Why does a mass on a spring bounce, and how fast?

How to use this simulation

1. Start with the defaults: a 1 kg block, a 25 N/m spring and a 0.20 m amplitude. The period readout says 1.257 s. Time one swing with a stopwatch and compare.
2. Raise the mass slider. A heavier block is lazier, so the period grows. Now raise the spring constant: a stiffer spring snaps the block back faster.
3. Change the amplitude. The maximum speed scales with it, but the period does not move at all.
4. Watch the x–t graph and the phase plot together. With zero damping the phase plot is a closed ellipse; add damping and it spirals into the centre.
5. Look at the three energy bars: kinetic falls as potential rises, and the total stays put.
6. Switch the arrangement to series, then parallel, and watch the effective spring constant and the period change.

The spring in your pen is a tiny clock

Press the spring inside a ballpoint pen and let go. It jumps up, drops, jumps again, and every bounce takes about the same time. Nobody is holding a stopwatch, yet there is a clock hidden in that coil.

How fast it ticks depends on just two things: how heavy the block is and how stiff the spring is. A heavy block is sluggish and swings slowly. A stiff spring snaps back and swings fast. The surprise is what does not matter: how far you pulled it. A small stretch and a big stretch take the same time per swing.

In this lesson we find the rule behind that hidden clock. It is called simple harmonic motion, and the same maths runs car suspensions, guitar strings, a swinging pendulum at small angles and even the vibration of atoms in a crystal.

Start from zero: what is an oscillation?

A child on a swing, the pendulum of a wall clock, a branch in the wind: each one moves back and forth about a fixed spot. That spot is the equilibrium position, the place where the object would rest if you left it alone.

Push the object away from equilibrium and a force pulls it back. We call it the restoring force. A spring does this whether you stretch it or squash it. But when the object returns it does not stop at equilibrium. It has speed, so inertia carries it past. On the other side the restoring force pulls it back again, and the cycle repeats.

Four numbers describe any oscillation: the displacement x from equilibrium, the amplitude A (the largest displacement), the period T (time for one full cycle) and the frequency f (cycles per second). They are linked by f = 1/T. A period of 0.5 s means a frequency of 2 Hz.

The words you must know

These terms come back in every question on this chapter. Learn the meaning and the unit together.

Hooke’s law: how hard the spring pulls

Pull a spring gently and it stretches a little; pull harder and it stretches more. Robert Hooke noticed that for small stretches the force and the extension are directly proportional. Double the force and you double the extension.

The force the spring returns points opposite to the displacement. Pull right and it pulls left. That is why the law carries a minus sign. The constant k is the spring constant: a big k means a stiff spring, a small k a soft one.

If 12 N stretches a spring by 0.08 m, then k = 12 ÷ 0.08 = 150 N/m. That means every metre of stretch would cost 150 N, even though in practice we stay with small stretches.

Mind the limit. Stretch a spring too far and it stays deformed, and Hooke’s law no longer holds. The law is true only inside the elastic limit.

Why this is simple harmonic motion

Once you know the force you know the acceleration. Newton’s second law says F = ma. Here the force is −kx, so ma = −kx, which gives a = −(k/m)x. Notice what happened: acceleration is proportional to displacement and points the opposite way. That condition is the definition of SHM.

The ratio k/m is a positive constant. Give it the name ω² and the law reads a = −ω²x. That ω is the angular frequency, and it follows that ω = √(k/m).

If acceleration were instead proportional to, say, x², the object would still oscillate but not in simple harmonic fashion. To get the clean sine or cosine wave, the acceleration must be exactly −ω²x.

What the period depends on

Since ω = 2π/T, we get T = 2π/ω, and substituting ω gives T = 2π√(m/k). Three facts hide in this formula. One: more mass means a longer period, by a square root. Two: a stiffer spring means a shorter period. Three: the amplitude A does not appear, so the period does not depend on it.

Four times the mass gives twice the period. With the defaults of 1 kg and 25 N/m, T = 1.257 s. Quadruple the mass and T becomes 2.0 times as long. Quadruple k instead and T becomes 0.5 times as long, which is half. Forget the square root and you will lose marks.

Every number in the table below was computed straight from the formula. Try any cell in the simulation to check it.

Another surprise: T does not depend on gravity g. Take a hanging spring to the Moon and its period is unchanged. A pendulum clock would run differently there.

The story of one swing: displacement, velocity, acceleration

Pull the block 0.20 m and release it. At the start the speed is zero but the force is greatest, so the acceleration is greatest. As the block moves in, the force falls but the speed climbs. At equilibrium the force and acceleration are zero while the speed is at its maximum. The block overshoots, slows, stops at the far end and comes back.

Mathematically the displacement is a cosine wave. Because we release from the end, the starting phase is φ = 0. Velocity is the rate of change of displacement and acceleration is the rate of change of velocity, so the velocity graph is a sine and the acceleration is the displacement flipped upside down.

Phase simply tells you which stage of the cycle you are in; the angle ωt + φ is the phase. Velocity runs a quarter of a cycle ahead of displacement (π/2 radians): when x is at a peak, v is zero, and when x is zero, v peaks.

The table lists four positions for the default setting (1 kg, 25 N/m, 0.20 m). Here ω = 5.00 rad/s, the maximum speed Aω = 1.00 m/s and the maximum acceleration Aω² = 5.00 m/s².

The energy swap

A stretched spring stores energy, called elastic potential energy. When the block moves, that energy turns into kinetic energy. At equilibrium the potential energy is zero and the kinetic is at its peak; at the ends it is the other way round. The total stays constant because we ignore friction and air drag.

With the defaults the total is ½ × 25 × 0.20² = 0.500 J. Watch the bars at the bottom of the simulation: when the green bar grows the red one shrinks, but the black total bar never changes length.

