Linear and angular relations

Rigid means every distance inside the body stays fixed, so when a door swings on its hinges every particle turns through the same angle in the same time. θ, ω and α therefore belong to the whole door. The distance travelled does not. Each particle runs round its own circle, and the radian is defined as arc divided by radius, so s = rθ. Swing the door through 60°, which is π/3 = 1.047 rad. The handle, 0.80 m from the hinge line, travels 0.80 × 1.047 = 0.84 m; a screw 0.05 m from the hinges travels 0.052 m. Same angle, sixteen times the path, because the handle is sixteen times farther from the axis.

Speed is the rate at which arc is covered. For a particle of a rigid body r is constant, so it passes straight through the derivative: v = ds/dt = r dθ/dt = rω. Differentiate again and aₜ = dv/dt = r dω/dt = rα. Both keep the sign convention of θ, positive along the direction of increasing angle, so the point speeds up when ω and α share a sign and slows when they differ. The step that made this work was dr/dt = 0. A bead sliding outward at 0.50 m s⁻¹ along a rod turning at 3.0 rad s⁻¹ has, at r = 0.20 m, a tangential velocity rω = 0.60 m s⁻¹ and a radial velocity of 0.50 m s⁻¹, so its speed is √(0.60² + 0.50²) = 0.78 m s⁻¹, not 0.60. v = rω is the whole speed only for points fixed in the body.

s = rθ needs no conversion factor only because the radian is a ratio of lengths: rad × m = m, and the rad quietly disappears. Revolutions and degrees do not disappear that way, which makes rpm the commonest source of wrong speeds in this topic. A disc drive spinning at 7200 rpm has ω = 7200 × 2π ÷ 60 = 754 rad s⁻¹, so a track 0.040 m from the spindle moves at 0.040 × 754 = 30.2 m s⁻¹. Multiply 7200 by 0.040 m and divide by 60 instead and you get 4.8 m s⁻¹, short by exactly 2π, because one revolution is 2π radians, not one. Degrees set the same trap: 90° s⁻¹ is 1.57 rad s⁻¹, not 90.

A point moving round a circle at constant speed still has a velocity that changes direction, and the rate of that change is the radial acceleration aᵣ = v²/r = ω²r, pointing at the axis. On one rigid body use the ω²r form, because ω is what every point shares. A child 0.50 m from the centre of a roundabout turning at 1.2 rad s⁻¹ has aᵣ = 1.2² × 0.50 = 0.72 m s⁻², and a child 2.0 m out has 2.88 m s⁻²: four times the radius, four times the acceleration. v²/r gives the same 2.88 m s⁻² with v = 2.4 m s⁻¹, but read at fixed v it seems to promise the opposite. That 1/r reading belongs to a car held at 20 m s⁻¹, which needs 8.0 m s⁻² on a 50 m bend and 2.0 m s⁻² on a 200 m one.

aₜ lies along the path and aᵣ across it, so they combine as perpendicular components: a = √(aₜ² + aᵣ²) = r√(α² + ω⁴). The magnitude is proportional to r, but the direction is fixed by tan φ = |α|/ω², measured from the inward radius, and r has cancelled. At any instant every point on the body accelerates at the same angle to its own radius, with a length that grows steadily outward. The angle does change in time. A turntable starting up at α = 2.0 rad s⁻² has φ = 82.9° when ω = 0.50 rad s⁻¹, almost purely tangential; it passes 45° at ω = √2.0 = 1.41 rad s⁻¹, and is down to 12.5° by ω = 3.0 rad s⁻¹, when the radial part has taken over.

For a fixed axis the vector statement is v⃗ = ω⃗ × r⃗, with ω⃗ along the axis by the right-hand rule and r⃗ drawn from any point on the axis. The cross product does two jobs. It makes v⃗ perpendicular to both the axis and the radius, which is the tangent to the circle. And its magnitude is ωr sin γ, where γ is the angle between r⃗ and the axis, so the distance that counts is r⊥ = r sin γ, measured perpendicular to the axis. Earth turns at 7.29 × 10⁻⁵ rad s⁻¹ and has radius 6.37 × 10⁶ m. A point on the equator moves at 464 m s⁻¹, but at latitude 40° it is only 6.37 × 10⁶ × cos 40° = 4.88 × 10⁶ m from the axis and moves at 356 m s⁻¹.