Physics I: Mechanics › Kinematics › full formula sheet

ω = ω₀ + αt     θ = ω₀t + ½αt²Say it: angular velocity equals initial angular velocity plus angular acceleration times time; angle equals initial angular velocity times time plus one half alpha t squared

Angular kinematics

Everything you know about linear motion, rewritten for spinning things — the same equations, new letters.

θ is angle (radians), ω = dθ/dt angular velocity (rad/s), α = dω/dt angular acceleration (rad/s²). Counterclockwise is positive by convention. Radians, not degrees!

Where it comes from

A potter’s wheel spins up; a figure skater pulls in her arms and whirls faster; a hard drive spins at 7200 rpm. Describing these with x, v, a would be torture — the natural coordinates are angle, angular velocity, angular acceleration. And here’s the gift: the equations are identical in form to the linear ones you already mastered.

Before reading on: a wheel spins up from rest with angular acceleration 3 rad/s². After 4 s, what’s its angular velocity? If you know v = v₀ + at, you already know the answer — translate it.
linear
v = v₀ + at     x = v₀t + ½at²
The motion you know.
angular
ω = ω₀ + αt     θ = ω₀t + ½αt²
Same shapes. x→θ, v→ω, a→α. Learn the mapping once; every linear result ports over.

Your prediction: ω = 0 + 3·4 = 12 rad/s — about 115 rpm. The rest of this page is the dictionary that makes every linear technique reusable.

Derivation

Define the angular quantities exactly the way the linear ones were defined, and the same derivations replay word for word:

ωavg
=
Δθ/Δt,    ω = dθ/dt
Step 1 — angular velocity. Same as v = Δx/Δt, with angle replacing position. Units: rad/s.
αavg
=
Δω/Δt,    α = dω/dt
Step 2 — angular acceleration. Same as a = Δv/Δt, with ω replacing v. Units: rad/s².
ω
=
ω₀ + αt
Step 3 — mirror v = v₀ + at. Rearranged α = (ω − ω₀)/t. Constant-α only.
θ
=
ω₀t + ½αt²
Step 4 — mirror x = v₀t + ½at². Same midpoint derivation: θ = ω̄t with ω̄ = (ω₀+ω)/2. ∎

The full dictionary: ω² = ω₀² + 2αθ mirrors v² = v₀² + 2aΔx, and θ = ½(ω₀+ω)t mirrors Δx = ½(v₀+v)t. Five linear equations, five angular twins.

How to use it

The procedure, every time:

  1. Convert everything to radians, rad/s, rad/s² first. rpm → rad/s: multiply by 2π/60. Degrees → radians: multiply by π/180.
  2. Translate the story. “Spins up from rest” → ω₀ = 0. “Slows to a stop” → ω = 0, α negative.
  3. Pick the equation missing the variable you don’t need — exactly like linear kinematics.
  4. Convert back if asked. θ in radians ÷ 2π = revolutions.

Unit conversions you’ll use constantly

1 rev = 2π rad ≈ 6.283 radfull turn in radians
ω (rad/s) = rpm × 2π/60Say it: radians per second equals rpm times two-pi over sixty1200 rpm = 125.7 rad/s
Common mistake: plugging degrees into θ = ω₀t + ½αt². The equations are derived in radians (s = rθ only works in radians). 360° where 2π belongs inflates every answer ~57×.
Common mistake: using rpm directly as ω. rpm is revolutions per minute; ω needs radians per second. 1200 rpm ≠ 1200 rad/s — it’s 125.7 rad/s.

Worked examples

Four problems, easiest first. In each one, read every step — the why of each move is the lesson.

Example 1 — the basic move: spin-up, α = 3 rad/s² from rest, t = 4 s

  1. List givens. ω₀ = 0, α = 3 rad/s², t = 4 s.
  2. Angular velocity: ω = 0 + 3·4 = 12 rad/s.
  3. Angle turned: θ = 0 + ½·3·16 = 24 rad.
  4. In revolutions: 24/(2π) ≈ 3.82 rev — nearly 4 full turns while speeding up.
Common mistake: θ = ωt = 12·4 = 48 rad — using the final ω as if it spun that fast the whole time. It sped up from 0; the average was 6 rad/s, hence 24 rad.
Your turn — α = 2 rad/s² from rest, t = 5 s. Final ω? Angle turned?

Answer: 10 rad/s; 25 rad. ω = 2 · 5 = 10 rad/s; θ = ½ · 2 · 25 = 25 rad.