Two things follow. Energy goes as the square of the amplitude, so doubling A quadruples E. And at any position, kinetic energy is total minus potential, so you can find speed without ever using the displacement formula.

Two springs together: series and parallel

When one spring will not do, we combine two. In series the springs are joined end to end; the same force passes through both and the extensions add. The combination is softer. In parallel the springs sit side by side; the extension is shared and the forces add. The combination is stiffer.

Either way the block behaves as if it hung on a single spring of effective constant k. After that, T = 2π√(m/k) works as always. Switch the arrangement in the simulation and watch the “effective spring constant” readout change.

There is a famous trap. For electrical resistors, series adds and parallel uses reciprocals. For spring constants it is the other way round: parallel adds directly, series adds the reciprocals. If you ever feel unsure, ask which arrangement shares the extension.

The hanging spring: where did gravity go?

Hang the block vertically and its weight mg pulls it down. It settles where the spring’s upward force equals the weight: kΔl = mg, so the stretch is Δl = mg/k. That is the new equilibrium.

Now pull the block a further x below it. The net force is −k(Δl + x) + mg = −kx. The mg and kΔl cancel. Gravity has vanished, so a hanging spring performs exactly the same SHM with the same T = 2π√(m/k).

A handy result follows: T = 2π√(Δl/g). You can find the period just by measuring the stretch, without knowing k. In the simulation choose “Vertical” and the “Stretch at rest” readout shows this mg/k.

Damping: why oscillations die away

In the real world oscillations do not last forever. Air resistance and friction inside the spring slowly drain energy, and the amplitude shrinks a little each swing. This is damped oscillation.

Turn on the damping slider and watch the phase ellipse spiral inward, since both speed and displacement are falling. With 1 kg and c = 0.5 N·s/m the amplitude halves in 2.77 s.

Light damping leaves the period almost unchanged and only shrinks the amplitude. Heavy damping stops the swinging altogether; the block just creeps back to equilibrium. A car’s shock absorber does exactly that job.

Hands-on experiments in the simulation

Each takes a minute. Start from the defaults, and press Reset whenever you want a clean slate.

• Experiment 1: quadruple the mass. Note how many times longer T becomes, then check the grid above.
• Experiment 2: sweep the amplitude from 0.05 m to 0.30 m. Only the peak speed and acceleration change; confirm that T stays put.
• Experiment 3: slow the speed to 0.25× and pause just as the block crosses the centre. Which energy bar is at its maximum?
• Experiment 4: choose series with k₁ equal to k₂. Is the effective k half? Now switch to parallel and see it double.
• Experiment 5: choose vertical and raise the mass. The stretch grows, yet the period formula is unchanged. Explain why in your own words.
• Experiment 6: raise the damping slowly and watch the shape of the phase plot. At what point does the swinging all but stop?

Worked problems

Six problems, step by step. Every number is computed in code, so you can enter the inputs and compare.

Mistakes everybody makes

Dodging these saves you half the marks you would otherwise lose.

• Thinking a bigger amplitude means a longer period. In SHM the period is independent of amplitude.
• Leaving the mass in grams. Always convert m to kilograms, or T comes out wrong.
• Writing T = 2π√(k/m). Mass goes on top and k underneath: √(m/k).
• Adding spring constants in series. Series adds reciprocals; parallel adds directly.
• Treating gravity as an extra force on a hanging spring and changing T. Measured from equilibrium, g drops out.
• Believing acceleration is greatest at equilibrium. There it is zero; the speed is greatest.
• Mixing up force and energy. The force is −kx, the potential energy is ½kx². They are different things.

Where you meet this in real life

Simple harmonic motion is hiding in plenty of places around you.

• Car and motorcycle suspension: springs absorb bumps and shock absorbers supply the damping.
• Mechanical watches: a balance wheel and hairspring tick at a steady rhythm.
• Spring balances: the stretch gives the weight, because F = kx.
• Buildings and bridges: wind and earthquakes make them sway, so engineers compute their natural frequency.
• Guitar strings and tuning forks: tiny SHM vibrations produce a precise note.
• Molecular bonds: the bond between two atoms vibrates like a spring, and infrared spectroscopy reads it.

Exam corner: board-style questions

Question: A spring of constant 100 N/m carries a 0.4 kg mass. The mass is pulled 0.05 m from equilibrium and released.

(a) Define simple harmonic motion. [1 mark]

(b) Why is the period of a spring-mass system independent of amplitude? [2 marks]. Hint: a = −ω²x and ω contains no A.

(c) Calculate the period. [3 marks]. Answer: T = 2π√(0.4 ÷ 100) = 0.397 s, with ω = 15.81 rad/s.

(d) Find the maximum speed and total energy, and say in what form all the energy exists at the centre. [4 marks]. Answer: Aω = 0.791 m/s and E = ½kA² = 0.125 J. At equilibrium PE is zero, so all 0.125 J is kinetic.

Extra: the maximum force is kA = 5.0 N and the maximum acceleration Aω² = 12.50 m/s².

Revision at a glance

Read this list once the night before the exam.

• Hooke’s law: F = −kx, valid for small stretches.
• In SHM the acceleration is proportional to displacement and opposite to it: a = −ω²x.
• ω = √(k/m), T = 2π√(m/k), f = 1/T.
• The period is independent of amplitude.
• Maximum speed Aω at the centre; maximum acceleration Aω² at the ends.
• Total energy E = ½kA²; kinetic and potential energy trade places.
• Series: 1/k = 1/k₁ + 1/k₂. Parallel: k = k₁ + k₂.
• Hanging spring: T = 2π√(Δl/g).
• Under damping the amplitude decays exponentially.

Frequently asked questions