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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.

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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.

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Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough
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.

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Frozen valuesUsing frozen parameters

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

Mass radius

0.35 m

Applied torque

4 N m

Moving mass

6 kg

1. Start from the inertia model

For this bounded rotor use , where and the moving mass is .

2. Insert the current radius

With , the ring contribution is , so the total moment of inertia is .

3. Compute the spin-up response

Then , so the angular acceleration is .

Current moment of inertia and angular acceleration

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?

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

Not with the same torque. Pushing the same mass outward raises , so keeping the same 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 .

Quick test

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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?

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

Accessibility

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.