Mass on a spring

Simple harmonic motion with optional damping and a driving force, so resonance can be measured.

Oscillations — Simple harmonic motion, damping, driving and resonance — what waves are made of.

Open it in The Lab The equations behind it

Experiments already set up on it

Period against mass
T² against m is a straight line of gradient 4π²/k.
- T = 2π√(m/k)
- Getting k from the gradient of T² against m
Period against stiffness
A stiffer spring oscillates faster: T ∝ 1/√k.
- T ∝ 1/√k
- Linearising a relationship that is not already straight
The shape of simple harmonic motion
Record x, v and a together and see how they line up.
- a = −ω²x: acceleration proportional to displacement and opposite to it
- Velocity is a quarter of a cycle ahead of displacement
- Energy moves back and forth between kinetic and elastic
Damped oscillation
Raise the damping towards 2√(mk) and watch the oscillation disappear.
- Light, critical and heavy damping
- Critical damping at c = 2√(mk)
- Exponential decay of the envelope
Resonance
Drive the mass across a range of frequencies and find the peak.
- Resonance when the driving frequency matches the natural frequency
- Damping lowers and broadens the peak
- Transients die away and leave a steady state
Hanging from a spring
Gravity stretches the spring by mg/k but leaves the period alone.
- A hanging spring oscillates about a new equilibrium, mg/k lower
- Hooke’s law from the static extension

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)
Displacement from equilibriumx (m)
Velocityv (m/s)
Speed|v| (m/s)
Accelerationa (m/s^2)
Extension of the springe (m)
Height of the massy (m)
Restoring forceF (N)
Total force from the springF_s (N)
Driving force nowF_d (N)
Measured periodT (s)
Frequencyf (Hz)
Steady amplitudeA (m)
Kinetic energyEₖ (J)
Elastic potential energyEₚ (J)
Total energyE (J)

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Massm (kg)
Spring constantk (N/m)
Initial displacementx₀ (m)
Initial velocityv₀ (m/s)
Damping constantc (kg/s)
Driving force amplitudeF₀ (N)
Driving frequencyf (Hz)
Settling time before measuring amplitudet_s (s)
Gravitational field strengthg (m/s^2)
Natural length of the springL₀ (m)

The equations behind it

What mass on a spring is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Work, energy and power

$W = F s cos θ$
Work done by a force
$E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}$
Kinetic energy
$Δ E_{p} = m g Δ h$
Change in gravitational potential energy — Near the surface, where g is constant.
$E_{elastic} = \frac{1}{2} k Δ x^{2}$
Elastic potential energy
$P = \frac{Δ W}{Δ t} = F v$
Power
$η = \frac{E _{useful}}{E _{input}}$
Efficiency

Simple harmonic motion

$a = - ω^{2} x$
The defining condition — Acceleration towards the centre, proportional to the displacement.
$T = \frac{1}{f} = \frac{2 π}{ω}$
Period, frequency and angular frequency
$x = x_{0} sin (ω t + ϕ)$
Displacement with time
$v = ω x_{0} cos (ω t + ϕ)$
Velocity with time
$v = \pm ω x_{0}^{2} - x^{2}$
Velocity at a displacement
$E_{T} = \frac{1}{2} m ω^{2} x_{0}^{2}$
Total energy of an oscillator
$T = 2 π \frac{L}{g}$
Period of a simple pendulum — Small swings only.
$T = 2 π \frac{m}{k}$
Period of a mass on a spring

Also in Oscillations

Pendulum
A simple pendulum with an exact or small-angle equation, optional damping, and the period measured for you.

Experiments already set up on it

Period against mass

T² against m is a straight line of gradient 4π²/k.

T = 2π√(m/k)
Getting k from the gradient of T² against m

Period against stiffness

A stiffer spring oscillates faster: T ∝ 1/√k.

T ∝ 1/√k
Linearising a relationship that is not already straight

The shape of simple harmonic motion

Record x, v and a together and see how they line up.

a = −ω²x: acceleration proportional to displacement and opposite to it
Velocity is a quarter of a cycle ahead of displacement
Energy moves back and forth between kinetic and elastic

Damped oscillation

Raise the damping towards 2√(mk) and watch the oscillation disappear.

Light, critical and heavy damping
Critical damping at c = 2√(mk)
Exponential decay of the envelope

Resonance

Drive the mass across a range of frequencies and find the peak.

Resonance when the driving frequency matches the natural frequency
Damping lowers and broadens the peak
Transients die away and leave a steady state

Hanging from a spring

Gravity stretches the spring by mg/k but leaves the period alone.

A hanging spring oscillates about a new equilibrium, mg/k lower
Hooke’s law from the static extension

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)

Displacement from equilibriumx (m)

Velocityv (m/s)

Speed|v| (m/s)

Accelerationa (m/s^2)

Extension of the springe (m)

Height of the massy (m)

Restoring forceF (N)

Total force from the springF_s (N)

Driving force nowF_d (N)

Measured periodT (s)

Frequencyf (Hz)

Steady amplitudeA (m)

Kinetic energyEₖ (J)

Elastic potential energyEₚ (J)

Total energyE (J)

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Massm (kg)

Spring constantk (N/m)

Initial displacementx₀ (m)

Initial velocityv₀ (m/s)

Damping constantc (kg/s)

Driving force amplitudeF₀ (N)

Driving frequencyf (Hz)

Settling time before measuring amplitudet_s (s)

Gravitational field strengthg (m/s^2)

Natural length of the springL₀ (m)

The equations behind it

What mass on a spring is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Work, energy and power

$W = F s cos θ$
Work done by a force
$E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}$
Kinetic energy
$Δ E_{p} = m g Δ h$
Change in gravitational potential energy — Near the surface, where g is constant.
$E_{elastic} = \frac{1}{2} k Δ x^{2}$
Elastic potential energy
$P = \frac{Δ W}{Δ t} = F v$
Power
$η = \frac{E _{useful}}{E _{input}}$
Efficiency

Simple harmonic motion

$a = - ω^{2} x$
The defining condition — Acceleration towards the centre, proportional to the displacement.
$T = \frac{1}{f} = \frac{2 π}{ω}$
Period, frequency and angular frequency
$x = x_{0} sin (ω t + ϕ)$
Displacement with time
$v = ω x_{0} cos (ω t + ϕ)$
Velocity with time
$v = \pm ω x_{0}^{2} - x^{2}$
Velocity at a displacement
$E_{T} = \frac{1}{2} m ω^{2} x_{0}^{2}$
Total energy of an oscillator
$T = 2 π \frac{L}{g}$
Period of a simple pendulum — Small swings only.
$T = 2 π \frac{m}{k}$
Period of a mass on a spring

Experiments already set up on it

Period against mass

Period against stiffness

The shape of simple harmonic motion

Damped oscillation

Resonance

Hanging from a spring

What you can measure

What you can change

The equations behind it

Work, energy and power

Simple harmonic motion

Also in Oscillations

Pendulum

Experiments already set up on it

Period against mass

Period against stiffness

The shape of simple harmonic motion

Damped oscillation

Resonance

Hanging from a spring

What you can measure

What you can change

The equations behind it

Work, energy and power

Simple harmonic motion

Also in Oscillations

Pendulum