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MechanicsIntermediateStarter track

Concept module

Angular Momentum

Treat angular momentum as rotational momentum on one compact rotor where mass radius and spin rate stay tied to the same readouts, response maps, and same-L conservation story.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 2.40 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s2.40 s

As the same mass moves outward, the moment of inertia rises and the same angular momentum would require less angular speed.

Graphs

Switch graph views without breaking the live stage and time link.

Rotation angle vs time

For a fixed setup on this bench, angle grows linearly because the rotor keeps one constant angular speed through the clip.

time (s): 0 to 2.4rotation angle (rad): 0 to 8

Hover or scrub to link the graph back to the stage.time (s) / rotation angle (rad)

Controls

Adjust the physical parameters and watch the motion respond.

Mass radius

r

0.55 m

Slide the same six equal masses inward or outward while keeping total mass fixed.

Angular speed

ω

2.4 rad/s

Set the current spin rate so you can see how angular momentum depends on both inertia and angular speed.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 2

Notice that rim speed is not identical to angular momentum. It changes with both radius and angular speed, but it does not include the full inertia story by itself.

Try this

Compare the compact same-L and wide same-L presets with the tangential-speed arrow visible. The faster spinner does not automatically make every other rotational quantity larger.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

r

Mass radius

0.55 m

Moving the same masses outward raises the moment of inertia sharply, so the same angular momentum can live behind a slower spin.

Graph: Angular momentum vs mass radiusGraph: Same-L angular speed vs mass radiusGraph: Rotation angle vs timeOverlay: Radius guideOverlay: Same-L compact reference

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompt to keep the conservation story honest. The best prompt should help you separate same-speed changes from same-angular-momentum changes.

ObservationPrompt 1 of 2

Graph: Rotation angle vs time

Notice that rim speed is not identical to angular momentum. It changes with both radius and angular speed, but it does not include the full inertia story by itself.

Try this

Compare the compact same-L and wide same-L presets with the tangential-speed arrow visible. The faster spinner does not automatically make every other rotational quantity larger.

Why it matters

This keeps straight-line speed cues from replacing the actual

L = I ω

idea.

Control: Mass radiusControl: Angular speedGraph: Rotation angle vs timeOverlay: Tangential-speed arrowEquation

r

Equation

ω

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Radius guide

Shows the current distance of the equal masses from the axis.

What to notice

The total mass is unchanged while the radius changes.
That changing radius is what makes the same-L compact and wide cases spin so differently.

Why it matters

The radius cue makes the inertia change visible instead of leaving it as an abstract variable.

Control: Mass radiusGraph: Angular momentum vs mass radiusGraph: Same-L angular speed vs mass radiusEquation

r

Challenge mode

Use the same rotor to prove that you can reason about angular momentum as a product story instead of treating conservation like a slogan.

0/3 solved

ConditionStretch

2 of 9 checks

Same L, different spin

Open compare mode and make Setup A compact and Setup B wide while keeping their angular momenta nearly matched.

Compare modeGraph-linkedGuided start2 hints

Suggested start

Clone the compact same-L state, then widen only Setup B until the angular-momentum badges still match but the spin-rate badges separate.

Matched

Open the Rotation angle vs time graph.

Rotation angle vs time

Matched

Keep the Same-L compact reference visible.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Keep Setup A compact between

0.26

and

0.3 m

Pending

Keep Setup A angular momentum between

5.3

and

5.6 kg m^{2} /s

Pending

Move Setup B outward between

0.9

and

1.0 m

Pending

Keep Setup B angular momentum between

5.3

and

5.6 kg m^{2} /s

Pending

Keep Setup A spinning between

5.7

and

6.1 rad/s

Pending

Bring Setup B between

0.85

and

1.05 rad/s

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the six equal masses sit at 0.55 m from the axis and rotate at 2.4 rad/s. That gives a moment of inertia of 2.27 kg m^2, an angular momentum of 5.44 kg m^2/s, a rim speed of 1.32 m/s, and a rotation angle of 0 rad. If that same angular momentum were packed to the compact reference radius, the spin would be 5.91 rad/s. At this mid-radius layout, the angular momentum sits between the compact and wide-layout extremes.

Equation detailsDeeper interpretation, notes, and worked variable context.

Rotational analogue of momentum

Linear momentum pairs mass with straight-line speed. Angular momentum pairs moment of inertia with angular speed.

p = m v ⟶ L = I ω

ω

Angular speed 2.4 rad/s

Equal-mass rotor model

On this bench the same moving mass stays on one ring, so sliding the masses outward changes angular momentum through the moment of inertia.

I = I_{hub} + M r^{2}

r

Mass radius 0.55 m

Angular momentum of this rotor

Angular momentum grows with either larger rotational inertia or larger angular speed.

L = I ω

r

Mass radius 0.55 m

ω

Angular speed 2.4 rad/s

Conserved-spin relation

If angular momentum stays fixed while the moment of inertia changes, the angular speed must change inversely.

