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UY1: Rotational Kinematics

Constant-angular-acceleration equations, derivation steps, and problem-solving checks for rotational motion.

Reading settings
  • University Physics Year 1
On this page

Enlarged diagram

Cluster role + pathway links

This page is the core concept page for constant-angular-acceleration tools in the UY1 mechanics pathway.

When this model is valid

Use the equations on this page only on intervals where α is approximately constant. In mixed-motion questions, split the timeline into separate constant-α segments and apply the equations per segment.

If a question gives torque as a function of angle/time, move to Torque & Angular Acceleration and use dynamics first.

At a glance

  • Prerequisites: rigid-body rotation, basic integration, sign convention for ω.
  • Outcomes: derive and apply the three constant-α equations.
  • Key result:
    ω = ω₀ + α t, θ = θ₀ + ω₀ t + (1/2)α t², ω² = ω₀² + 2α(θ-θ₀)
Common traps (SUVAT for rotation)
  • These equations require constant α over the interval. - Angles must be in radians (degrees will break the calculus-based derivation). - Keep your sign convention consistent: negative α means slowing down in the chosen positive sense.

Setup

Model:

  • Rotation about a fixed axis.
  • Angular acceleration is constant over the time interval.

Signs and units:

  • Choose one positive direction (usually counterclockwise).
  • θ in rad, ω in rad s⁻¹, α in rad s⁻².

Core method

Start from constant angular acceleration:

α = dω/dt = const

Integrate from 0 to t:

∫_ω₀^ωdω = ∫₀^tα dt ⇒ ω = ω₀ + α t

Then use ω = dθ/dt:

dθ/dt = ω₀ + α t

Integrate again:

θ-θ₀ = ω₀ t + (1/2)α t²

To remove t, combine the first two equations:

ω² = ω₀² + 2α(θ-θ₀)

Non-trivial point: this is the exact rotational analog of linear SUVAT, with the mapping

θ ↔ x, ω ↔ v, α ↔ a

Useful interpretation (often faster than memorizing):

  • For constant α, the average angular speed is
    ω_avg = (ω₀ + ω)/2
    so angular displacement is Δθ = ω_avgt.
  • This is the rotational twin of Δ x = v_avgt in linear motion.

Worked example

Worked example 1

A flywheel speeding up uniformly

Problem

A flywheel speeds up uniformly from ω₀ = 8 rad s⁻¹ to ω = 20 rad s⁻¹ in 4.0 s. Find its angular acceleration and angular displacement.

Show full solution

  1. Find the angular acceleration

    Method

    Use α = (ω-ω₀)/t.

    Reason

    Uniform speeding-up means constant α.

    Working

    Answer

    α = (20-8)/4.0 = 3.0 rad s⁻².

  2. Find the angular displacement

    Method

    Use Δθ = ω₀t + (1/2)α t².

    Reason

    The constant-acceleration equations carry over with θ, ω and α.

    Working

    Answer

    Δθ = (8)(4.0) + (1/2)(3.0)(4.0)².

  3. State the answer

    Working

    Answer

    α = 3.0 rad s⁻² and Δθ = 56 rad.

Practice set

Check your understanding 1

A disc starts from rest with α = 5 rad s⁻² for 3 s. Find ω.

Show hint

Use ω = ω₀ + α t.

Show answer

ω = ω₀ + α t = 0 + (5)(3) = 15 rad s⁻¹.

Check your understanding 2

A motor slows uniformly from 30 to 10 rad s⁻¹ in 8 s. Find α.

Show hint

The sign must come out negative.

Show answer

α = (10-30)/8 = -2.5 rad s⁻². The negative sign shows it is slowing down.

Check your understanding 3

A wheel turns through 40 rad while speeding up uniformly from 6 to 14 rad s⁻¹. Find α.

Show hint

Use the equation without time, ω² = ω₀² + 2αΔθ.

Show answer

ω² = ω₀² + 2αΔθ gives 196 = 36 + 2α(40), so α = 160/80 = 2.0 rad s⁻².

Summary and next steps

For constant α, the three kinematics equations are fast and reliable. Choose signs once, check constancy of α, and verify units at each step.

Fast revision route:

  1. Identify known variables from {θ,ω,ω₀,α,t}.
  2. Pick the equation that avoids introducing extra unknowns.
  3. Run one sign/unit sanity check before finalizing.

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Syllabus and review details

No official syllabus alignment is listed for this lesson.

Last reviewed:

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