---
title: "Uniform Circular Motion Observatory Interactive Lab | Sandbox Physics"
description: "How can constant speed still require acceleration? Velocity includes direction."
canonical_url: "https://sandboxphysics.com/labs/circular-motion-observatory/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/circular-motion-observatory/"
last_modified: "2026-11-04"
---

M007 · An orbit and its projections

# Uniform Circular Motion Observatory

**How can constant speed still require acceleration?** Velocity includes direction. Even at constant speed it turns toward the next tangent, so acceleration points inward. At the same speed a tighter circle needs more acceleration. Each orthogonal projection repeats as a simple harmonic motion with the same period as the orbit.

Drag a rotating marker. Compare tangent velocity, inward acceleration, period and two harmonic projections as radius and turning rate change.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

An ideal motor prescribes uniform motion in the horizontal laboratory plane. Positive angle is counterclockwise from the positive horizontal axis. The arm, bearings, marker size and vector lengths are schematic. Radius and signed angular velocity change between trials; parameter edits preserve the current angle and restart the clock, without simulating the adjustment. The graphs contain exact model coordinates and velocity components, not camera observations or perspective screen positions. Orthogonal projections are harmonic, while the speed is constant. At zero angular velocity there is no unique rotation period; the stationary state hides its time plots. No motor torque, structural load, drag or transient dynamics is modeled.

Physics tutorial

## Speed stays constant; velocity turns

**Background** A rotating marker advances along each new tangent. Its speed can stay fixed while its velocity changes direction. The inward acceleration measures that change.

**Why it matters** OpenStax University Physics Volume 1, Sections 6.3 and 15.3 connect centripetal acceleration with the harmonic projections of uniform circular motion.

Start with the essentials

- **Focus question**: What changes when the radius changes at the same speed?
- **One-sentence intuition**: A tighter circle turns velocity faster. Each orthogonal projection oscillates once per revolution.

### Core mathematical model

#### Two coordinates of one orbit

$$
\theta=\omega t+\varphi_0,\qquad x=R\cos\theta,\quad y=R\sin\theta
$$

Axes are fixed in the horizontal orbit plane. Positive angle is counterclockwise from the positive horizontal axis.

#### Tangent velocity

$$
\mathbf v=R\omega(-\sin\theta,\cos\theta),\qquad v=R|\omega|
$$

Reversing the motor changes the velocity direction without changing its magnitude at a given angle.

#### Acceleration points inward

$$
\mathbf a=-R\omega^2(\cos\theta,\sin\theta),\qquad a_c=R\omega^2=\frac{v^2}{R}
$$

The speed is constant, but the velocity vector rotates. No separate outward acceleration is needed in this laboratory frame.

#### One period for the orbit and both projections

$$
T=\frac{2\pi}{|\omega|},\qquad\ddot x=-\omega^2x,\quad\ddot y=-\omega^2y
$$

These period statements assume nonzero angular velocity. At rest the experiment hides its constant time plots.

### Common difficulties

#### Acceleration can change direction alone

**Typical misconception** Constant speed means zero acceleration.

**Better mental model** Compare two neighboring velocity arrows. Their difference points inward even though their lengths agree.

#### A projection is a coordinate

**Typical misconception** Tilting the camera changes the harmonic law.

**Better mental model** The graphs use fixed laboratory axes. Perspective changes the picture, not the underlying coordinates.

### Run the experiment

1. 01

   #### Watch one quarter turn

   Pause and scrub through a quarter revolution.

   **What to observe:** One coordinate reaches zero as the other reaches an extreme.

2. 02

   #### Keep the speed, change the bend

   Compare Tight and quick with Wide, same speed.

   **What to observe:** Both move at the same speed; the tighter circle needs three times the inward acceleration.

3. 03

   #### Reverse the motor

   Choose Reverse the motor, then inspect the same angle.

   **What to observe:** The tangent velocity reverses. Acceleration still points toward the centre.

Physical conclusion

Radius and signed angular velocity determine the whole motion. The apparatus and coordinate plots show the same analytic trajectory, using different views.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)