Rotational Kinematics

You’ll be able to

Define angular velocity and acceleration as derivatives ω = dθ/dt and α = dω/dt
Apply the constant-angular-acceleration equations
Relate linear and angular quantities with v = rω and a_t = rα

Angles have their own kinematics

Rotational motion mirrors translation exactly, with angular position θ replacing x. The angular velocity is the derivative ω = dθ/dt, and the angular acceleration is α = dω/dt = d²θ/dt². Everything you learned for x, v, a carries over — you differentiate to go down the chain and integrate to go back up, just with angular variables measured in radians.

Angular velocity and acceleration

ω = dθ/dt α = dω/dt = d²θ/dt²

Angular position θ in radians, ω in rad/s, α in rad/s². The direct analogs of v = dx/dt and a = dv/dt.

Linking angular to linear

A point at radius r on a rotating body traces an arc, so its linear speed is v = rω and its tangential acceleration is a_t = rα. It also has a centripetal acceleration a_c = ω²r = v²/r directed inward, present whenever it moves in a circle. When α is constant, the rotational kinematic equations ω = ω₀ + αt and θ = θ₀ + ω₀t + ½αt² hold — the exact analogs of the linear ones.

Linking linear and angular motion

v = rω a_t = rα a_c = ω²r = v²/r

Multiply an angular quantity by the radius to get the corresponding linear one at that point. The centripetal term a_c points toward the center.

Worked example

A wheel rotates so that its angular position is θ(t) = 2t³ − t (radians, seconds). Find ω(t) and α(t), and evaluate both at t = 1 s.

1.Differentiate θ to get angular velocity: ω(t) = dθ/dt = 6t² − 1.
2.Differentiate again for angular acceleration: α(t) = dω/dt = 12t.
3.At t = 1: ω(1) = 6(1) − 1 = 5 rad/s.
4.At t = 1: α(1) = 12(1) = 12 rad/s².

Answer: ω(t) = 6t² − 1, α(t) = 12t; at t = 1 s, ω = 5 rad/s and α = 12 rad/s²

Checkpoint

A wheel rotates with angular position θ(t) = 2t³ radians. What is its angular velocity at t = 2 s?

Checkpoint

A point on the rim of a wheel of radius 0.5 m moves as the wheel spins at ω = 8 rad/s. What is the point's linear (tangential) speed?

Tip

Set up a two-column analogy sheet: x↔θ, v↔ω, a↔α, m↔I, F↔τ, p↔L. Every translational relationship has a rotational twin, so once you know one side you can write the other by swapping symbols.

Answer the 2 checkpoints as you read.