Physics · Mechanics

Rotational Inertia / Moment of Inertia

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Angular MomentumCarry the same inertia picture into rotational motion that keeps going.

Key takeaway

The same total mass can spin up very differently when that mass is distributed at different radii.
Moving equal masses outward raises $I$ strongly because each contribution is weighted by $m r^{2}$ .
Torque changes the response through $α = τ / I$ , but the rotor layout is what sets its moment of inertia.

Common misconception

If two rotors have the same total mass and the same applied torque, they must respond the same way.

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

Carry rotational inertia forward

Keep this idea moving

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Rolling response bridgeMechanics

Rolling Motion

Roll a sphere, cylinder, hoop, or custom mass distribution down one incline and see how rolling without slipping ties translation, rotation, and rotational inertia to the same honest run.

Rotational momentum bridgeMechanics

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.

Linear inertia contrastMechanics

Momentum and Impulse

Push one cart with a timed force pulse and watch momentum, impulse, and force-time area stay tied to the same motion, readouts, and graphs.

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Reference and support

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Equation snapshotMoment of inertia from mass placement · Rotational responseShow 3 equations

Track how the layout changes $I$ first, then read the spin-up that follows from $α = τ / I$ .

Moment of inertia from mass placement
$I = \sum m_{i} r_{i}^{2}$
Each piece of mass counts more when it sits farther from the axis because the weighting grows like $r^{2}$ .
Rotational response
$α = \frac{τ}{I}$
For the same torque, a larger moment of inertia gives a smaller angular acceleration.
Bounded constant-torque spin-up
$ω = α t, θ = \frac{1}{2} α t^{2}$
For each fixed setup, the rotor starts from rest under one constant torque, so angular speed grows linearly while rotation angle curves upward.

Why it behaves this way

Explanation

Rotational inertia, or moment of inertia, measures how hard it is to change an object's spin about an axis. Two rotors can have the same total mass and still respond very differently if one keeps more of that mass far from the axis.

This bench keeps the comparison clean. The same six equal masses and the same motor-style torque stay in play while you slide the masses inward or outward. That means the main thing changing is mass distribution, so you can see directly how layout changes spin-up.

Key ideas

01Moment of inertia depends on where the mass is, not just how much mass there is. Mass farther from the axis counts much more strongly because each contribution is weighted by

r^{2}

02For the same applied torque, angular acceleration follows

α = τ / I

. Larger moment of inertia means slower spin-up, a shallower angular-speed graph, and less rotation at the same time.

03Rotational inertia is the rotational analogue of mass in linear motion. The same total mass can behave like an easy-to-spin compact rotor or a hard-to-spin spread-out rotor.

Worked examples

Solve the live rotor

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the current mass radius and torque as evidence. First calculate the moment of inertia from the live layout, then use that same setup to predict angular acceleration, angular speed, and rotation angle.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current layout, what moment of inertia does this equal-mass rotor have, and what angular acceleration should the current torque produce?

r

Mass radius

0.35 m

τ

Applied torque

4 N m

M

Total moving mass

6 kg

1. Write 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. Use $\tau/I$ to find the spin-up

Then

α = τ / I = 4/1.18

, so the angular acceleration is

3.38 rad/ s^{2}

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.

Quick test

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Accessibility

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The simulation shows a rotor viewed from the front, 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, mark 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 for the same live layout.

Graph summary

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

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

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Step 3 of 5

Rotational Mechanics

Rotational Inertia / Moment of Inertia appears later in this track, so it is cleaner to start from the beginning first.

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