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.

Speed $1.8\,\mathrm{m\,s^{-1}}$

Inward acceleration $2.16\,\mathrm{m\,s^{-2}}$

Rotation period $5.24\,\mathrm{s}$

Horizontal projection $0.75\,\mathrm{m}$

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.

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 misconceptionConstant speed means zero acceleration.

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

A projection is a coordinate

Typical misconceptionTilting the camera changes the harmonic law.

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

Run the experiment

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.
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.
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.