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Concept module

Rotational Inertia / Moment of Inertia

Keep the same total mass and torque, then slide equal masses inward or outward to see why moment of inertia makes some rotors much harder to spin up than others.

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

Rotational Inertia / Moment of Inertia

Keep the same total mass and the same motor torque, then slide the equal masses inward or outward to see how rotational inertia alone reshapes the spin-up.

Graphs

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

Angular speed vs time

For each fixed layout, angular speed rises linearly because the same torque keeps producing one constant angular acceleration.

time (s): 0 to 2.4angular speed (rad/s): 0 to 16

omega

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

Controls

Adjust the physical parameters and watch the motion respond.

Applied torque

τ

4 N m

Set the motor twist without changing the total mass or the mass layout.

Mass radius

r

0.35 m

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

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.

Graph readingPrompt 1 of 2

Notice on the inertia map that spreading the masses outward does not raise

I

linearly. The

r^{2}

weighting makes the resistance grow faster than the radius itself.

Try this

Hover or scrub the inertia map from

r = 0.3 m

toward

r = 1.0 m

and compare how quickly the curve climbs.

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

0.35 m

Sliding the same masses outward raises the rotational inertia sharply because each contribution is weighted by $r^2$.

Graph: Moment of inertia vs mass radiusGraph: Angular acceleration vs mass radiusGraph: Angular speed vs timeGraph: Rotation angle vs timeOverlay: Radius guideOverlay: 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.

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What to notice

Keep the bench focused on one idea at a time: the same total mass, the same torque, and a changing distance from the axis.

Graph readingPrompt 1 of 2

Notice on the inertia map that spreading the masses outward does not raise

I

linearly. The

r^{2}

weighting makes the resistance grow faster than the radius itself.

Try this

Hover or scrub the inertia map from

r = 0.3 m

toward

r = 1.0 m

and compare how quickly the curve climbs.

Why it matters

The square dependence is why small outward shifts can matter a lot for rotational response.

Control: Mass radiusGraph: Moment of inertia vs mass radiusOverlay: Radius guideEquation

r

Guided overlays

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

2 visible

Overlay focus

Radius guide

Shows how far the equal masses sit from the axis right now.

What to notice

The same masses can move a long way without changing total mass.
That distance from the axis is the key variable moment of inertia cares about.

Why it matters

Mass distribution is not abstract here. The radius cue makes the changing geometry visible.

Control: Mass radiusGraph: Moment of inertia vs mass radiusGraph: Angular acceleration vs mass radiusEquation

r

Challenge mode

Use the same live rotor to prove that you can reason about mass distribution and rotational response instead of treating moment of inertia like a memorized label.

0/3 solved

TargetCore

2 of 5 checks

Compact and quick

Keep the motor near the baseline torque, then make the rotor spin up sharply by changing only the mass layout.

Graph-linkedGuided start2 hints

Suggested start

Use the spin-up map to keep the angular-acceleration target tied to the same live layout.

Pending

Open the Angular acceleration vs mass radius graph.

Angular speed vs time

Matched

Keep the Radius guide visible.

Matched

Keep the torque between

3.9

and

4.1 N m

4 N m

Pending

Bring the mass radius between

0.22

and

0.27 m

0.35 m

Pending

Make angular acceleration land between

4.2

and

5.4 rad/ s^{2}

3.38 rad/s^2

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

At t = 0 s, the motor applies 4 N m while the six equal masses sit at 0.35 m from the axis. That gives a moment of inertia of 1.18 kg m^2, so the angular acceleration is 3.38 rad/s^2, the angular speed is 0 rad/s, and the rotation angle is 0 rad. Most of the mass stays close to the axis, so the same torque spins the rotor up quickly.

Equation detailsDeeper interpretation, notes, and worked variable context.

Moment of inertia from mass placement

Each piece of mass contributes more strongly when it sits farther from the axis because the weighting grows like $r^2$.

I = \sum m_{i} r_{i}^{2}

r

Mass radius 0.35 m

Equal-mass rotor model

On this bench the same moving mass stays on one ring, so sliding that ring outward raises the rotational inertia quickly.

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

r

Mass radius 0.35 m

Rotational response

For the same torque, a larger moment of inertia means a smaller angular acceleration.

α = \frac{τ}{I}

r

Mass radius 0.35 m

τ

Applied torque 4 N m

Bounded constant-torque spin-up

Starting from rest, the rotor's angular speed grows linearly while the angle curves upward for each fixed setup.

