What is Hooke's law?

Hang a mass on a spring and it stretches. Hang twice the mass and it stretches twice as far. Three times the mass, three times the stretch. That is Hooke's law: F = k × x.

F is the force pulling the spring, in newtons. A 100 g mass pulls with about 1 N. x is the extension, how much longer the spring gets. k is the spring constant, the stiffness: how many newtons it takes to stretch the spring by one metre. Plot force against extension and you get a straight line whose steepness is k.

Stretching a spring stores energy: E = ½ k x². On the chart it is the area of the triangle under the line. That energy is what flings a catapult and bounces a mass back up.

Join springs end to end (in series) and each one carries the whole load, so the stretches add: two springs act like one spring half as stiff. Hang them side by side (in parallel) and they share the load: two act like one twice as stiff.

Pull the mass down and let go, and it bounces. One swing takes T = 2π √(m ÷ k), and the surprise is what's missing: the size of the swing. Big bounce or small, the timing is the same. That's why springs keep time in watches, and it is the start of Waves and resonance.

A spring balance is Hooke's law turned into a tool. Hang a bag of onions on the hook and the spring stretches in proportion to the pull. Because the stretch is proportional to the force, every extra kilogram moves the pointer the same distance, so the scale can be printed with evenly spaced marks. Richard Salter was making spring balances in England by about 1770, and the hanging balance is still used in markets and for weighing luggage.

Our balance reads 10 kg over 10 cm of stretch, so its spring has k = 98.1 N ÷ 0.1 m ≈ 980 N/m.

Here's the catch: a spring feels force, not mass. Take it to the Moon, where gravity is about a sixth as strong, and 6 kg of onions reads just 1 kg. A kitchen pan balance, which compares two masses, would read 6 kg anywhere.

Inside a click pen (see PenClear), a little spring sits round the front of the refill. Press the button and you squeeze it about 8 mm, so it pushes back with about 2 N, the weight of a small apple. The click mechanism locks the refill out. Click again, the lock lets go, and the squeezed spring snaps the tip back inside.

Every wheel of a car sits under a coil spring (see CarClear). A small hatchback puts about 225 kg on each one. With a spring of k = 22,000 N/m, Hooke's law says it squeezes x = F ÷ k ≈ 10 cm. Each passenger adds a little more: about 8 mm per person on this corner. Load the boot and the car sits lower in a perfectly straight-line way.

Hit a speed breaker and the spring squeezes, soaking up the jolt, then gives the energy back. On its own it would keep bouncing about 1.5 times a second, T = 2π √(m ÷ k). So every spring has a partner, the shock absorber or damper: oil forced through small holes turns the bounce into heat. Motorbikes use the same pair in their forks and rear shocks, and a washing machine hangs its drum on springs and dampers too (see MotorcycleClear and WasherClear).

A bungee cord is a very long, soft spring. Jump from a 43 m bridge on a 12 m cord and you fall freely for 12 m. Then the cord starts to stretch and pull. By the bottom, all the energy of your fall, m g h, has been stored in the cord as ½ k x². Heavier jumpers stretch it further, so crews pick a stiffer cord for them.

Every note of a harmonium comes from a thin brass tongue, fixed at one end over a slot (see HarmoniumClear). Push its tip and it bends in proportion to the push, F = k × x: it is a tiny diving-board spring. Let go and it swings back and forth, just like the mass on a spring in chapter 1, at f = (1/2π) √(k ÷ m).

So a reed maker has two knobs. Stiffness: a tongue's k grows with its thickness cubed and falls with its length cubed. Twice as thick is 8 times stiffer. Mass: a dab of solder at the tip adds mass and lowers the note. Scraping metal off the tip raises it.

A guitar string is a spring too, a very stiff one. Turning the tuning peg winds the string round its post and stretches it by a few millimetres. By Hooke's law the tension rises in step, about 13.5 N for every millimetre on a thin steel E string. The note climbs with the square root of the tension (see GuitarClear). A piano does the same with far thicker wire: each string pulls with 700 to 900 N (see PianoClear), and a tabla player tightens the drumhead's straps for the same reason.

Hooke's law is a promise with small print: it holds only for small stretches. Engineers write it for materials as stress = E × strain. Stress is force per area; strain is stretch per length; E is Young's modulus, the stiffness of the stuff itself. Steel's is about 200 GPa. Rubber's is roughly 0.001 to 0.1 GPa, thousands of times floppier.

Steel. Pull a steel bar and it obeys Hooke's law perfectly, but only up to about 0.1 % stretch. That is the elastic limit, or yield point. Let go before it and the bar returns exactly to its length. Go past it and the atoms start to slide: it stays stretched (plastic deformation). Keep pulling and it can stretch 20 % or more, thin in the middle (necking) and finally break. That is why an over-stretched spring never goes back.

Rubber never had a straight line. A band stretches easily at first, then stiffens sharply near six times its length. And it pulls back less than you pulled it out: that lost energy becomes heat. Stretch a band fast against your lip and feel it warm up. A bungee cord is rubber, which is why crews test and change cords so often.

Myth-buster: “A stiffer spring stores more energy.” Not for the same force! Hang the same weight on a soft and a stiff spring: the soft one stretches further, and energy is F² ÷ 2k, so it stores more. That is why archers and catapult builders chase long, springy draws. Only when you stretch both the same distance does the stiff one win.