HomeConcept libraryOscillations and WavesMotion and Circular Motion

OscillationsIntroStarter track

Concept module

Uniform Circular Motion

Track a particle moving at constant speed around a circle and connect radius, angular speed, tangential speed, centripetal acceleration, and the inward-force requirement to the same live state.

Interactive lab

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

Time

0.00 s / 8.00 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s8.00 s

Uniform circular motion

Drag the particle to set the phase on the orbit.

Graphs

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

x and y projections

Shows the horizontal and vertical projections as sinusoidal views of the same rotation.

time (s): 0 to 8displacement (m): -2.2 to 2.2

x(t)y(t)

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

Controls

Adjust the physical parameters and watch the motion respond.

Radius

R

1.2 m

Controls the size of the orbit.

Angular speed

ω

1.4 rad/s

Controls how quickly the particle sweeps around the circle.

Phase

ϕ

0.3 rad

Sets the starting angle on the orbit.

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 3

Notice that the x and y projections trace oscillations even though the particle itself moves in a circle.

Try this

Turn on the projection guides, then scrub the graph so the marker and the projected coordinates stay synchronized.

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

Radius

1.2 m

Sets the size of the orbit, so it changes the circle itself and the size of the x(t) and y(t) projections.

Graph: x and y projectionsGraph: Velocity componentsOverlay: Radius vectorOverlay: Projection guides

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

Use the orbit, vectors, and graphs as one system. The prompt changes when a different representation becomes the clearest thing to watch.

Graph readingPrompt 1 of 3

Graph: x and y projections

Notice that the x and y projections trace oscillations even though the particle itself moves in a circle.

Try this

Turn on the projection guides, then scrub the graph so the marker and the projected coordinates stay synchronized.

Why it matters

This is the direct visual bridge from circular motion to simple harmonic motion.

Graph: x and y projectionsOverlay: Projection guidesEquation

R

Equation

ω

Equation

ϕ

Guided overlays

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

4 visible

Overlay focus

Radius vector

Shows the line from the center to the particle.

What to notice

The centripetal acceleration always points back along this inward direction.

Why it matters

It anchors the geometric meaning of inward acceleration.

Control: RadiusControl: PhaseGraph: x and y projectionsGraph: Angle over timeEquation

R

Equation

ϕ

Challenge mode: centripetal force

Use this circular-motion lab for centripetal-force practice. These first tasks lock radius, speed, angular speed, period, and inward acceleration to the same live orbit before you branch into broader UCM targets.

0/4 solved

ConditionStretch

0 of 5 checks

Compare the spin

Open compare mode from the reference orbit and make Setup B complete turns noticeably faster than Setup A while both setups keep nearly the same radius.

Compare modeGraph-linkedGuided start2 hints

Suggested start

Start both setups from the same reference state, then edit Setup B only.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Keep the angle graph active.

x and y projections

Pending

Keep Setup A radius between

1.1

and

1.3 m

Pending

Keep Setup B radius between

1.1

and

1.3 m

Pending

Raise Setup B angular speed above

2.6 rad/s

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

At t = 0 s, the particle is at 0.3 rad with x = 1.15 m and y = 0.35 m. Its tangential speed stays 1.68 m/s while the centripetal acceleration of 2.35 m/s² points inward.

Equation detailsDeeper interpretation, notes, and worked variable context.

Angle in time

The angular position increases linearly when angular speed is constant.

θ (t) = ω t + ϕ

\omega sets how quickly the particle sweeps around the circle.

\phi chooses the starting angle.

ω

Angular speed 1.4 rad/s

ϕ

Phase 0.3 rad

Horizontal projection

The horizontal component of the motion is a cosine projection of the circle.

x (t) = R cos (θ) = R cos (ω t + ϕ)

R

Radius 1.2 m

ϕ

Phase 0.3 rad

Vertical projection

The vertical component is the matching sine projection of the same circular motion.

y (t) = R sin (θ) = R sin (ω t + ϕ)

R

Radius 1.2 m

ϕ

Phase 0.3 rad

Tangential speed

The speed stays constant for uniform circular motion and depends on both radius and angular speed.

v = ω R

R

Radius 1.2 m

ω

Angular speed 1.4 rad/s

Centripetal acceleration

The inward acceleration needed to keep the motion turning.

a_{c} = ω^{2} R = \frac{v ^{2}}{R}

R

Radius 1.2 m

ω

Angular speed 1.4 rad/s

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 4 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 3 of 30 / 3 complete

Motion and Circular Motion

Earlier steps still set up Uniform Circular Motion.

