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Imagine riding in a car cruising at a steady 30 miles per hour, when suddenly the driver cuts the steering wheel hard to the left. Your speedometer hasn't budged by a single digit, yet your body violently lurches toward the passenger door.
Even though your speed is completely constant, you are unmistakably accelerating. How can an object accelerate without speeding up or slowing down?
Before we calculate that inward pull, how do we measure how fast something travels around a circle in the first place?

Tangential Speed and Angular Velocity

When an object travels along a circular path of radius rr, the physical distance it covers along the rim is the arc length, denoted by ss. A full circle has a circumference of 2Ο€r2\pi r, which corresponds to an angle of 2Ο€2\pi radians.
Because angle in radians is defined as the ratio of arc length to radius (ΞΈ=s/r\theta = s / r), the distance traveled along the perimeter is simply s=rΞΈs = r\theta.
If the object sweeps through an angle Δθ\Delta\theta in time Ξ”t\Delta t, its rotational rate is its angular velocity, Ο‰=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}, measured in radians per second (extrad/s ext{rad/s}).
To find its linear speed along the edgeβ€”called tangential speed (vv)β€”we take the rate of change of arc length: v=Ξ”sΞ”t=rΔθΔt=rΟ‰v = \frac{\Delta s}{\Delta t} = \frac{r\Delta\theta}{\Delta t} = r\omega.
Notice what this equation reveals: if two people sit on the same spinning merry-go-round, they share the exact same angular velocity Ο‰\omega, but the person sitting twice as far from the center moves with twice the tangential speed.
Now that we can track speed along the rim, what happens when we look at velocity as a vector?

Acceleration Without Changing Speed

Velocity is a vector with both a magnitude (speed) and a direction. Because acceleration is defined as a⃗=Δv⃗Δt\vec{a} = \frac{\Delta\vec{v}}{\Delta t}, changing either the speed or the direction creates acceleration.