Unit 6 · Topic 01

Angular kinematics

Rotation about a fixed axis is described by θ, ω, and α. An axial sign convention is set with the right-hand rule.

Unit 6Angular kinematicsOpen navigation

Concept 01

Describing rotation

Essential The minimum you should retain

Angular coordinate θ specifies orientation, and its change is angular displacement.

UnderstandInterpret and connect

Every point in a rigid body turns through the same angle in the same interval.

DeepenFormulation and conditions

One revolution is 2π rad; radians connect arc and angle without extra factors.

ExploreConnections for further study

The axial direction associated with rotation follows the right-hand rule.

Worked example

Constant angular acceleration

A rotor has ω₀=2.0 rad/s and α=3.0 rad/s² for 4.0 s.

Given
  • ω₀=2.0 rad/s
  • α=3.0 rad/s²
  • t=4.0 s
Target

Find ω_f and Δθ.

  1. Angular speed

    ω_f=ω₀+αt=14 rad/s.

  2. Angle

    Δθ=ω₀t+(1/2)αt².

  3. Substitution

    Δθ=2(4)+(1/2)(3)(16)=32 rad.

  4. Interpretation

    Positive α increases initially positive ω.

Conclusion

ω_f=14 rad/s and Δθ=32 rad.

Mathematical relation

Angular definitions

ω=dθdt,α=dωdt \omega=d\theta/dt,\quad \alpha=d\omega/dt
Represents

Instantaneous angular rates.

Physical interpretation

ω changes θ and α changes ω.

DeepenVariables, conditions, and checks

Variables

θ
angular position; usual unit: rad
ω
angular velocity; usual unit: rad/s
α
angular acceleration; usual unit: rad/s²

Conditions of application

  • Axis and sign are declared.

Dimensional check

rad, rad/s, and rad/s².

A schematic disk relates rotation sense to the z axis.axis+z ⊙right hand

Counterclockwise rotation viewed from +z corresponds to an axial vector toward +z by the right-hand rule.

Concept 02

Angular rates

Essential The minimum you should retain

ω measures the change of θ and α measures the change of ω.

UnderstandInterpret and connect

Positive ω with negative α describes positive rotation slowing at that instant.

DeepenFormulation and conditions

ω=dθ/dt \omega=d\theta/dt and α=dω/dt \alpha=d\omega/dt are instantaneous rates in rad/s and rad/s².

ExploreConnections for further study

The sign of α tells how ω changes, not the current rotation direction by itself.

Concept 03

Uniformly accelerated rotation

Essential The minimum you should retain

With constant α, ω changes linearly in time.

UnderstandInterpret and connect

Area under ω(t) represents angular displacement.

DeepenFormulation and conditions

Constant-α equations mirror constant-acceleration linear kinematics.

ExploreConnections for further study

Using ω²=ω₀²+2αΔθ can avoid time.

Mathematical relation

Constant angular acceleration

ω=ω0+αt;Δθ=ω0t+12αt2;ω2=ω02+2αΔθ \omega=\omega_0+\alpha t;\ \Delta\theta=\omega_0t+\tfrac12\alpha t^2;\ \omega^2=\omega_0^2+2\alpha\Delta\theta
Represents

Angular evolution for constant α.

Physical interpretation

Signs follow the axial convention.

DeepenVariables, conditions, and checks

Variables

t
time; usual unit: s
Δθ
angular displacement; usual unit: rad

Conditions of application

  • α remains constant over the interval.

Dimensional check

Every angular term has radians.

Area under ω(t) is Δθ

An increasing straight line of angular velocity versus time.slope α
  • ω=2+3t

The constant slope is α, and algebraic area between two times gives angular displacement.

Concept 04

Revolutions and radians

Essential The minimum you should retain

Frequency counts revolutions per second and ω=2πf.

UnderstandInterpret and connect

Converting rpm requires minutes to seconds and revolutions to radians.

DeepenFormulation and conditions

The radian is dimensionless physically, though retained in pedagogical display.

ExploreConnections for further study

Angular phase can grow through many turns while orientation repeats.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Assuming every point in a rigid body has the same linear speed.

They share ω; v=rω v=r\omega depends on r.

Using degrees directly in s=rθ s=r\theta .

The direct form requires θ in radians.