Angular Displacement (θ = ω₀t + ½αt²)

\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

θ =

ω₀ =

α =

t =

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Under constant angular acceleration, the angle turned splits into a coasting term ω₀t and a spin-up term ½αt² — exactly parallel to the linear displacement formula. A washing machine drum accelerating from rest at 15 rad/s² turns through ½ × 15 × 2² = 30 rad ≈ 4.8 full revolutions in its first two seconds. Solving for t requires the quadratic formula, so formula.expert solves this one for θ, ω₀, and α.

Angular Displacement (θ = ω₀t + ½αt²)

\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

Where

$\theta$ = Angular displacement
$\omega_0$ = Initial angular velocity
$\alpha$ = Angular acceleration
$t$ = Time

Missing one of these? Work it out first, then come back

Angular displacement — Work (W = Fd cos θ), Torque
Initial angular velocity — Angular Acceleration, Angular Velocity (ω = θ/t)
Angular acceleration — Angular Acceleration, Newton's Second Law for Rotation (τ = Iα)
Time — Speed, Distance & Time, Final Velocity (Uniform Acceleration)

Part of this formula set

The rotational equations of motion

13 formulasThe angular twins of the linear equations — θ, ω and α in place of d, v and a, and torque, moment of inertia and Iω in place of F, m and p.