A ladybug rides a turntable. Drag her anywhere on the disk, or use the sliders. Every point on a rigid turntable shares one angular velocity ω, but the speed each point actually travels at depends on how far out it sits: v = ωr. The green arrow is her velocity, always along the rim; the pink one is her centripetal acceleration, always straight at the axle.
What to try. Park the ladybug at the rim and note her speed. Now slide her halfway in without touching the rotation slider. The turntable is turning exactly as fast as it was — the period readout does not budge — but her own speed has halved. That is the whole content of v = ωr: ω belongs to the turntable, v belongs to the point. Turning on velocity all along the spoke makes the same point in one picture: a row of arrows sharing one ω, each one longer than the last.
Then hold the radius fixed and double ω. The velocity arrow doubles, as you would expect. The pink acceleration arrow does not: it grows by a factor of four, because a = ω2r. On the graph you can watch the same thing happen to the two slopes — the green line tilts up twice as fast, the pink one four times as fast.
Drag the ladybug all the way to the center. Her speed and her acceleration both fall to zero even though the turntable is still spinning, which is why the axle is the only place on a merry-go-round where nothing pushes on you. Finally, drag ω through zero to the negative side: the turntable reverses, the velocity arrow flips with it, and the acceleration arrow does not — it still points inward, because the sign of ω is squared away.
What this leaves out. ω changes the instant you move the slider, so there is no angular acceleration and no tangential component to the acceleration; the pink arrow is purely centripetal. Nothing here depends on the ladybug's mass, and no force is drawn — this is kinematics only. The disk is rigid and she never slips on it, which is the assumption that lets every point share a single ω.