Hooke’s Law Explained

Why does a spring pull back twice as hard when you stretch it twice as far? That simple pattern is Hooke’s law. It links force, stiffness, and stretch in one short formula. A spring with a stiffness of 200 N/m, stretched 10 centimeters, pulls back with exactly 20 newtons. This guide shows what the law means, how to measure a spring, and where the rule breaks down.

Hooke’s Law Quick Reference

Keep this table handy when you work a spring problem. Every row uses the same formula, just solved for a different unknown. Convert all lengths to meters before you plug them in.

Two unit shortcuts save time. One newton per centimeter equals 100 N/m. One pound-force per inch equals about 175.1 N/m, which helps with spring specs written in US units.

What Does Hooke’s Law Actually Say?

Hooke’s law says the force a spring exerts grows in direct proportion to how far you stretch or squeeze it. Double the stretch, and the force doubles. Triple it, and the force triples.

The formula is F = -kx. Here F is the spring force in newtons, and x is the distance from the resting length in meters. The letter k is the spring constant, a single number that describes how stiff the spring is.

A stiff spring has a large k, so a tiny stretch creates a big force. A soft spring has a small k, so it stretches a lot under a light load. The same law also covers squeezing, since compression is just a negative stretch.

Spring constant (k)

The stiffness of a spring, measured in newtons per meter. It tells you the force needed for each meter of stretch.

Displacement (x)

How far the spring has moved from its natural, resting length. It is measured from rest, not from the floor or a table.

Restoring force

The pull or push a spring makes as it tries to return to its resting length.

Elastic limit

The largest stretch a material can take and still return fully to its original shape.

Elastic potential energy

The energy stored inside a stretched or squeezed spring, measured in joules.

How Do You Find a Spring Constant?

Divide the force on the spring by the stretch it causes, with the stretch in meters. The answer is k in newtons per meter. One careful measurement is often enough for a simple spring.

Say a 5 newton pull stretches a spring by 2 centimeters. First convert 2 centimeters to 0.02 meters. Then 5 divided by 0.02 gives a spring constant of 250 N/m.

A hanging mass works as the pull, too. A 0.5 kilogram mass weighs about 4.9 newtons. On that 250 N/m spring, it stretches 4.9 divided by 250, or 0.0196 meters. That is just under 2 centimeters.

Real springs span a huge range. A college physics text finds a car suspension constant near 65,300 N/m, since an 80 kilogram person sinks the car only 1.20 centimeters. To solve for force, stiffness, or stretch in one step, use the Hooke’s law calculator for spring force and stiffness.

Why Is There a Minus Sign in F = -kx?

The minus sign means the spring always pushes or pulls opposite to the way you moved it. Stretch it to the right, and it pulls left. Squeeze it, and it pushes back out.

This opposite force is why physicists call it a restoring force. It always points back toward the resting position. The idea follows the action and reaction pairs covered in our guide to Newton’s laws and how forces act.

For everyday sizing, most people drop the sign and work with sizes only. A spring stretched 0.05 meters with k = 200 N/m pulls back with 10 newtons. The sign simply reminds you which way that 10 newtons points.

The restoring force also makes springs bounce. A mass on a spring overshoots the rest point, gets pulled back, and repeats the motion. For a 0.5 kilogram mass on a 200 N/m spring, one full bounce takes about 0.314 seconds. A stiffer spring gives a shorter bounce, and a heavier mass gives a longer one.

How Much Energy Does a Stretched Spring Store?

A stretched spring stores energy equal to one half times k times the stretch squared. In symbols, that is PE = 1/2 k x^2, with the answer in joules. The spring gives this energy back when released.

Take the 200 N/m spring stretched 0.1 meters. Half of 200 times 0.01 equals 1.0 joule. On a force graph, that energy is the triangle under the line, as the chart below shows.

That squared term matters for safety. A spring pulled a little further holds far more energy than you expect. For how stored energy turns into motion, see our guide on kinetic vs potential energy.

When Does Hooke’s Law Stop Working?

Hooke’s law stops working once a material passes its linear range. Past that point, force and stretch no longer rise together in a straight line. Push further still, and the part bends for good.

A college physics text notes that most materials behave elastically when deformation stays under about 0.1 percent. Coil springs reach much larger stretches because the coiled wire twists only a little. That clever shape is why a spring obeys the law over many centimeters.

Signs You Have Left the Linear Range

Watch for three clues during a test. The force per centimeter starts to change, the spring stays longer after you let go, or the coils open up unevenly. Any one of these means your k value no longer applies.

Test a spring gently to stay safe. Take readings at small, even steps, such as 1, 2, and 3 centimeters. When each step adds the same force, you are still inside the Hooke’s law range.

Rubber bands are a helpful contrast. Their force curve bends almost from the start, so one k value fits them poorly. Coil springs, steel wires, and building beams under light loads suit the law much better.

Questions People Ask About Hooke’s Law

Where These Numbers Come From

This article is general physics education for ideal springs within their elastic range. Check real spring ratings with the maker before any load-bearing use. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.