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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.

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Stable links

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

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

Premium unlocks saved study tools, exact-state sharing, and the richer review surfaces that support this guided flow.

View plans
Frozen valuesFrozen at 0.00

For the current state, what is the horizontal projection at ?

Time

0 s

Radius

1.2 m

Angular speed

1.4 rad/s

Phase

0.3 rad

1. Identify the projection relation

Use the horizontal projection of circular motion: .

2. Substitute the current values

.

3. Compute the current projection

The current angle is , so the horizontal position is .

Current horizontal projection

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?

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

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

Accessibility

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.