Example 2 — spinning down: ω₀ = 3.5 rad/s, α = −0.5 rad/s²

  1. Time to stop: 0 = 3.5 − 0.5t → t = 3.5/0.5 = 7 s.
  2. Angle while stopping: θ = 3.5·7 + ½(−0.5)·49 = 24.5 − 12.25 = 12.25 rad.
  3. In revolutions: 12.25/6.283 ≈ 1.95 rev — just under two full turns.
  4. Check via ω² = ω₀² + 2αθ: 0 = 12.25 − 2·0.5·θ → θ = 12.25 ✓.
Common mistake: θ = 3.5·7 = 24.5 rad — using initial ω as the whole-trip average. Slowing means the average was 1.75 rad/s; the −½αt² term corrects it.
Your turn — ω₀ = 2 rad/s, α = −0.25 rad/s². Time to stop? Angle?

Answer: 8 s; 8 rad. t = 2/0.25 = 8 s; θ = 2 · 8 − ½ · 0.25 · 64 = 16 − 8 = 8 rad.

Example 3 — the turbine: 0 → 1200 rpm in 30 s

  1. Convert: 1200 rpm · 2π/60 = 40π ≈ 125.66 rad/s.
  2. Angular acceleration: α = (125.66 − 0)/30 ≈ 4.19 rad/s².
  3. Angle: θ = ½·4.19·900 ≈ 1885 rad.
  4. In revolutions: 1885/6.283 ≈ 300 rev. Check: average 600 rpm · 0.5 min = 300 rev ✓.
Common mistake: α = 1200/30 = 40 “rad/s²” — using rpm as ω. The real α is 4.19 rad/s²; the rpm version is off by 2π/60 ≈ 0.105.
Your turn — 0 → 900 rpm in 20 s. Final ω? α? Revolutions?

Answer: 94.25 rad/s; 4.71 rad/s²; 150 rev. ω = 900 · 2π/60 = 30π ≈ 94.25 rad/s; α = 94.25/20 ≈ 4.71 rad/s²; θ = ½ · 4.71 · 400 ≈ 942.5 rad ≈ 150 rev.

Before reading on: a Ferris wheel turns once every 12 s at constant speed. What is ω? After 30 s, how many radians has it turned — and is that a whole number of revolutions?

Example 4 — constant spin: one revolution per 12 s

  1. Angular velocity: ω = 2π/12 = π/6 ≈ 0.5236 rad/s. (α = 0 — no speeding up.)
  2. After 30 s: θ = ωt = 0.5236·30 ≈ 15.71 rad.
  3. Revolutions: 15.71/6.283 = 2.5 rev — two and a half turns, as 30/12 = 2.5 confirms ✓.
  4. The pattern. Constant ω → θ = ωt, the angular version of x = vt.
Common mistake: ω = 12/2π = 1.91 “rad/s” — inverting the fraction. ω = angle/time = 2π/12, not 12/2π. Units check: rad/s should be smaller than 1 here (a slow wheel).
Your turn — one revolution per 10 s; angle after 25 s, in rad and rev?

Answer: 15.71 rad = 2.5 rev. ω = 2π/10 = 0.6283 rad/s; θ = 0.6283 · 25 = 15.71 rad; 15.71/6.283 = 2.5 rev.

Memorization tips

  • Say it aloud: “omega equals omega-naught plus alpha t.” If you can say the linear version, you can say this one.
  • Port, don’t relearn. Every linear kinematics technique — choosing equations, sign conventions, unit checks — works here with x→θ, v→ω, a→α.
  • Radians or ruin. The equations demand radians. Convert rpm and degrees before the physics, convert back after.
  • 2π rad = 1 rev; rpm × 2π/60 = rad/s. Tattoo these two conversions on the inside of your eyelids.
  • Sign = spin direction. Counterclockwise positive is the convention; clockwise spins get negative ω and α.
  • α = 0 → θ = ωt. Constant spin is the angular version of cruising — the simplest case, and a good check.

Final challenge

Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and Angular kinematics is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

How is angular kinematics related to linear kinematics?

They are the same mathematics with different letters: theta replaces x, omega replaces v, alpha replaces a. Every linear equation (v = v0 + at, x = v0t + 1/2 at^2, v^2 = v0^2 + 2a delta-x) has an exact angular twin.

Why must angles be in radians?

The equations are derived using s = r theta and v = omega r, which hold only when theta is in radians. Degrees introduce a pi/180 factor everywhere; radians keep the formulas clean.

How do I convert rpm to rad/s?

Multiply by 2pi/60: omega (rad/s) = rpm x 2pi/60. So 1200 rpm = 125.7 rad/s. (rpm counts revolutions per minute; omega needs radians per second.)

What is the sign convention for rotation?

Counterclockwise is positive by standard convention; clockwise is negative. As with linear motion, set it before computing and keep every sign consistent.

When is angular acceleration zero?

When the spin rate is constant (alpha = 0), e.g. a steadily turning Ferris wheel. Then theta = omega t, the angular version of x = vt — the simplest case.

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