Rotational dynamics

A flywheel driven by a motor or a falling mass, with the turns counted for you: τ = Iα, measured rather than assumed.

Rigid body mechanics — Forces, momentum and energy acting on bodies, and what happens when they collide.

Open it in The Lab The equations behind it

Experiments already set up on it

Angular acceleration against torque
The headline: α against τ is straight, with 1/I for a gradient.
- τ = Iα, so α against τ is straight with gradient 1/I
- The intercept is the friction torque, not an error
- Getting α from a timing: θ = ½αt², so α = 2θ/t²
Does a heavier wheel accelerate less?
I ∝ M for a fixed radius, so α should go as 1/M for the same torque.
- I = ½MR² is proportional to M
- α against 1/M is the straight-line version
Driven by a falling mass
The classic apparatus — and a trap: the hanging mass is accelerated too, so it adds mr².
- The torque on the wheel is mgr, not mg
- The tension is less than the weight while the mass accelerates
- α = (mgr − τ_f)/(I + mr²): the mass is part of what you are accelerating
Measuring the friction
Let the string run off and watch it slow: the only torque left is friction.
- With no drive, α = −τ_f/I is a constant deceleration
- ω against t after the string runs off is a straight line of gradient −τ_f/I

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)
Angle turnedθ (°)
Angle turnedθ (rad)
Turns completedN
Angular velocityω (rad/s)
Rotation raten (rpm)
Angular accelerationα (rad/s^2)
Speed of the rimv_R (m/s)
Moment of inertiaI (kg m^2)
Net torqueτ (N m)
Driving torqueτ_a (N m)
Angular momentumL (kg m^2/s)
Tension in the stringT (N)
Distance fallenh (m)
Speed of the hanging massv (m/s)
Rotational kinetic energyEₖ (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.

Mass of the wheelM (kg)
Radius of the wheelR (m)
Radius of the axler (m)
Applied torqueτ (N m)
Hanging massm (kg)
Length of stringh (m)
Friction at the bearingτ_f (N m)
Gravitational field strengthg (m/s^2)
Turns to timeN

The equations behind it

What rotational dynamics 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

Rotation

$τ = F r sin θ$
Torque
$ω = ω_{0} + α t$
Angular velocity after a time
$θ = ω_{0} t + \frac{1}{2} α t^{2}$
Angle turned through
$ω^{2} = ω_{0}^{2} + 2 α θ$
Angular velocity from an angle
$τ = I α$
Newton’s second law for rotation
$L = I ω$
Angular momentum
$E_{k} = \frac{1}{2} I ω^{2}$
Rotational kinetic energy

Also in Rigid body mechanics

Experiments already set up on it

Angular acceleration against torque

The headline: α against τ is straight, with 1/I for a gradient.

τ = Iα, so α against τ is straight with gradient 1/I
The intercept is the friction torque, not an error
Getting α from a timing: θ = ½αt², so α = 2θ/t²

Does a heavier wheel accelerate less?

I ∝ M for a fixed radius, so α should go as 1/M for the same torque.

I = ½MR² is proportional to M
α against 1/M is the straight-line version

Driven by a falling mass

The classic apparatus — and a trap: the hanging mass is accelerated too, so it adds mr².

The torque on the wheel is mgr, not mg
The tension is less than the weight while the mass accelerates
α = (mgr − τ_f)/(I + mr²): the mass is part of what you are accelerating

Measuring the friction

Let the string run off and watch it slow: the only torque left is friction.

With no drive, α = −τ_f/I is a constant deceleration
ω against t after the string runs off is a straight line of gradient −τ_f/I

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)

Angle turnedθ (°)

Angle turnedθ (rad)

Turns completedN

Angular velocityω (rad/s)

Rotation raten (rpm)

Angular accelerationα (rad/s^2)

Speed of the rimv_R (m/s)

Moment of inertiaI (kg m^2)

Net torqueτ (N m)

Driving torqueτ_a (N m)

Angular momentumL (kg m^2/s)

Tension in the stringT (N)

Distance fallenh (m)

Speed of the hanging massv (m/s)

Rotational kinetic energyEₖ (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.

Mass of the wheelM (kg)

Radius of the wheelR (m)

Radius of the axler (m)

Applied torqueτ (N m)

Hanging massm (kg)

Length of stringh (m)

Friction at the bearingτ_f (N m)

Gravitational field strengthg (m/s^2)

Turns to timeN

The equations behind it

What rotational dynamics 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

Rotation

$τ = F r sin θ$
Torque
$ω = ω_{0} + α t$
Angular velocity after a time
$θ = ω_{0} t + \frac{1}{2} α t^{2}$
Angle turned through
$ω^{2} = ω_{0}^{2} + 2 α θ$
Angular velocity from an angle
$τ = I α$
Newton’s second law for rotation
$L = I ω$
Angular momentum
$E_{k} = \frac{1}{2} I ω^{2}$
Rotational kinetic energy

Rotational dynamics

Experiments already set up on it

Angular acceleration against torque

Does a heavier wheel accelerate less?

Driven by a falling mass

Measuring the friction

What you can measure

What you can change

The equations behind it

Work, energy and power

Rotation

Also in Rigid body mechanics

Inclined plane & friction

Collisions

Moments & equilibrium

Circular motion

Rotational dynamics

Experiments already set up on it

Angular acceleration against torque

Does a heavier wheel accelerate less?

Driven by a falling mass

Measuring the friction

What you can measure

What you can change

The equations behind it

Work, energy and power

Rotation

Also in Rigid body mechanics

Inclined plane & friction

Collisions

Moments & equilibrium

Circular motion