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Rotation & Angular Dynamics · Physics

How hard is it to spin something? That’s moment of inertia.

Moment of inertia depends on mass distribution and axis choice. Learn its definition, build intuition, and use standard results for common shapes.

This topic

Moment of Inertia

Define I precisely, understand how geometry controls it, and learn standard I values you’ll use repeatedly.

Definition
What moment of inertia measures
Moment of inertia is the rotational analogue of mass. It measures resistance to angular acceleration about a specific axis.
  • Core meaning: “rotational inertia”
  • Axis dependence: same object, different I
  • Units: kg·m² and what they imply
  • Why I matters in τ = Iα and rotational energy
Start nano-lesson Always name the axis explicitly.
Model
Definition from mass distribution
The definition builds I by weighting each mass element by the square of its distance to the axis. Farther mass contributes much more.
  • Discrete masses: I = Σ m r²
  • Continuous bodies: I = ∫ r² dm
  • Why the r² weighting is so important
  • Choosing r correctly for the axis
Start nano-lesson Distance to the axis is the key variable.
Intuition
How shape and radius affect I
Moving mass outward increases I rapidly. This explains why a hollow cylinder can be much harder to spin than a solid disk of the same mass and radius.
  • Concentrated vs spread-out mass
  • Same M, same R, different I for different shapes
  • Qualitative comparisons without calculation
  • Everyday examples: doors, wheels, flywheels
Start nano-lesson Outward mass dominates because of r².
Toolkit
Standard moments of inertia (common shapes)
Many problems rely on standard results for common rigid bodies. Learn the most-used forms and how to recognize which one applies.
  • Point mass at r: I = mr²
  • Solid disk/cylinder about center: I = ½MR²
  • Thin hoop about center: I = MR²
  • Rod about center vs about end (preview)
Start nano-lesson Match the shape and the axis.
Practice
Practice & Exercises
Practice choosing the correct axis and I expression, comparing shapes qualitatively, and using I in basic rotational dynamics.
  • Compute I for simple discrete-mass systems
  • Use standard I values for common bodies
  • Qualitative “which spins easier” comparisons
  • Use I inside τ = Iα in short problems
  • Exam-style moment of inertia sets
Open practice sets Always write the axis in words.