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Physics 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.

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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.

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Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

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.

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Frozen valuesUsing frozen 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?

Make a prediction before you reveal the next step.

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 against the live bench.

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

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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?

Use the live bench to test the result before moving on.

Accessibility

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 this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

System-total conservationMechanics