Circular motion

A conical pendulum whirling on a string: the tension holds the bob up and pulls it round, and only one speed keeps the angle steady.

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

Period against the height of the cone
The headline: T² against h = L cos θ is straight, with gradient 4π²/g.
- T = 2π√(h/g), where h is the depth of the bob below the pivot
- Getting g from the gradient of T² against h
- Timing ten turns and dividing
Does a longer string go round more slowly?
Keep the angle and lengthen the string: T² against L is still straight.
- h = L cos θ, so at a fixed angle T² ∝ L
- The same gradient, scaled by cos θ
Does the mass matter?
Change the bob from 10 g to 2 kg. The period should not move at all.
- Mass cancels from T cos θ = mg and T sin θ = mω²r
- A null result is a result
The wrong speed
Set it going too slowly and the bob rises and falls instead of circling.
- Only one speed holds the string at a steady angle
- Anything else is a wobbling orbit between two heights

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 from the verticalθ (°)
Turns completedN
Angular velocity about the axisω (rad/s)
Speed of the bobv (m/s)
Radius of the circler (m)
Depth of the bob below the pivoth (m)
TensionT (N)
Centripetal forceF_c (N)
Measured periodT_p (s)
Frequencyf (Hz)
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.

Length of the stringL (m)
Mass of the bobm (kg)
Angle of the stringθ (°)
Angular speed about the axisω₀ (rad/s)
Gravitational field strengthg (m/s^2)
Turns to timeN

The equations behind it

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

Forces and momentum

$F_{net} = ma$
Newton’s second law
$p = m v$
Momentum
$F_{net} = \frac{Δ p}{Δ t}$
Force as a rate of change of momentum
$J = F Δ t = Δ p$
Impulse
$F_{f} \leq μ_{s} N$
Static friction — At most this, until it slips.
$F_{f} = μ_{d} N$
Dynamic friction

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.

Also in Rigid body mechanics

Experiments already set up on it

Period against the height of the cone

The headline: T² against h = L cos θ is straight, with gradient 4π²/g.

T = 2π√(h/g), where h is the depth of the bob below the pivot
Getting g from the gradient of T² against h
Timing ten turns and dividing

Does a longer string go round more slowly?

Keep the angle and lengthen the string: T² against L is still straight.

h = L cos θ, so at a fixed angle T² ∝ L
The same gradient, scaled by cos θ

Does the mass matter?

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

Mass cancels from T cos θ = mg and T sin θ = mω²r
A null result is a result

The wrong speed

Set it going too slowly and the bob rises and falls instead of circling.

Only one speed holds the string at a steady angle
Anything else is a wobbling orbit between two heights

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 from the verticalθ (°)

Turns completedN

Angular velocity about the axisω (rad/s)

Speed of the bobv (m/s)

Radius of the circler (m)

Depth of the bob below the pivoth (m)

TensionT (N)

Centripetal forceF_c (N)

Measured periodT_p (s)

Frequencyf (Hz)

Kinetic energyEₖ (J)

Mechanical energyE (J)

The equations behind it

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

Forces and momentum

$F_{net} = ma$
Newton’s second law
$p = m v$
Momentum
$F_{net} = \frac{Δ p}{Δ t}$
Force as a rate of change of momentum
$J = F Δ t = Δ p$
Impulse
$F_{f} \leq μ_{s} N$
Static friction — At most this, until it slips.
$F_{f} = μ_{d} N$
Dynamic friction

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.

Circular motion

Experiments already set up on it

Period against the height of the cone

Does a longer string go round more slowly?

Does the mass matter?

The wrong speed

What you can measure

What you can change

The equations behind it

Forces and momentum

Circular motion

Also in Rigid body mechanics

Inclined plane & friction

Collisions

Rotational dynamics

Moments & equilibrium

Circular motion

Experiments already set up on it

Period against the height of the cone

Does a longer string go round more slowly?

Does the mass matter?

The wrong speed

What you can measure

What you can change

The equations behind it

Forces and momentum

Circular motion

Also in Rigid body mechanics

Inclined plane & friction

Collisions

Rotational dynamics

Moments & equilibrium