Physics I: Mechanics › Kinematics › full formula sheet

v = ωr     at = αrSay it: tangential speed equals angular speed times radius; tangential acceleration equals angular acceleration times radius

Tangential quantities

The bridge between spinning and moving — how a wheel’s rotation becomes the road speed of its rim.

v is the tangential (rim) speed (m/s), at the tangential acceleration (m/s²), ω angular velocity (rad/s), α angular acceleration (rad/s²), r the radius. Radians required.

Before this lesson: Angular kinematics

Where it comes from

Two bugs ride a spinning record — one near the center, one near the edge. Same angular velocity ω (they stay on the same spoke), yet the outer bug is hauling: it covers far more vinyl per second. The wheel’s spin is one number; the speed you’d feel standing at each radius is another. Tangential quantities translate between them.

Before reading on: a record spins at 3.5 rad/s. Bug A sits at 5 cm, bug B at 10 cm. Bug B moves twice as fast — why? And what is each bug’s speed in m/s? Estimate, then check.
bug A (r = 0.05 m)
v = 3.5 · 0.05 = 0.175 m/s
Same ω, small radius, small speed.
bug B (r = 0.10 m)
v = 3.5 · 0.10 = 0.35 m/s
Double the radius, double the speed — v = ωr is a straight proportion.

This is also how cars work: the engine spins the axle at ω, and the tire’s rim moves at v = ωr — which, for rolling without slipping, is the car’s speed.

Derivation

Start from arc length — the distance a point at radius r actually travels — and differentiate. Each derivative peels off one more tangential quantity:

s
=
rθ
Step 1 — arc length. Angle θ (in radians) times radius = distance traveled along the rim. (This is why radians are mandatory.)
v = ds/dt
=
r · dθ/dt = ωr
Step 2 — differentiate once. r is constant, so it slides out; dθ/dt is ω. Tangential speed: v = ωr.
at = dv/dt
=
r · dω/dt = αr
Step 3 — differentiate again. dω/dt is α. Tangential acceleration: at = αr. ∎

The full picture: every point on a spinning body has two accelerations — tangential at = αr (along the motion, from speeding up) and centripetal ac = ω²r (inward, from turning). Uniform spin (α = 0) leaves only the centripetal one.

How to use it

The procedure, every time:

  1. Identify r. Which radius? The rim, the bug’s spot, the tire — the point whose motion you care about.
  2. Get ω in rad/s (convert from rpm first!). Then v = ωr.
  3. Get α in rad/s², then at = αr — the rate the rim speed is changing.
  4. Don’t forget the inward partner: ac = ω²r always acts too. at changes the speed; ac changes the direction.

Rolling without slipping

vcar = ωwheel · rtirethe contact point is instantaneously at rest — so the axle moves at the rim speedSay it: road speed equals wheel spin times tire radius

Tire r = 0.33 m spinning at ω = 90 rad/s → v = 29.7 m/s (≈ 107 km/h). The speedometer is just an ω-meter in disguise.

Common mistake: v = ωr with ω in rpm or degrees. ω = 100 rpm, r = 0.5 m → v = 50 m/s?? No: convert first (ω = 10.47 rad/s → v = 5.24 m/s). The formula only speaks radians.
Common mistake: confusing at with ac. Spinning up a wheel: rim points gain both — at = αr along the tangent (speed increasing) and ac = ω²r inward (direction turning). Different directions, different formulas, both real.

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: wheel r = 0.4 m, ω = 10 rad/s

  1. List givens. r = 0.4 m, ω = 10 rad/s.
  2. Apply v = ωr. = 10 · 0.4 = 4 m/s.
  3. Picture it. The rim sweeps 4 meters of arc every second — while the hub doesn’t translate at all.
  4. Units. (rad/s)·m = m/s ✓ (radians are dimensionless).
Common mistake: v = ω/r = 25 m/s — dividing instead of multiplying. Bigger radius means more arc per radian, so v must grow with r. If your formula shrinks with r, flip it.
Your turn — r = 0.25 m, ω = 8 rad/s. Rim speed?

Answer: 2 m/s. 8 · 0.25 = 2 m/s.

