Rotational Kinematics
Angular velocity, acceleration, displacement and time — the rotational twins of the motion equations.
Angles in radians, and constant angular acceleration throughout — these are the rotational twins of the straight-line equations.
190.99 rpm · 3.1831 rev/s
Rotational motion uses the same equations as straight-line motion with angular quantities substituted in. A wheel starting from rest at 4 rad/s² reaches ω = 0 + 4 × 5 = 20 rad/s after five seconds — about 191 rpm — having turned through 50 radians, or just under eight full revolutions.
One set of equations, two coordinate systems
There is nothing new to learn here. Every linear quantity has an angular counterpart, and swapping them turns each kinematic equation into its rotational form: displacement x becomes angle θ, velocity v becomes angular velocity ω, acceleration a becomes angular acceleration α. So v = v₀ + at becomes ω = ω₀ + αt, and the rest follow the same way.
The advantage of angular variables is that every point on a rigid rotating body shares them. A point near the rim of a wheel moves much faster than one near the hub, so there is no single v for the wheel — but there is a single ω, because the whole thing sweeps the same angle in the same time. That is what makes rotation tractable.
Radians are not optional
These equations only work in radians, and the reason is the bridge to linear motion: v = ωr, a = αr, and s = θr all hold only when the angle is in radians. Substituting degrees breaks that relationship by a factor of about 57. Convert with 1 revolution = 2π radians = 360°, and remember that rpm is revolutions per minute, so converting it to rad/s means multiplying by 2π and dividing by 60.
constant angular acceleration only; every angle in radians
- 1 List what you know in angular terms. A wheel starting from rest gives ω₀ = 0; a stated 4 rad/s² gives α, and five seconds gives t.
- 2 Convert any rpm or degrees into radians. Multiply revolutions by 2π; for rpm, multiply by 2π and divide by 60.
- 3 Pick the equation missing the variable you neither know nor want. For the final angular velocity with no interest in θ, use ω = ω₀ + αt.
- 4 Substitute and solve. 0 + 4 × 5 = 20 rad/s.
- 5 Convert back if the question asks in everyday units. 20 rad/s × 60 ÷ 2π = 190.99 rpm, and 50 radians ÷ 2π = 7.96 revolutions.
Linear and angular counterparts
The right-hand column is the left-hand one with the substitutions applied.
| Linear | Angular | Link |
|---|---|---|
| Displacement x (m) | Angle θ (rad) | s = θr |
| Velocity v (m/s) | Angular velocity ω (rad/s) | v = ωr |
| Acceleration a (m/s²) | Angular acceleration α (rad/s²) | a = αr |
| v = v₀ + at | ω = ω₀ + αt | — |
| x = v₀t + ½at² | θ = ω₀t + ½αt² | — |
| v² = v₀² + 2ax | ω² = ω₀² + 2αθ | — |
The conversions that trip people up
Revolutions per minute is the unit most machinery is quoted in and the one these equations cannot use directly. A motor at 3000 rpm is turning at 3000 × 2π ÷ 60 = 314 rad/s. Going the other way, multiply by 60 and divide by 2π. Forgetting the 60 is a factor-of-sixty error that usually announces itself in the answer.
Two further points. The equations assume constant angular acceleration, exactly as their linear counterparts assume constant acceleration — a motor spinning up under varying torque needs calculus instead. And angular acceleration α is not the same thing as centripetal acceleration: α describes the rate at which the rotation speeds up, while centripetal acceleration v²/r points inward and exists even at a perfectly steady spin.