Physics · Mechanics

Uniform Circular Motion

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Circular Orbits / Orbital SpeedLet gravity supply the inward turn

Key takeaway

Constant speed can still mean acceleration when the velocity direction keeps changing.
Velocity points tangent to the circle, while centripetal acceleration points inward along the radius.
Angular speed controls the period, tangential speed, and inward acceleration together with radius.
The x(t) and y(t) graphs are projections of the same circular motion, not separate motions.

Common misconception

Do not aim the velocity arrow toward the center. The inward arrow is acceleration; the velocity arrow stays tangent to the path.

Speed tells you only how fast the particle moves. Velocity includes direction as well as size.

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Orbit bridgeGravity and Orbits

Circular Orbits and Orbital Speed

See why a circular orbit needs the right sideways speed, how gravity supplies the centripetal acceleration, and how source mass and radius together set orbital speed and period on one bounded live model.

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Damping / Resonance

Explore how damping removes energy, how driving frequency changes amplitude, and why resonance becomes dramatic near the natural frequency.

Vector comparisonMechanics

Projectile Motion

Launch a projectile, watch the trajectory form, and connect the range, height, and component motion to the launch settings.

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Reference and support

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Equation snapshotOne turning state · Centripetal accelerationShow 3 equations

Keep the speed, inward-acceleration, and projection equations together so each formula points back to the same particle.

One turning state
$v = ω R$
The speed stays constant for uniform circular motion and depends on both radius and angular speed.
Centripetal acceleration
$a_{c} = ω^{2} R = \frac{v ^{2}}{R}$
The inward acceleration needed to keep the motion turning.
Horizontal projection
$x (t) = R cos (θ) = R cos (ω t + ϕ)$
The horizontal component of the motion is a cosine projection of the circle.

Why it behaves this way

Explanation

Uniform circular motion is the clearest example of why constant speed does not mean zero acceleration. The speed stays the same, but the velocity keeps changing because its direction keeps turning.

This lab keeps the orbit, vectors, and time graphs tied to one live state. Change the radius, angular speed, or phase, then compare the circle, the tangent velocity, the inward acceleration, and the x(t) and y(t) projections. You are seeing the same motion in several representations, not learning separate formulas.

Key ideas

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

02Centripetal acceleration points inward because the velocity direction is changing continuously, even when the speed stays constant.

03The x and y coordinates of uniform circular motion behave like sine and cosine oscillations, which is why this motion connects directly to simple harmonic motion.

Worked examples

Solve the live motion

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the current orbit and inspected time, not a separate diagram. First read the x-coordinate from the same radius, angular speed, and phase that the stage is using. Then calculate the inward acceleration from that same circular motion.

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

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Example 1 of 2

Frozen valuesFrozen at 0.00

At $t = 0 s$ , what is the particle's x-coordinate?

t

Time

0 s

R

Radius

1.2 m

ω

Angular speed

1.4 rad/s

ϕ

Phase

0.3 rad

1. Write the x-projection formula

Use the horizontal projection of circular motion:

x (t) = R cos (ω t + ϕ)

2. Substitute the live values

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

3. Evaluate the current x-coordinate

The current angle is

θ = 0.3

, so the horizontal position is

x = 1.15 m

Current x-coordinate

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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a particle moving around a circle centered on visible x and y axes. Optional overlays can show the radius vector, the tangent velocity vector, the inward centripetal-acceleration vector, the angular marker, and the x and y projection guides.

Changing radius, angular speed, or phase updates the same orbit, vectors, readouts, and graphs together. The stage and graphs are synchronized views of one live circular motion.

Graph summary

The projection graphs show x(t) and y(t), the velocity graph shows the x and y components of the tangent velocity, and the angle graph shows angular position over time.

These are different views of the same circular motion, not separate datasets. The graphs and the orbit stage always describe the same instant.

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Starter track

Step 3 of 3

Motion and Circular Motion

Uniform Circular Motion appears later in this track, so it is cleaner to start from the beginning first.

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