Angular Velocity, Period & Frequency

Fix a rigid body to an axle and follow any point on it. Its angle θ, measured in radians from a fixed reference line in the plane perpendicular to the axle, changes by the same Δθ as every other point's, because a rigid body cannot bend: the angle between any two of its radius lines is frozen. So one function θ(t) describes the entire rotation, and its rate belongs to the body rather than to any one point. A vinyl record at 33⅓ rpm shows it. A speck on the label and a speck on the outer groove both sweep 2π rad every 1.80 s, so both turn at 2π/1.80 s = 3.49 rad s⁻¹, even though the outer speck travels much farther each turn. Distance travelled depends on radius; angle turned does not. Keep θ in radians: only then is arc length simply radius times angle, a simplicity every later relation built on ω inherits.

The average angular velocity over an interval, ω_avg = Δθ/Δt, is the slope of the straight chord joining two points on the θ–t graph. It uses two readings and ignores everything between them. Shrink the interval and the chord pivots towards the tangent; its limiting slope is the instantaneous angular velocity ω = dθ/dt. Take a wind-turbine rotor starting from rest with θ(t) = 0.050t³ rad. At t = 4.0 s the derivative gives ω = 0.15t² = 2.40 rad s⁻¹. Averages that start at 4.0 s close in on that value: 3.05 rad s⁻¹ over the next 1.0 s, 2.46 rad s⁻¹ over 0.10 s, 2.41 rad s⁻¹ over 0.010 s. The average over the first 4.0 s is only θ/t = 3.2 rad ÷ 4.0 s = 0.80 rad s⁻¹, a third of the final rate, because the rotor spent most of that time turning slowly. Current angle divided by elapsed time is always an average from t = 0, never the rate now.

Choose a direction along the axle and call it +z. Seen from the +z end looking back at the body, counterclockwise turning makes θ grow, so ω_z = dθ/dt is positive; clockwise turning makes it negative. The right-hand rule turns that sign into a direction in space: curl the fingers of your right hand the way the body turns, and your thumb points along ω⃗. The arrow lies on the axis because the axis is the only line the whole body agrees on; its points are moving in every direction around the circle. Two cases fix the convention. A car driving north has wheels whose tops move north, so each wheel's ω⃗ points west, to the driver's left, and swings round to east when the car reverses. The Earth turns eastward, so its ω⃗ runs along the axis from the South Pole out through the North Pole, towards Polaris.

When ω is constant the body repeats itself, and three numbers describe the same motion. The period T is the time for one revolution, Δθ = 2π, so T = 2π/|ω|. The frequency f = 1/T counts revolutions per second, in hertz, and ω = 2πf. Machines quote a fourth form, revolutions per minute. A car engine idling at 750 rpm makes 750/60 = 12.5 revolutions each second, so f = 12.5 Hz, T = 1/12.5 s = 80 ms and ω = 2π × 12.5 = 78.5 rad s⁻¹. The trap is that the radian is dimensionless: rad s⁻¹, Hz and rev s⁻¹ all reduce to s⁻¹, so a dimensional check cannot catch a missing or doubled 2π. Carry the unit words anyway, rad s⁻¹ for ω and Hz or rev s⁻¹ for f, and let them tell you when a 2π is owed.

T = 2π/|ω| assumes ω never changes, yet plenty of rotations repeat without being uniform. A playground roundabout that gets a shove each time its handle comes past speeds up, then coasts more slowly until the next push. Model it as ω(t) = ω̄[1 + k cos(2πt/T)] with T = 4.0 s, k = 0.50 and ω̄ = 2π/T = 1.57 rad s⁻¹. At its fastest ω = 2.36 rad s⁻¹ and at its slowest 0.785 rad s⁻¹, so 2π/ω(t) would swing between 2.7 s and 8.0 s. Yet every revolution takes exactly 4.0 s: integrating ω over any 4.0 s window adds precisely 2π to θ, because the cosine term integrates to zero over a whole cycle. The period of an uneven rotation is fixed by the average angular velocity over one full turn, T = 2π/ω_avg. Shorter windows depend on where they start: from t = 0.5 s to 2.5 s the average is only 1.22 rad s⁻¹.

Angular velocities about one shared axle combine like one-dimensional velocities: take components along the same +z and add or subtract them with their signs. Periods do not combine that way, so convert to ω first. A clock's hands share an axle and both turn clockwise, so take +z into the wall and both components are positive: ωₘᵢₙᵤₜₑ = 2π/3600 s = 1.745 × 10⁻³ rad s⁻¹ and ωₕₒᵤᵣ = 2π/43 200 s = 1.454 × 10⁻⁴ rad s⁻¹. The minute hand gains on the hour hand at the difference, 1.600 × 10⁻³ rad s⁻¹, and must gain a full 2π between one overlap and the next, which takes 2π/(1.600 × 10⁻³ rad s⁻¹) = 3927 s, or 65.45 minutes, not the 60 minutes that comparing periods suggests. Had the hour hand turned the other way, the rates would add and the hands would meet every 55.38 minutes.