ω = α t, θ = \frac{1}{2} α t^{2}

τ

Applied torque 4 N m

Progress

Not startedMastery: NewLocal-first

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Let the live model runChange one real controlOpen What to notice

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

Stable links

Starter track

Step 3 of 50 / 5 complete

Rotational Mechanics

Earlier steps still set up Rotational Inertia / Moment of Inertia.

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

Previous step: Static Equilibrium / Centre of Mass.

Start from track beginning

Short explanation

What the system is doing

Rotational inertia, or moment of inertia, tells you how stubbornly a rotating object is about changing its spin. The same total mass can feel easy or hard to spin up depending on how far that mass sits from the axis.

This bench keeps the same six equal masses and the same motor-style torque while you slide the masses inward or outward. That keeps the comparison honest: the big change is mass distribution, not the amount of mass or the cause of the rotation.

Key ideas

01Moment of inertia depends on where the mass is, not just how much mass there is. Each piece contributes through $mr^2$, so mass farther from the axis counts much more strongly.

02For the same applied torque, angular acceleration follows $\alpha = \tau/I$. Larger moment of inertia means slower spin-up.

03Rotational inertia is the rotational analogue of mass in linear motion. Later it becomes the bridge to rolling motion and angular momentum.

Live inertia checks

Solve the exact state on screen.

Solve the inertia response you are actually watching. The first example reads the current mass layout, and the second follows that same setup through the current inspected time.

Live valuesFollowing current parameters

For the current setup, what moment of inertia does this equal-mass rotor have and what angular acceleration follows from the current torque?

r

Mass radius

0.35 m

τ

Applied torque

4 N m

M

Moving mass

6 kg

1. Start from the inertia model

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 current radius

With

r = 0.35 m

, the ring contribution is

M r^{2} = 0.73 kg m^{2}

, so the total moment of inertia is

I = 1.18 kg m^{2}

3. Compute the spin-up response

Then

α = τ / I = 4/1.18

, so the angular acceleration is

3.38 rad/ s^{2}

Current moment of inertia and angular acceleration

I = 1.18 kg m^{2}, α = 3.38 rad/ s^{2}

Most of the same mass stays close to the axis here, so the rotor keeps a relatively small moment of inertia and responds quickly to the torque.

Mass-distribution checkpoint

Can you double the mass radius without changing total mass and still keep the same angular acceleration?

Prediction prompt

Try pushing the masses outward while keeping the motor torque fixed, then predict whether the same spin-up can survive.

Check your reasoning

Not with the same torque. Pushing the same mass outward raises

I

, so keeping the same

α

would require a larger torque.

Mass distribution is the heart of moment of inertia. If the same mass sits farther from the axis, the same motor twist now has to change a more resistant layout.

Common misconception

Two objects with the same mass must respond the same way to the same torque.

Equal mass does not guarantee equal rotational response. A spread-out object can have a much larger moment of inertia than a compact one.

What matters is both the amount of mass and how far that mass sits from the axis, because the resistance to spin-up is weighted by $r^{2}$ .

Quick test

Compare cases

Question 1 of 3

These checks ask whether you can reason with mass distribution and rotational response, not just repeat the phrase moment of inertia.

Two rotors have the same total mass and the same applied torque. Rotor A keeps the masses near the axis, while Rotor B keeps them near the rim. 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 rotor seen face-on, with six equal masses attached to spokes around a central hub. A curved torque arrow near the hub indicates a steady motor-style twist, and the masses can slide inward or outward while the total mass stays fixed.

Optional overlays can show the current mass radius, label the equal masses, and add a ghost compact reference layout. The linked readout and graphs report moment of inertia, angular acceleration, angular speed, and rotation angle on the same bounded bench.

Graph summary

The angular-speed graph is linear and the rotation-angle graph is curved because each fixed layout experiences one constant angular acceleration under the current torque. Compact layouts climb faster, while spread-out layouts rise more slowly.

The moment-of-inertia map rises strongly with mass radius, while the angular-acceleration map falls for the same torque. Together they show that spreading the same mass outward increases rotational resistance and slows spin-up.

Carry rotational ideas 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.

Rolling response bridgeMechanics

Rotational Inertia / Moment of Inertia

See each variable before you move it.

Radius guide

Compact and quick

Rotational Mechanics

What the system is doing

Solve the exact state on screen.

For the current setup, what moment of inertia does this equal-mass rotor have and what angular acceleration follows from the current torque?

Two rotors have the same total mass and the same applied torque. Rotor A keeps the masses near the axis, while Rotor B keeps them near the rim. Which statement is correct?

Keep moving through the concept map.

Rolling Motion

Angular Momentum

Momentum and Impulse