ω_{same L} = \frac{L}{I}

r

Mass radius 0.55 m

ω

Angular speed 2.4 rad/s

Rim-speed link

Tangential speed helps you see that changing radius and spin rate also changes how fast the masses move through space.

v_{rim} = r ω

r

Mass radius 0.55 m

ω

Angular speed 2.4 rad/s

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 3 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 5 of 50 / 5 complete

Rotational Mechanics

Earlier steps still set up Angular Momentum.

1. Torque2. Static Equilibrium / Centre of Mass3. Rotational Inertia / Moment of Inertia4. Rolling Motion+1 more steps

Previous step: Rolling Motion.

Start from track beginning

Short explanation

What the system is doing

Angular momentum is the rotational analogue of linear momentum. Linear momentum keeps track of how hard it is to change straight-line motion with $p = m v$ ; angular momentum keeps track of rotational motion with $L = I ω$ .

This bench stays bounded on purpose. The same six equal masses rotate about one axis while you change only the mass radius and the angular speed. That makes the two ingredients of angular momentum visible without drifting into a giant rigid-body or orbital-mechanics system.

Key ideas

01Angular momentum combines rotational inertia and spin rate through $L = I\omega$. A rotor can have large angular momentum because it spins quickly, because its mass sits far from the axis, or because both happen together.

02Moment of inertia is the rotational analogue of mass in this setting. Moving the same mass outward raises $I$, so the same angular momentum can show up with a smaller angular speed.

03Conserving angular momentum explains dramatic spin changes. If external torque is negligible and $I$ drops, $\omega$ must rise so that $L$ stays the same.

Live angular-momentum checks

Solve the exact state on screen.

Use the live rotor you are actually looking at. The first example computes the current angular momentum directly, and the second asks what the same angular momentum would demand from a compact reference layout.

Live valuesFollowing current parameters

For the current setup, what moment of inertia and angular momentum does the rotor have?

r

Mass radius

0.55 m

ω

Angular speed

2.4 rad/s

M

Moving mass

6 kg

1. Build the current inertia

For this bounded rotor use

I = I_{hub} + M r^{2}

, where

I_{hub} = 0.45 kg m^{2}

and the moving mass is

M = 6 kg

2. Insert the live radius

With

r = 0.55 m

, the ring contribution is

M r^{2} = 1.82 kg m^{2}

, so the total moment of inertia is

I = 2.27 kg m^{2}

3. Multiply by the live spin rate

Then

L = I ω = 2.27 \times 2.4

, so the angular momentum is

5.44 kg m^{2} /s

Current moment of inertia and angular momentum

I = 2.27 kg m^{2}, L = 5.44 kg m^{2} /s

This mid-radius layout carries angular momentum through a balanced mix of spin rate and mass distribution.

Conservation checkpoint

Can a rotor spin more slowly after its mass moves outward and still keep the same angular momentum?

Prediction prompt

Try matching the same-L compact and wide presets before you answer. Focus on whether

I

and

ω

can trade against each other.

Check your reasoning

Yes. If the moment of inertia increases when the mass moves outward, the angular speed can decrease while the product

I ω

stays nearly the same.

Conservation of angular momentum is a product story, not a single-variable story. Wider layouts can carry the same

L

with less spin because their moment of inertia is larger.

Common misconception

A faster-spinning object always has more angular momentum than a slower one.

Angular momentum depends on both $I$ and $ω$ . A wide, slow rotor can carry the same angular momentum as a compact, fast rotor.

If you hold angular speed fixed while moving mass outward, the angular momentum increases because the moment of inertia increased. That is not conservation; it means some external torque or work must have changed the state.

Quick test

Compare cases

Question 1 of 4

These checks ask whether you can reason with angular momentum as a rotational momentum, not just repeat the phrase conservation.

Two rotors carry the same angular momentum. Rotor A keeps the same mass close to the axis, while Rotor B keeps it far from the axis. Which statement is correct?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a face-on rotor with six equal masses attached to spokes around a central hub. The masses can move inward or outward together, and the rotor spins at an adjustable angular speed while a curved arrow near the hub marks the rotational direction.

Optional overlays can show the current radius, a tangential-speed arrow on one mass, equal-mass labels, and a compact reference ring that reports the angular speed needed to carry the same angular momentum there. The linked readout and graphs report moment of inertia, angular momentum, rim speed, and the same-L spin response on the same bench.

Graph summary

The rotation-angle graph is linear in time because each fixed setup keeps one constant angular speed through the clip. Faster setups make steeper lines, while slower wide-layout same-L setups sweep out less angle over the same time.

The angular-momentum map rises with radius when angular speed is held fixed because the moment of inertia increases. The same-L spin map falls with radius because keeping angular momentum fixed requires lower angular speed at larger radius.

Carry rotational momentum forward

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

System-total conservationMechanics

Angular Momentum

See each variable before you move it.

Radius guide

Same L, different spin

Rotational Mechanics

What the system is doing

Solve the exact state on screen.

For the current setup, what moment of inertia and angular momentum does the rotor have?

Two rotors carry the same angular momentum. Rotor A keeps the same mass close to the axis, while Rotor B keeps it far from the axis. Which statement is correct?

Keep moving through the concept map.

Conservation of Momentum

Momentum and Impulse

Uniform Circular Motion