Angular Acceleration

Take two readings of angular velocity, ω₁ at t₁ and ω₂ at t₂, each signed with the positive sense you chose for θ. The average angular acceleration is ᾱ = (ω₂ − ω₁)/(t₂ − t₁): radians per second gained per second, written rad s⁻². Readings often arrive in rpm, so convert before dividing: 1 rpm = 2π/60 rad s⁻¹ = 0.105 rad s⁻¹, so 1 rpm per second is 0.105 rad s⁻². A cordless drill that reaches 2400 rpm from rest in 0.80 s has ω₂ = 251 rad s⁻¹ and ᾱ = 251/0.80 = 314 rad s⁻². That number uses only the two end readings. A drill that reaches full speed in the first 0.10 s and then holds it has exactly the same ᾱ over the 0.80 s as one that ramps evenly, although during that first tenth of a second its angular acceleration averaged eight times as much, 2.51 × 10³ rad s⁻².

The instantaneous angular acceleration is the limit of the average as the interval closes: α = dω/dt, the slope of the ω–t graph at one instant, and since ω = dθ/dt it is also d²θ/dt². Watch the limit arrive for a ceiling fan starting up with ω(t) = 20(1 − e^(−t/4.0)) rad s⁻¹. From t = 2.0 s, the average over the next 1.0 s is 2.68 rad s⁻², and over the next 0.10 s it is 3.00 rad s⁻²; differentiating gives α(t) = 5.0e^(−t/4.0) rad s⁻², so α(2.0) = 3.03 rad s⁻². Two facts come with the derivative. By the mean value theorem every interval contains an instant where α equals that interval's ᾱ, so a measured average is a value the rotor really passed through. And a rigid body has one α: the hub and the blade tips of the fan share it, even though their linear accelerations do not.

On a fixed axis, angular velocity is a vector along it, ω⃗ = ω k̂, with k̂ fixed by the right-hand rule: curl your fingers the positive way round and your thumb gives k̂. Because k̂ does not change, α⃗ = dω⃗/dt = α k̂ lies on the axis too, and its sign says which way ω⃗ is changing, not which way the body turns. Whether the rotation is speeding up follows from d(ω²)/dt = 2ωα = 2ω⃗ ⋅ α⃗. When ω and α share a sign, α⃗ is parallel to ω⃗ and the rotation spins up; when their signs differ, α⃗ is antiparallel and it winds down. An office chair seen from above, counterclockwise positive, with ω = −3.0 rad s⁻¹ and α = −0.60 rad s⁻² turns clockwise and speeds up, since ωα = +1.8 rad² s⁻³; with α = +0.60 rad s⁻² it would be slowing. Choose clockwise as positive instead and both numbers change sign, while their product, and the verdict, stay the same.

A rotor that reverses must pass through ω = 0, and at that instant it is still accelerating. Take the figure's default drive, ω(t) = 5.0 − 11e^(−t/1.5) rad s⁻¹, which starts the rotor at −6.0 rad s⁻¹. Differentiating gives α(t) = 7.33e^(−t/1.5) rad s⁻²: positive throughout, and shrinking. ω reaches zero at t = 1.5 ln(11/5.0) = 1.18 s, where α = 3.33 rad s⁻², not zero. Before that instant ωα is negative and the rotor is slowing; after it ωα is positive and the rotor is speeding up, all under one sign of α and while α itself falls. A shrinking α means ω is changing more gently, not that the body is slowing. An average taken across a reversal counts the whole change in ω: going from −6.0 to +2.0 rad s⁻¹ is Δω = +8.0 rad s⁻¹, not the 4.0 rad s⁻¹ drop in speed.

Tachometers and encoders report ω at a fixed spacing h, so α has to be estimated from differences. The forward difference [ω(t + h) − ω(t)]/h is really the average over the next interval: it describes the instant t + h/2 better than t, and its error shrinks only in proportion to h. The central difference [ω(t + h) − ω(t − h)]/(2h) straddles t symmetrically, its leading errors cancel, and what remains shrinks as h². Try both on the fan above with h = 0.50 s at t = 2.0 s. Forward: [ω(2.5) − ω(2.0)]/0.50 = 2.85 rad s⁻², 6% low. Central: [ω(2.5) − ω(1.5)]/1.0 = 3.04 rad s⁻², within 0.3% of α(2.0) = 3.03 rad s⁻². Shrinking h is not free: differencing subtracts two nearly equal readings and divides by a small number, so the effect of reading noise grows as 1/h.

Everything above assumed a fixed axis, where k̂ is constant and α⃗ = α k̂. Let the axis turn and ω⃗ changes direction even when its length does not. The bookkeeping matches circular motion, where a velocity of fixed size v whose direction turns at rate ω changes at rate vω, the familiar v²/r. Likewise a spin vector of fixed length ω whose direction turns at rate Ω changes at rate ωΩ, at right angles to itself. A car wheel spinning at a steady 60 rad s⁻¹ while the car rounds a bend at 0.25 rad s⁻¹ keeps its spin rate, yet |α⃗| = 60 × 0.25 = 15 rad s⁻², horizontal and perpendicular to the axle; for a car driving forward and turning left it points toward the back of the car. The scalar α of a fixed axis is only the component of α⃗ along the axis, α⃗ ⋅ k̂. The sideways part is why a spinning bicycle wheel fights the hands that try to tilt it, and it returns when torque and angular momentum become vectors.