Measuring Forces with Springs

You cannot see a force, so how do you measure it?

A force is simply a push or a pull. You feel forces all the time — the pull of a heavy school bag, the push of the wind, the tug of a dog on its lead. But forces are invisible. You cannot hold one in your hand or pour it into a measuring jug. So how do scientists put a number on a force?

The clever trick is to find something that responds to a force in a steady, predictable way — and a spring does exactly that. The harder you pull a spring, the more it stretches. If we can measure the stretch, we can work out the force. Force is measured in a unit called the newton (N). One newton is roughly the force you feel holding a small apple in your hand.

How a spring turns force into a stretch

Take an ordinary metal coil spring and hang it from a hook so it dangles freely. Now hang a small weight on the bottom. The spring stretches a little. Hang a second, identical weight, and it stretches more — in fact, it stretches twice as far. Add a third weight, and it stretches three times as far.

This neat pattern was discovered by the English scientist Robert Hooke in 1676, and it is now called Hooke's law:

The extension of a spring is directly proportional to the force pulling on it.

"Extension" just means how much longer the spring has become compared to its natural, unstretched length. "Directly proportional" means that if you double the force, you double the extension; triple the force, triple the extension. The relationship is a straight line, which is why springs are so wonderfully useful for measuring.

The equation: F = k × x

We can write Hooke's law as a simple equation:

F = k × x

where:

• F is the force, in newtons (N)
• x is the extension, in metres (m)
• k is the spring constant, in newtons per metre (N/m)

The spring constant k describes how stiff the spring is. A stiff spring (like the heavy ones in a car's suspension) has a large k — it takes a big force to stretch it even a little. A floppy spring (like a Slinky) has a small k and stretches easily.

Worked example 1. A spring has a spring constant of k = 40 N/m. You pull it so it stretches by x = 0.15 m. What force are you applying?

F = k × x = 40 N/m × 0.15 m = 6 N

Worked example 2. You hang a force of 12 N on a spring and it stretches by 0.30 m. What is its spring constant?

k = F ÷ x = 12 N ÷ 0.30 m = 40 N/m

Notice that this is the same spring as in example 1 — the spring constant stays the same no matter how much you stretch it (as long as you do not overstretch it). That constancy is exactly what makes the spring trustworthy as a measuring tool.

The newton-meter: a ready-made spring scale

A newton-meter (also called a spring balance or spring scale) is a spring sealed inside a tube with a scale printed alongside it. When you hang or pull something on the hook, a pointer slides down the scale and shows the force directly in newtons. Fishing scales, luggage scales, and the force meters in your science lab all work this way.

Because weight is itself a force — the pull of gravity on an object's mass — a newton-meter also measures weight. On Earth, every kilogram of mass is pulled down with a force of about 9.8 N. So a 2 kg bag of sugar would stretch a spring scale to read about 19.6 N. To understand why mass turns into a downward force at all, see our lesson on gravity explained.

The elastic limit: when springs stop obeying the rules

Hooke's law is reliable — but only up to a point. Every spring has an elastic limit. Stretch it gently and it springs back to its original length every time; that is the elastic behaviour we rely on. But pull too hard, past the elastic limit, and the metal is permanently bent out of shape. The spring stays stretched even after you let go, and the neat F = k × x relationship breaks down.

You have probably seen this with an old, overstretched spring or a Slinky that has gone wavy and will not coil back up properly. Once a spring is stretched past its elastic limit, it can no longer give accurate force readings.

Why this matters

Measuring force is the starting point for understanding motion. Once you can measure the size of a push or a pull, you can predict how objects will speed up, slow down, or change direction. This connects directly to Newton's laws of motion, which describe exactly how forces change the way things move.

Try it yourself! 🧪

Build your own spring scale.

1. Hang a rubber band or a soft spring from a hook or nail so it dangles freely. Tape a strip of paper or a ruler next to it.
2. Mark the position of the bottom of the spring with nothing attached — this is your "zero".
3. Hang a known mass on the bottom. A 100 g mass weighs about 1 N. Mark where the bottom of the spring is now.
4. Add masses one at a time (100 g, 200 g, 300 g…), marking the new position each time.

Look at your marks: are they evenly spaced? If they are, your spring is obeying Hooke's law — each extra newton adds the same extra stretch. Now you can use your scale to "weigh" any small object: hang it on, read off where the bottom lands, and you have measured a force you could never see.