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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.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 1.88 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s1.88 s

Rolling Motion

One bounded incline keeps rolling without slipping honest: gravity pulls the same way for every setup, while the inertia factor decides how strongly the object translates, spins, and splits its energy.

Graphs

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

Distance down ramp vs time

Distance curves upward because the center speeds up as the object rolls down the same incline.

time (s): 0 to 1.88distance down ramp (m): 0 to 4

Hover or scrub to link the graph back to the stage.time (s) / distance down ramp (m)

Controls

Adjust the physical parameters and watch the motion respond.

Slope angle

θ

12°

Change the downhill gravity component without changing the same no-slip rolling model.

Radius

r

0.22 m

Change the roller size. In this bounded model it mostly changes the angular-speed requirement, not the center-of-mass acceleration for a fixed shape.

Mass distribution

k

0.5

Slide from a center-loaded sphere-like response toward a rim-loaded hoop-like response, or land exactly on the named shape presets.

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 speed-link graph that the two curves sit exactly on top of each other. The center speed and the rotational speed are not separate stories here.

Try this

Turn on the no-slip overlay, then shrink the radius and watch how the same center speed now demands a larger angular speed.

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.

θ

Slope angle

12°

Steeper slope angle increases the downhill component of gravity for every shape on the same bench.

Graph: Distance down ramp vs timeGraph: No-slip speed linkGraph: Acceleration vs inertia factorOverlay: Energy split barOverlay: Static-friction torque

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.

Hide

What to notice

Keep the bench honest: one incline, one no-slip rule, and one shape-dependent inertia factor.

Graph readingPrompt 1 of 2

Notice on the speed-link graph that the two curves sit exactly on top of each other. The center speed and the rotational speed are not separate stories here.

Try this

Turn on the no-slip overlay, then shrink the radius and watch how the same center speed now demands a larger angular speed.

Why it matters

Rolling without slipping is the bridge that keeps the translational and rotational representations conceptually honest.

Control: RadiusGraph: No-slip speed linkOverlay: No-slip linkEquation

r

Guided overlays

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

3 visible

Overlay focus

No-slip link

Shows the live relation $v_{\mathrm{cm}} = r\omega$ right next to the rolling object.

What to notice

The center speed and angular speed are not independent once the object rolls without slipping.
Smaller radius demands larger angular speed for the same center speed.

Why it matters

This keeps the translational and rotational views tied to the same honest state instead of treating spin as a decorative add-on.

Control: RadiusGraph: No-slip speed linkEquation

r

Challenge mode

Use one compact incline to prove that you can reason about rolling response instead of memorizing which shape wins.

0/3 solved

TargetCore

2 of 5 checks

Fast run, same ramp

Keep the ramp near its baseline angle and tune the roller so it reaches the bottom in under about

1.85 s

Graph-linkedGuided start2 hints

Suggested start

Use the acceleration map to stay on the same incline while you move toward a more center-loaded shape.

Pending

Open the Acceleration vs inertia factor graph.

Distance down ramp vs time

Matched

Keep the Mass layout cue visible.

Matched

Keep the slope angle between

11. 5^{\circ}

and

12. 5^{\circ}

12°

Pending

Bring the inertia factor between

0.4

and

0.43

0.5

Pending

Make the full-ramp travel time land between

1.78

and

1.86 s

1.88 s

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

At t = 0 s, the solid cylinder rolls down a 12° incline with radius 0.22 m. Rolling without slipping, its center-of-mass acceleration is 1.36 m/s², the current speed is 0 m/s, the angular speed is 0 rad/s, and the rotation angle is 0 rad. The kinetic energy split is 0 J translational plus 0 J rotational, with static friction 0.82 N providing the rolling torque. The solid cylinder sits in the middle: some of the downhill pull still builds rotation, but less than the hoop needs.

Equation detailsDeeper interpretation, notes, and worked variable context.

Shape through inertia factor

The bounded bench summarizes each shape or custom mass distribution by one dimensionless factor $k = I/(mr^2)$.

I = k m r^{2}

k

Inertia factor 0.5

r

Radius 0.22 m

Rolling acceleration on the incline

The same slope angle drives every setup, but a larger inertia factor leaves less of that downhill pull available for center-of-mass acceleration.

a_{cm} = \frac{g sin θ}{1 + k}

θ

Slope angle 12°

k

Inertia factor 0.5

Rolling without slipping

Translation and rotation are locked together. The contact point does not slide, so the linear and angular kinematics must stay consistent.

v_{cm} = r ω, a_{cm} = r α

r

Radius 0.22 m

Energy split in rolling

The loss of gravitational potential energy becomes both translational and rotational kinetic energy.

m g h = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}

θ

Slope angle 12°

k

Inertia factor 0.5

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 3 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

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

Stable links

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

Short explanation

What the system is doing

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 $\omega = v/r$.

Live rolling checks

Solve the exact state on screen.

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

Live valuesFollowing current 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?

Prediction prompt

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

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

Compare cases

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?

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 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 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.

Carry the spin language forwardMechanics

Rolling Motion

See each variable before you move it.

No-slip link

Fast run, same ramp

Rotational Mechanics

What the system is doing

Solve the exact state on screen.

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 moving through the concept map.

Angular Momentum

Uniform Circular Motion

Momentum and Impulse

Rolling Motion

See each variable before you move it.

No-slip link

Fast run, same ramp

Rotational Mechanics

What the system is doing

Solve the exact state on screen.

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 moving through the concept map.

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?