Orbits & gravitation

Put a satellite in orbit and measure its period against its radius, for Kepler’s third law.

Fields — Action at a distance: gravitational, electric and magnetic fields and the paths in them.

Open it in The Lab The equations behind it

Experiments already set up on it

Kepler’s third law
T² against r³ is a straight line of gradient 4π²/GM.
- T² = 4π²r³/GM for a circular orbit
- Getting the mass of the central body from the gradient
Field strength with distance
g against 1/r² is a straight line of gradient GM.
- g = GM/r²
- An inverse-square law plotted straight
An elliptical orbit
Launch faster than circular speed and watch energy move between forms.
- Kinetic and potential energy trade off around an orbit
- Total energy and angular momentum are constant
Getting away
How far out a satellite reaches, as the launch speed approaches escape.
- v_esc = √(2GM/r)
- Why the furthest point runs away to infinity

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)
Measured periodT (s)
Distance apartr (m)
Speedv (m/s)
xx (m)
yy (m)
Semi-major axisa (m)
Eccentricitye (1)
Closest approachr_p (m)
Furthest pointr_a (m)
Field strength hereg (m/s^2)
Escape speed herev_{esc} (m/s)
Kinetic energyE_k (J)
Potential energyE_p (J)
Total energyE (J)
Angular momentumL (kg*m^2/s)

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 central bodyM (kg)
Mass of the satellitem (kg)
Radius of the central bodyR (m)
Starting distancer_0 (m)
Launch speed (× circular)f (1)
Launch angle from the tangentφ (°)
Gravitational constantG (N*m^2/kg^2)

The equations behind it

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

Circular motion

$v = \frac{2 π r}{T} = ω r$
Speed round a circle
$ω = \frac{2 π}{T}$
Angular velocity
$a = \frac{v ^{2}}{r} = ω^{2} r = \frac{4 π ^{2} r}{T ^{2}}$
Centripetal acceleration
$F = \frac{m v ^{2}}{r} = m ω^{2} r$
Centripetal force — Not a new force — it is whatever is pointing inwards.

Gravitational fields

$F = \frac{G m _{1} m _{2}}{r ^{2}}$
Newton’s law of gravitation
$g = \frac{F}{m} = \frac{GM}{r ^{2}}$
Gravitational field strength
$V_{g} = - \frac{GM}{r}$
Gravitational potential — Zero at infinity, so it is negative everywhere else.
$E_{p} = - \frac{GM m}{r}$
Gravitational potential energy
$g = - \frac{Δ V _{g}}{Δ r}$
Field as a potential gradient
$v_{esc} = \frac{2 GM}{r}$
Escape speed
$v_{orbit} = \frac{GM}{r}$
Orbital speed
$T^{2} = \frac{4 π ^{2}}{GM} r^{3}$
Kepler’s third law

Also in Fields

Experiments already set up on it

Kepler’s third law

T² against r³ is a straight line of gradient 4π²/GM.

T² = 4π²r³/GM for a circular orbit
Getting the mass of the central body from the gradient

Field strength with distance

g against 1/r² is a straight line of gradient GM.

g = GM/r²
An inverse-square law plotted straight

An elliptical orbit

Launch faster than circular speed and watch energy move between forms.

Kinetic and potential energy trade off around an orbit
Total energy and angular momentum are constant

Getting away

How far out a satellite reaches, as the launch speed approaches escape.

v_esc = √(2GM/r)
Why the furthest point runs away to infinity

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)

Measured periodT (s)

Distance apartr (m)

Speedv (m/s)

xx (m)

yy (m)

Semi-major axisa (m)

Eccentricitye (1)

Closest approachr_p (m)

Furthest pointr_a (m)

Field strength hereg (m/s^2)

Escape speed herev_{esc} (m/s)

Kinetic energyE_k (J)

Potential energyE_p (J)

Total energyE (J)

Angular momentumL (kg*m^2/s)

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 central bodyM (kg)

Mass of the satellitem (kg)

Radius of the central bodyR (m)

Starting distancer_0 (m)

Launch speed (× circular)f (1)

Launch angle from the tangentφ (°)

Gravitational constantG (N*m^2/kg^2)

The equations behind it

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

Circular motion

$v = \frac{2 π r}{T} = ω r$
Speed round a circle
$ω = \frac{2 π}{T}$
Angular velocity
$a = \frac{v ^{2}}{r} = ω^{2} r = \frac{4 π ^{2} r}{T ^{2}}$
Centripetal acceleration
$F = \frac{m v ^{2}}{r} = m ω^{2} r$
Centripetal force — Not a new force — it is whatever is pointing inwards.

Gravitational fields

$F = \frac{G m _{1} m _{2}}{r ^{2}}$
Newton’s law of gravitation
$g = \frac{F}{m} = \frac{GM}{r ^{2}}$
Gravitational field strength
$V_{g} = - \frac{GM}{r}$
Gravitational potential — Zero at infinity, so it is negative everywhere else.
$E_{p} = - \frac{GM m}{r}$
Gravitational potential energy
$g = - \frac{Δ V _{g}}{Δ r}$
Field as a potential gradient
$v_{esc} = \frac{2 GM}{r}$
Escape speed
$v_{orbit} = \frac{GM}{r}$
Orbital speed
$T^{2} = \frac{4 π ^{2}}{GM} r^{3}$
Kepler’s third law

Orbits & gravitation

Experiments already set up on it

Kepler’s third law

Field strength with distance

An elliptical orbit

Getting away

What you can measure

What you can change

The equations behind it

Circular motion

Gravitational fields

Also in Fields

Charged particle in a field

Electromagnetic induction

AC generator

Electric charges & fields

Parallel-plate capacitor

Orbits & gravitation

Experiments already set up on it

Kepler’s third law

Field strength with distance

An elliptical orbit

Getting away

What you can measure

What you can change

The equations behind it

Circular motion

Gravitational fields

Also in Fields

Charged particle in a field

Electromagnetic induction

AC generator

Electric charges & fields

Parallel-plate capacitor