Mini Physics
Part of Mini EducationSwitch subject
SearchBookmarks

UY1: Moment Of Inertia

Moment of inertia from first principles, axis theorems, and how to use I in rotational dynamics.

Reading settings
  • University Physics Year 1
On this page

Enlarged diagram

Cluster role + pathway links

This page is the core concept page for inertia and axis choice in the UY1 mechanics pathway.

How to choose the right axis workflow

For most UY1 questions, solve in this order:

  1. Name the physical axis in words (“about axle through centre”, “about one end”, etc.).
  2. Write/recall I for a CM axis first.
  3. Apply axis theorems only if the asked axis differs.

This avoids the most common mistake: using a correct formula about the wrong axis.

At a glance

  • Prerequisites: rotational kinematics, integration, center of mass.
  • Outcomes: compute and interpret I for different axes and mass distributions.
  • Key result:
    I = ∑ᵢ mᵢ rᵢ² or I = ∫ r² dm
Common traps (axis + geometry)
  • I is not intrinsic like mass: it depends on the chosen axis.
  • In I = ∫ r² dm, r is the perpendicular distance to the axis, not “distance from the origin”.
  • Don’t mix I_CM (about a CM axis) with I about a shifted axis: use the parallel-axis theorem.

Setup

Model and assumptions:

  • Rigid body rotating about a fixed axis.
  • Distance r means perpendicular distance from each mass element to that axis.

Useful definitions:

Kᵣₒₜ = (1/2)Iω², k = square root of (I/M)

where k is radius of gyration.

Geometry reminder (common choice):

  • If the rotation axis is the z-axis through the origin, then the perpendicular distance is r = square root of (x² + y²), so
    Iz = ∫ (x² + y²) dm
  • If the axis is shifted or tilted, compute the perpendicular distance to that axis first, then integrate.

Core method

Start from particle kinetic energy and sum over body:

K = ∑ᵢ(1/2)mᵢ vᵢ² = ∑ᵢ(1/2)mᵢ(rᵢω)² = (1/2)ω²∑ᵢ mᵢ rᵢ²

So define

I = ∑ᵢ mᵢ rᵢ²

For continuous mass:

I = ∫ r² dm

Two key theorems:

  1. Parallel axis theorem:

    I = I_CM + MD²

    D is distance between parallel axes.

  2. Perpendicular axis theorem (planar lamina only):

    Iz = Iₓ + Iy

Intuition: I is a weighted average of r², so moving mass farther out increases I strongly.

  • Same mass and same outer radius R: a hoop has larger I than a solid disk because more mass sits near r ≈ R.
  • Same M,R: a thin spherical shell has larger I than a solid sphere for the same reason.
  • Radius of gyration k = square root of (I/M) is a single “effective radius” that tells you where the mass “acts” for rotation.

Worked example

Worked example 1

Parallel-axis theorem for a rod

Problem

A uniform rod of mass M = 3.0 kg and length L = 1.2 m rotates about a perpendicular axis through one end. Find its moment of inertia.

Show full solution

  1. Start from the centre value

    Method

    Use I_CM = (1/12)ML².

    Reason

    The parallel-axis theorem needs the moment of inertia about a parallel axis through the centre of mass.

    Working

    Answer

    I_CM = (1/12)ML².

  2. Shift the axis

    Method

    Add MD² with D = L/2.

    Reason

    The end is half a length from the centre.

    Working

    Answer

    I = (1/12)ML² + M(L/2)² = (1/3)ML².

  3. State the answer

    Working

    Answer

    I = (1/3)(3.0)(1.2)² = 1.44 kg m².

Practice set

Check your understanding 1

A 2.0 kg point mass is 0.40 m from an axis. Find its moment of inertia.

Show hint

For a single particle, I = mr².

Show answer

I = mr² = (2.0)(0.40)² = 0.32 kg m².

Check your understanding 2

Compare the moment of inertia of a hoop with that of a uniform disc of the same mass and radius, about their axes.

Show hint

Write both standard formulas.

Show answer

Iₕₒₒₚ = MR² and I_disc = (1/2)MR², so the hoop’s is twice as large: all its mass is at radius R.

Check your understanding 3

A flat lamina has Iₓ = 0.30 and Iy = 0.50 kg m² about two perpendicular axes in its plane. Find Iz about the perpendicular axis through their intersection.

Show hint

Use the perpendicular-axis theorem for a planar body.

Show answer

By the perpendicular-axis theorem, Iz = Iₓ + Iy = 0.80 kg m².

Summary and next steps

Moment of inertia quantifies rotational resistance and is axis-dependent. Compute it by summing/integrating r² dm, then use axis theorems to shift or combine results.

Revision priority:

  1. Memorize a compact table of standard I_CM values you actually use.
  2. Practice one parallel-axis and one perpendicular-axis application.
  3. Cross-check with dynamics/energy on a rolling or torque problem.

Spotted an error? Report a correction

Syllabus and review details

No official syllabus alignment is listed for this lesson.

Last reviewed:

Back to top