Pendulum

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

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 length
The standard experiment: T² against L is a straight line of gradient 4π²/g.
- T = 2π√(L/g), so T² ∝ L
- Getting g from the gradient of a straight-line graph
Does amplitude matter?
The period creeps up as the swing widens: T ≈ T₀(1 + θ₀²/16).
- The small-angle approximation and where it breaks down
- A systematic error that grows with amplitude
Does mass matter?
Change the bob from 10 g to 2 kg. The period should not move at all.
- Mass cancels in the equation of motion
- A null result is still a result
Measuring g
Time swings at several lengths and get g from the gradient.
- g from 4π²/gradient
- Why small amplitudes matter for this measurement
Damped swing
With damping the amplitude decays; the period hardly changes.
- Exponential decay of amplitude
- Light damping barely changes the period
- Energy leaving a system

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)
Angleθ (°)
Angleθ (rad)
Angular velocityω (rad/s)
Angular accelerationα (rad/s^2)
Speed of the bobv (m/s)
Horizontal positionx (m)
Vertical positiony (m)
Height above the lowest pointh (m)
TensionT (N)
Measured periodT (s)
Frequencyf (Hz)
Kinetic energyEₖ (J)
Potential energyEₚ (J)
Mechanical 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.

LengthL (m)
Mass of the bobm (kg)
Release angleθ₀ (°)
Initial angular velocityω₀ (°/s)
Gravitational field strengthg (m/s^2)
Dampingb (1/s)

The equations behind it

What pendulum 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

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

Experiments already set up on it

Period against length

The standard experiment: T² against L is a straight line of gradient 4π²/g.

T = 2π√(L/g), so T² ∝ L
Getting g from the gradient of a straight-line graph

Does amplitude matter?

The period creeps up as the swing widens: T ≈ T₀(1 + θ₀²/16).

The small-angle approximation and where it breaks down
A systematic error that grows with amplitude

Does mass matter?

Change the bob from 10 g to 2 kg. The period should not move at all.

Mass cancels in the equation of motion
A null result is still a result

Measuring g

Time swings at several lengths and get g from the gradient.

g from 4π²/gradient
Why small amplitudes matter for this measurement

Damped swing

With damping the amplitude decays; the period hardly changes.

Exponential decay of amplitude
Light damping barely changes the period
Energy leaving a system

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)

Angleθ (°)

Angleθ (rad)

Angular velocityω (rad/s)

Angular accelerationα (rad/s^2)

Speed of the bobv (m/s)

Horizontal positionx (m)

Vertical positiony (m)

Height above the lowest pointh (m)

TensionT (N)

Measured periodT (s)

Frequencyf (Hz)

Kinetic energyEₖ (J)

Potential energyEₚ (J)

Mechanical energyE (J)

The equations behind it

What pendulum 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 length

Does amplitude matter?

Does mass matter?

Measuring g

Damped swing

What you can measure

What you can change

The equations behind it

Work, energy and power

Simple harmonic motion

Also in Oscillations

Mass on a spring

Experiments already set up on it

Period against length

Does amplitude matter?

Does mass matter?

Measuring g

Damped swing

What you can measure

What you can change

The equations behind it

Work, energy and power

Simple harmonic motion

Also in Oscillations

Mass on a spring