Learning objectives
- Calculate average angular velocity from an angle–time interval.
- Interpret instantaneous angular velocity as graph slope.
- Convert radians per second and RPM.
- Relate angular velocity, period, frequency and tangential speed.
Equations and conventions
ωavg = Δθ/Δt. Instantaneous ω = dθ/dt. Counterclockwise is positive. RPM = ω × 60/(2π). Frequency = |ω|/(2π). Period = 2π/|ω|. Tangential speed = r|ω|.
For constant angular acceleration: ω = ω₀ + αt and θ = ω₀t + ½αt². Angle is unwrapped so completed turns remain visible.
Model assumptions
The motor prescribes angular velocity and acceleration; torque, rotational inertia and drag are outside this lesson. The carousel is rigid, so all attached points share θ, ω and α. Tangential speed differs with radius.
The optical sensor logs positive-direction crossings of angle multiples of 2π. Clockwise crossings are visible in the animation but are not added to this one-direction pulse list. The tachometer displays speed magnitude in RPM, while the caption retains signed ω.
Interval cursors evaluate the mathematical motor program at selected times, even when the current animation has not reached them. Narration slows the demonstration using a time-scale factor; all physical-state equations and graph points still use the same simulation clock.