AkCalculators

Rotational kinematics — equations, examples and practical guidance

Rotational motion with constant angular acceleration is the angular analogue of linear uniformly accelerated motion. Many problems in mechanics — wheels, flywheels, rotating turbines, and rotating machinery — are conveniently described using four primary scalar quantities:

• Angular displacement Δθ (radians)
• Angular velocity ω (rad/s) — instantaneous rate of change of angle
• Initial angular velocity ω₀ (rad/s)
• Angular acceleration α (rad/s²) — rate of change of angular velocity
• Time t (s)

When angular acceleration α is constant, three primary relations connect these variables (direct rotational analogues of linear kinematics):

1. Angular velocity as a function of time
   ω = ω₀ + α t
2. Angular displacement
   Δθ = ω₀ t + ½ α t²
3. Velocity–displacement relation
   ω² = ω₀² + 2 α Δθ

These relations are sufficient to solve for an unknown when two other independent quantities are provided. For example, given ω₀ and α you can compute ω(t) or Δθ(t) for a chosen time; given ω and ω₀ and Δθ, you can find α using the third relation.

Units and conversion

Use SI units whenever possible. Angular displacements must be in radians when used directly in these equations. If you prefer degrees, convert degrees → radians before computing (rad = deg × π/180). The calculator supports degree inputs and converts them internally.

Worked examples

Example 1 — Constant acceleration from rest:
A flywheel starts from rest (ω₀ = 0) and accelerates at α = 2.0 rad/s² for t = 5.0 s. Final angular velocity ω = ω₀ + α t = 0 + 2×5 = 10 rad/s. Angular displacement Δθ = ω₀ t + ½ α t² = 0 + 0.5×2×25 = 25 rad (≈ 4.0 revolutions).

Example 2 — Finding α from velocities and displacement:
Suppose a wheel's angular velocity increases from 3.0 rad/s to 9.0 rad/s while sweeping Δθ = 18 rad. Use ω² = ω₀² + 2 α Δθ → α = (ω² − ω₀²) / (2 Δθ) = (81 − 9) / 36 = 2.0 rad/s².

Signs and directions

Angular quantities are signed. Choose a positive direction (commonly counterclockwise). If rotation direction reverses, signs of ω, ω₀ and α will reflect that. Be careful when interpreting Δθ: whether it represents net rotation or absolute angular distance depends on the sign and context.

When constant-α is valid

These equations assume α is constant over the interval. Many real-world systems approximate constant angular acceleration for short intervals (for example, near-uniform torque, or controlled motor ramping). For variable α(t), integrate numerically: ω(t) = ω₀ + ∫ α(t) dt and Δθ = ∫ ω(t) dt.

Practical notes

• When solving, verify units and convert degrees-to-radians where needed.
• If multiple variables are provided, use the combination that avoids solving a quadratic when possible (e.g., use ω = ω₀ + α t rather than solving Δθ = ω₀ t + ½ α t² for t if a linear relation suffices).
• For solving time from Δθ = ω₀ t + ½ α t², you may need to solve a quadratic equation; check both roots and use the physically meaningful positive root (and consistent sign).
• Export iteration tables and steps for lab reports using CSV export.

This calculator is intended for students and engineers to rapidly check rotational kinematics problems, prepare lab notes, and print step-by-step derivations.