1. Vectors and Components2. Projectile Motion3. Uniform Circular Motion

Previous step: Projectile Motion.

Start from track beginning

Short explanation

What the system is doing

Uniform circular motion is the cleanest place to see why constant speed does not mean zero acceleration. The particle keeps turning, so its velocity changes direction even when the speed stays fixed.

Open Model Lab pairs the orbit, vectors, and time graphs so you can connect radius, angular speed, tangent velocity, and inward acceleration to the same live state instead of treating them as separate formulas.

Key ideas

01Velocity is always tangent to the path, not pointed toward the center.

02Centripetal acceleration points inward because the motion keeps turning, even when the speed stays constant.

03The x and y projections of circular motion behave like oscillations, which is why uniform circular motion connects naturally to simple harmonic motion.

Live circular-motion examples

Solve the exact state on screen.

Use the current orbit and time state directly. In live mode the substitution updates from the real controls and inspected time.

Live valuesLive at t = 0.00 s

For the current state, what is the horizontal projection at $t = 0 s$ ?

t

Time

0 s

R

Radius

1.2 m

ω

Angular speed

1.4 rad/s

ϕ

Phase

0.3 rad

1. Identify the projection relation

Use the horizontal projection of circular motion:

x (t) = R cos (ω t + ϕ)

2. Substitute the current values

x (0) = 1.2 cos (1.4 \cdot 0 + 0.3)

3. Compute the current projection

The current angle is

θ = 0.3

, so the horizontal position is

x = 1.15 m

Current horizontal projection

x (0) = 1.15 m

The positive x projection means the particle is on the right side of the orbit, which matches the positive part of the x(t) graph.

Vector checkpoint

At the very top of the circle, which direction must the velocity vector point, and which direction must the centripetal acceleration point?

Prediction prompt

Decide before you turn on the vectors whether either arrow can point straight up at that instant.

Check your reasoning

At the top of the circle, the velocity must be tangent to the orbit, so it points sideways. The centripetal acceleration still points inward, so it points straight toward the center.

Velocity follows the tangent, while centripetal acceleration follows the inward radius. They are perpendicular in uniform circular motion.

Common misconception

If the particle moves at constant speed, its acceleration must be zero.

Speed tells you how fast the particle moves, but velocity also includes direction.

In uniform circular motion the direction changes continuously, so there must be a nonzero inward acceleration even when the speed readout stays constant.

Quick test

Misconception check

Question 1 of 4

Use the orbit, vectors, and graph meaning together. These questions check whether you can explain what must be true in uniform circular motion.

A particle moves around a circle at constant speed. Which statement must still be true?

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 particle moving around a circular path centered on visible x and y axes. Optional overlays can show the radius vector, tangent velocity, inward acceleration, angular marker, and axis projections.

When the user changes radius, angular speed, or phase, the orbit, graphs, vectors, and readouts all update together.

Graph summary

The graphs show the x and y projections, the velocity components, and the angular position over time.

They are different representations of the same live circular motion rather than separate datasets.

Follow this motion next

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.

Orbit bridgeMechanics

Uniform Circular Motion

See each variable before you move it.

Radius vector

Compare the spin

Motion and Circular Motion

What the system is doing

Solve the exact state on screen.

For the current state, what is the horizontal projection at $t = 0 s$ ?

A particle moves around a circle at constant speed. Which statement must still be true?

Keep moving through the concept map.

Circular Orbits and Orbital Speed

Damping / Resonance

Projectile Motion

Uniform Circular Motion

See each variable before you move it.

Radius vector

Compare the spin

Motion and Circular Motion

What the system is doing

Solve the exact state on screen.

For the current state, what is the horizontal projection at t=0s?

A particle moves around a circle at constant speed. Which statement must still be true?

Keep moving through the concept map.

Circular Orbits and Orbital Speed

Damping / Resonance

Projectile Motion

For the current state, what is the horizontal projection at $t = 0 s$ ?