first if a length is given in centimetres. Keep the one question to hand: is it still in the straight-line region, and what does force equals spring constant times extension give me?

Order the spring experiment

Put the steps of investigating how a spring stretches into a sensible order.

• Hang the spring and measure its starting length
• Add a known weight to stretch it
• Measure the new length and find the extension
• Repeat, adding equal weights each time
• Plot force against extension and read the straight part

Work out the stretching force

A spring has a spring constant of 40 newtons per metre and is stretched by an extension of 0.3 metres, still below its limit. Using force equals the spring constant times the extension, what is the force, in newtons?

The spring run

Five quick decisions about springs. Three lives.

Asking a favour of a neighbour

Leave physics aside for a screen. Think about a helpful neighbour and how much you can reasonably ask of them. ⭐ For small favours, the help grows in step with the ask. Ask them to water one plant and they will; ask them to water twice as many and they happily do twice the work. Within reason, a bigger ask gets a proportionately bigger response. ⭐⭐ But there is a threshold. Ask far beyond what is reasonable, water fifty plants every day for a month, and the neat rule breaks down. The extra help no longer keeps pace with the ask; you have gone past the point where doubling the request doubles the help. Within the reasonable range, no harm is done: ask a small favour, they help, and next week the goodwill is fully back, ready to be asked again. ⭐ Push far past the threshold, though, and something changes that does not simply reset. The relationship is strained in a way that does not spring back to how it was, even after you stop asking. ⚠️ Hold that picture. The force on a spring is the size of the ask, the stretch is the help given, and the neat proportional rule holds only up to a threshold. Below it the spring springs back each time; far beyond it, it is left changed for good.

Finish each spring line

The increase in length of a spring when a force stretches it is its _____. The stiffness of a spring, the force needed for each metre of stretch, is the spring _____. The point beyond which extension is no longer proportional to force is the limit of _____. Stretching that fully returns to the original shape when the force is removed is _____ deformation. Stretching that leaves the object permanently changed is _____ deformation.

Three springs on the bench

Three situations arise while testing springs. For each, choose the response the spring rule supports.

• A spring of spring constant 50 newtons per metre is stretched 0.2 metres, well within its limit, and you need the force.
• A spring is loaded so heavily that, when the weights are removed, it stays longer than it started.
• Two springs are tested and one gives a much steeper straight line on the force-extension graph than the other.

Explain how a spring behaves

Explain how the extension of a spring depends on the force, using the spring rule and the idea of the limit of proportionality. Explain the difference between elastic and inelastic deformation, and how a force-extension graph shows the spring constant.

• State that extension is proportional to force below the limit
• Give the spring rule in words and say what the spring constant means
• Explain what happens at and beyond the limit of proportionality
• Explain the difference between elastic and inelastic deformation
• Explain how the steepness of the graph shows the stiffness