Rotational Kinematics Equations
Simple Explanation
When angular acceleration is constant, three equations connect angular displacement, initial and final angular velocity, angular acceleration, and time — directly mirroring the straight-line "suvat" equations, with θ, ω, and α replacing displacement, velocity, and acceleration.
Why Do We Need It?
These equations let any two known rotational quantities be combined to find the rest, without needing to track the motion moment by moment — exactly how the linear kinematics equations work for straight-line motion.
See It
A straight rising line showing angular velocity increasing steadily with time, reaching 17 rad/s at t=4 seconds
Formula
Rotational Kinematics Equations
ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ
The rotational counterparts of the straight-line "suvat" equations — they relate angular displacement, initial and final angular velocity, angular acceleration, and time, for constant angular acceleration.
- ω₀
- — initial angular velocity, in radians per second (rad/s)
- ω
- — final angular velocity, in radians per second (rad/s)
- α
- — constant angular acceleration, in radians per second squared (rad/s²)
- t
- — time elapsed, in seconds (s)
- θ
- — angular displacement over that time, in radians (rad)
When to use it: Whenever a rotation has constant angular acceleration, and any three of the five quantities (ω₀, ω, α, t, θ) are known — pick the equation missing the one unknown that is not needed.
Worked Example
Use the rotational kinematics equations
A flywheel starts at ω₀=5 rad/s and accelerates at α=3 rad/s² for t=4 s. Find its final angular velocity and angular displacement.
Why Does This Work?
These equations follow directly from the definitions of ω and α by the same reasoning used to derive the linear suvat equations — ω=ω₀+αt comes from integrating a constant α over time, and θ=ω₀t+½αt² comes from integrating that resulting ω(t) over time.
Real-Life Example
A car engine's flywheel reaching operating speed
When a car engine starts, its flywheel accelerates from rest up to its running speed at a roughly constant angular acceleration.
Engineers use these equations to predict exactly how many rotations the flywheel completes and how fast it is spinning at any moment during that startup.
Practice
A wheel starts at ω₀=2 rad/s and accelerates at α=4 rad/s² for t=3 s. Find its angular displacement θ.
HardCommon mistake
Using these equations when angular acceleration is NOT constant — like the linear suvat equations, they only apply during a phase of constant α.
Quick Review
- ω = ω₀+αt; θ = ω₀t+½αt²; ω² = ω₀²+2αθ.
- Only valid for constant angular acceleration.
- Directly mirror the linear "suvat" equations.