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Mechanics · Physics

Rotation & Angular Dynamics

Go beyond straight-line motion: learn torque, moment of inertia, rolling, and angular momentum — the tools behind wheels, machines, sports, and orbital motion.

Rotation & Angular Dynamics topics

Core topics in this track

Work through in order for mastery, or jump to what you need right now.

Start here
Rotational Kinematics
Describe rotation using angular variables and connect them to linear motion.
  • Rotation as angular motion of rigid bodies
  • Angular position and reference axes
  • Right-hand rule for rotation direction
  • Relationship between angular and linear variables
Start topic Multiple nano-lessons inside.
Core variables
Angular Displacement, Velocity, and Acceleration
Build fluency with θ, ω, α and constant angular acceleration.
  • Angular displacement (θ)
  • Angular velocity (ω)
  • Angular acceleration (α)
  • Constant angular acceleration equations
  • Tangential vs angular quantities
Start topic Connect rotation to linear motion.
Rigid bodies
Rigid Body Rotation
Understand rotation about a fixed axis and how it differs from particle motion.
  • Definition of a rigid body
  • Rotation about a fixed axis
  • Comparison with particle motion
  • Translational vs rotational motion
Start topic Essential setup for dynamics.
Key concept
Torque
The rotational “push”: direction matters and depends on the moment arm.
  • Definition of torque as a vector
  • Lever arm and moment arm
  • Direction of torque (right-hand rule)
  • Net torque and rotational equilibrium
Start topic Opens the door to rotational dynamics.
Core law
Rotational Dynamics (τ = Iα)
Newton’s second law for rotation: how torque causes angular acceleration.
  • Newton’s second law for rotation
  • Moment of inertia as rotational mass
  • Applying τ = Iα to rigid bodies
  • Comparison with F = ma
Start topic From forces to torques.
Quantity
Moment of Inertia
How mass distribution controls rotational resistance.
  • Definition and physical meaning
  • Moment of inertia for point masses
  • Calculating I for common shapes
  • Dependence on mass distribution
Start topic A must for τ = Iα problems.
Theorems
Parallel-Axis & Perpendicular-Axis Theorems
Compute I about different axes quickly and correctly.
  • Parallel-axis theorem
  • Perpendicular-axis theorem
  • Choosing the correct axis
  • Applications to composite objects
Start topic Work smarter with I.
Energy
Rotational Kinetic Energy
Rotation stores energy — and rolling objects store both translational and rotational KE.
  • Definition of rotational kinetic energy
  • Comparison with translational KE
  • Total kinetic energy of rolling objects
  • Energy storage in rotation
Start topic Energy methods get powerful here.
Work & power
Work and Power in Rotational Motion
Use torque to compute rotational work, energy change, and power output.
  • Work done by torque
  • Angular work-energy theorem
  • Rotational power
  • Torque–power–angular velocity relationship
Start topic Connect rotation to real machines.
Rolling
Rolling Motion Without Slipping
Use the rolling constraint to relate v and ω and solve rolling energy problems.
  • Rolling constraint condition
  • Relationship between linear and angular speed
  • Energy analysis of rolling motion
  • Rolling on inclines
Start topic Classic exam topic.
Momentum
Angular Momentum
The rotational cousin of linear momentum — crucial for conservation and gyroscopes.
  • Definition of angular momentum
  • Angular momentum of a particle
  • Angular momentum of a rigid body
  • Relation between torque and angular momentum
Start topic From τ to dL/dt.
Conservation
Conservation of Angular Momentum
When external torque is zero, angular momentum stays constant.
  • Conditions for conservation
  • Isolated rotational systems
  • Applications (skater, collapsing star)
  • Comparison with linear momentum conservation
Start topic Big idea, big applications.
Rigid-body L
Angular Momentum of a Rigid Body
Explore how L depends on axis choice and symmetry.
  • Angular momentum about different axes
  • Rotational symmetry
  • Connection to moment of inertia
  • System-level angular momentum
Start topic Axis choice matters.
Preview
Gyroscopic Motion (Intro)
A qualitative introduction to precession and torque-induced rotation.
  • Gyroscope basics
  • Precession
  • Torque-induced rotation
  • Qualitative real-world examples
Start topic Conceptual but powerful.
Practice
Practice & Exercises
Reinforce rotation with mixed problems and skill builders.
  • Angular kinematics drills (θ, ω, α)
  • Torque and rotational equilibrium sets
  • τ = Iα and rolling motion problems
  • Moment of inertia + axis theorem practice
  • Angular momentum and conservation sets
  • Exam-style mixed rotation questions
Open practice sets Move from familiarity to fluency.