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Physics MechanicsIntermediateStarter track

Concept module

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.

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Starter track

Step 4 of 50 / 5 complete

Rotational Mechanics

Earlier steps still set up Rolling Motion.

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

Previous step: Rotational Inertia / Moment of Inertia.

Start from track beginning

Why it behaves this way

Explanation

Rolling without slipping ties translational motion and rotational motion to the same live state. The center of mass speeds up down the incline while the object also spins fast enough to keep the contact point from sliding.

This bench stays bounded on purpose. One rigid roller moves on one incline with a no-slip constraint, so the key comparison stays honest: the same gravity component acts down the slope, while shape and mass distribution decide how strongly that motion must also build rotation.

Key ideas

01Rolling without slipping means the center-of-mass speed and angular speed stay locked by $v_{\mathrm{cm}} = r\omega$.

02Different shapes roll differently because their rotational inertia changes how the same downhill pull is shared between translation and rotation. A convenient summary is $k = I/(mr^2)$.

03For one incline angle, the center-of-mass acceleration becomes $a_{\mathrm{cm}} = g\sin\theta/(1 + k)$. Smaller inertia factor means faster rolling.

04Changing radius at the same shape barely changes the center-of-mass acceleration in this bounded model, but it does change the required angular speed because $\omega = v/r$.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the live incline, shape preset, and inspected time to solve the rolling state you are actually looking at.

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Frozen valuesUsing frozen parameters

For the current setup, what acceleration should this solid cylinder have, and how long should it take to travel the full $2.4 m$ ramp?

θ

Slope angle

12 °

k

Inertia factor

0.5

1. Start from the rolling-without-slipping acceleration rule

For this bounded incline use

a_{cm} = \frac{g sin θ}{1 + k}

, with

g = 9.81 m/ s^{2}

and

k = 0.5

2. Insert the live slope and shape response

With

θ = 1 2^{\circ}

, the center-of-mass acceleration becomes

a_{cm} = 1.36 m/ s^{2}

for the current shape or custom mass distribution.

3. Turn acceleration into a ramp time

The ramp length is

L = 2.4 m

, so from rest use

L = \frac{1}{2} a t^{2}

. That gives a travel time of

1.88 s

Current acceleration and ramp time

a_{cm} = 1.36 m/ s^{2}, t_{bottom} = 1.88 s

The solid cylinder sits between the sphere and hoop cases, so its acceleration and travel time stay in the middle as well.

No-slip checkpoint

Can you make one roller reach the bottom sooner without touching the slope angle?

Make a prediction before you reveal the next step.

Try keeping the incline fixed, then move from a rim-heavy shape toward a center-loaded shape and predict what happens before you touch the controls.

Check your reasoning against the live bench.

Yes. Lowering the inertia factor lets more of the same downhill pull accelerate the center of mass, so the travel time shrinks even though the no-slip link still has to be satisfied.

Rolling races are really about how translation and rotation share the same gravity-driven budget. Smaller rotational inertia leaves less of that budget trapped in spin-up.

Common misconception

A heavier-looking or larger-looking rolling object must always win the race down the incline.

In this bounded no-slip model the key race variable is not the total mass. It is how the mass is distributed relative to the axis, summarized here by the inertia factor $k$ .

Radius matters for angular speed and rotational bookkeeping, but for the same shape on the same incline it does not by itself change the center-of-mass acceleration formula.

Quick test

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Question 1 of 4

These checks ask whether you can reason through rolling motion, not just recite which shape wins.

A solid sphere and a hoop start from rest on the same incline with the same radius. Which statement is correct?

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

Accessibility

The simulation shows one roller descending a single incline from left to right. The wheel interior changes to represent a solid sphere, solid cylinder, hoop, or custom mass distribution, while the center point and spoke lines rotate with the motion.

Optional overlays can show the no-slip relation between center speed and angular speed, a cue for the mass layout and inertia factor, an energy-split bar, and the uphill static-friction force together with the rolling torque. The readout card reports the same live time, slope, acceleration, travel time, and spin state.

Graph summary

The distance curve bends upward because the center of mass is accelerating down the incline. The no-slip speed-link graph keeps the center speed and $r ω$ on top of each other whenever rolling without slipping is active.

The energy graph shows translational, rotational, and total kinetic energy together. The acceleration map falls as the inertia factor rises, which is why a hoop rolls more slowly than a sphere on the same incline.

Carry rolling ideas 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.

Carry the spin language forwardMechanics

Rolling Motion

Rotational Mechanics

Explanation

Step through the frozen example

For the current setup, what acceleration should this solid cylinder have, and how long should it take to travel the full $2.4 m$ ramp?

A solid sphere and a hoop start from rest on the same incline with the same radius. Which statement is correct?

Keep this idea moving

Angular Momentum

Uniform Circular Motion

Momentum and Impulse

Rolling Motion

Rotational Mechanics

Explanation

Step through the frozen example

For the current setup, what acceleration should this solid cylinder have, and how long should it take to travel the full 2.4m ramp?

A solid sphere and a hoop start from rest on the same incline with the same radius. Which statement is correct?

Keep this idea moving

Angular Momentum

Uniform Circular Motion

Momentum and Impulse

For the current setup, what acceleration should this solid cylinder have, and how long should it take to travel the full $2.4 m$ ramp?