Example 2 — speeding up: α = 2 rad/s², r = 0.5 m

  1. List givens. α = 2 rad/s², r = 0.5 m.
  2. Apply at = αr. = 2 · 0.5 = 1 m/s².
  3. Read it. Every second, each rim point’s tangential speed grows by 1 m/s — along the tangent, in the direction of spin.
  4. And the inward part? If ω = 6 rad/s right now, ac = 36 · 0.5 = 18 m/s² inward — much bigger, and perpendicular.
Common mistake: reporting at = 1 m/s² as “the acceleration” full stop. It’s only the tangential piece — the total acceleration is the vector sum with ac, pointing diagonally inward-forward.
Your turn — α = 4 rad/s², r = 0.2 m. Tangential acceleration?

Answer: 0.8 m/s². 4 · 0.2 = 0.8 m/s² along the tangent.

Example 3 — the record: 33⅕ rpm, r = 0.15 m

  1. Convert first! ω = 33.333 · 2π/60 ≈ 3.4907 rad/s.
  2. Rim speed: v = 3.4907 · 0.15 ≈ 0.524 m/s.
  3. Sanity via circumference: 33.333 rev/min = 0.5556 rev/s; each rev is 2π·0.15 = 0.9425 m; 0.5556 · 0.9425 = 0.5236 m/s ✓.
  4. The lesson, twice: convert before the formula, and the circumference route agrees — two independent paths, one answer.
Common mistake: v = 33.333 · 0.15 = 5 m/s — rpm straight into the formula. Ten times too big, and the units (rev·m/min) aren’t m/s. Convert first, always.
Your turn — 45 rpm record, r = 0.12 m. Rim speed?

Answer: 0.566 m/s. ω = 45 · 2π/60 = 4.7124 rad/s; v = 4.7124 · 0.12 ≈ 0.566 m/s.

Before reading on: a merry-go-round spins at ω = 0.8 rad/s; you stand at r = 6 m. Your inward acceleration is ω²r. Estimate it — closer to 1, 4, or 10 m/s²?

Example 4 — the inward partner: ω = 0.8 rad/s, r = 6 m

  1. Tangential speed first: v = 0.8 · 6 = 4.8 m/s.
  2. Centripetal: ac = ω²r = 0.64 · 6 = 3.84 m/s², inward.
  3. Compare with v²/r: 23.04/6 = 3.84 m/s² ✓ — same answer, two forms.
  4. Feel it. ~0.4 g inward — a brisk merry-go-round, holding on required.
Common mistake: ac = ωr = 4.8 m/s² — that’s the speed v, not an acceleration (units m/s). The centripetal formula squares ω; the tangential one doesn’t.
Your turn — ω = 1.5 rad/s, r = 2 m. Centripetal acceleration?

Answer: 4.5 m/s² inward. 1.5² · 2 = 2.25 · 2 = 4.5 m/s².

Memorization tips

  • Say it aloud: “tangential speed is omega times r.” Spin × radius = road speed.
  • r is the lever. Same ω, bigger r → bigger v, bigger at, bigger ac. Radius multiplies everything.
  • Two accelerations, two jobs: at = αr changes the speed (along motion); ac = ω²r changes the direction (inward). Name which one the question asks for.
  • Convert before you compute. rpm → rad/s (×2π/60), degrees → radians (×π/180). The formulas only speak radians.
  • v = ωr grows with r — if your expression shrinks with radius, you divided by mistake.
  • Rolling without slipping: vcar = ωwheelrtire. The speedometer is an ω-meter in disguise.

Final challenge

Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and Tangential quantities 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

What is the difference between tangential and angular velocity?

Angular velocity omega (rad/s) is the spin rate — same for every point on a rigid body. Tangential velocity v = omega r (m/s) is the actual linear speed of a point at radius r — bigger farther out. Same spin, different speeds.

What is tangential acceleration?

a_t = alpha r: the rate at which a point's tangential speed changes, directed along the tangent. It's nonzero only while the spin rate changes (alpha != 0) — e.g. a wheel spinning up.

How do tangential and centripetal acceleration differ?

Tangential acceleration (alpha r) points along the motion and changes the speed; centripetal acceleration (omega^2 r) points inward and changes the direction. A spinning-up wheel has both, perpendicular to each other.

Why do the formulas need radians?

They come from s = r theta, which holds only when theta is in radians. Plug in degrees or rpm and every answer is wrong — convert first, compute second.

How does v = omega r explain a car's speed?

For rolling without slipping, the tire's contact point is instantaneously at rest, so the axle (and car) moves at the rim's tangential speed: v_car = omega_wheel x r_tire. Spin the wheels faster, drive faster — linearly.

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