Unit 6 · Topic 03

Moment of inertia

Moment of inertia quantifies mass distribution about a specific axis.

Unit 6Moment of inertiaOpen navigation

Concept 01

Mass and squared distance

Essential The minimum you should retain

For particles, I=Σmᵢrᵢ².

UnderstandInterpret and connect

Moving mass away from the axis increases I without changing total mass.

DeepenFormulation and conditions

Each rᵢ is perpendicular distance to the chosen axis.

ExploreConnections for further study

I is not an axis-free universal property of the object.

Mathematical relation

Moment of inertia

I=Σmiri2=∫r2dm I=\sum_i m_i r_i^2=\int r^2dm
Represents

Mass distribution about an axis.

Physical interpretation

Distant mass is weighted quadratically.

DeepenVariables, conditions, and checks

Variables

I
moment of inertia; usual unit: kg·m²
r_i
perpendicular distance to axis; usual unit: m

Conditions of application

  • The axis is specified.

Dimensional check

kg·m².

Two distributions compare masses near and far from their axes.mass near: lower Imass far: higher ImmmmO_AO_B

With equal total mass and outer radius, placing more mass far from the axis increases I.

Concept 02

Adding a distribution

Essential The minimum you should retain

For a continuous body, I=∫r²dm.

UnderstandInterpret and connect

Density converts dm into an integral over geometry.

DeepenFormulation and conditions

Standard values always correspond to a particular shape and axis.

ExploreConnections for further study

Replacing an extended body by a point mass at its CM usually loses the distribution.

Mathematical relation

Common moments of inertia

Iaro=MR2;Idisco=12MR2;Iesfera=25MR2;Ibarra,cm=112ML2;Ibarra,ext=13ML2 I_{aro}=MR^2;\ I_{disco}=\tfrac12MR^2;\ I_{esfera}=\tfrac25MR^2;\ I_{barra,cm}=\tfrac1{12}ML^2;\ I_{barra,ext}=\tfrac13ML^2
Represents

Standard values for the stated axes.

Physical interpretation

Geometry and axis determine the coefficient.

DeepenVariables, conditions, and checks

Variables

M
total mass; usual unit: kg
R,L
geometric dimensions; usual unit: m

Conditions of application

  • Uniform bodies and described symmetry axes.

Dimensional check

kg·m².

Concept 03

Comparing geometries

Essential The minimum you should retain

Ring, disk, sphere, and rod have different moments for equal mass and size.

UnderstandInterpret and connect

Mass farther out explains why a ring has greater I than a disk.

DeepenFormulation and conditions

Tabulated expressions apply only to the stated axis.

ExploreConnections for further study

Radius of gyration summarizes distribution through I=Mk_g².

Concept 04

Moving the axis

Essential The minimum you should retain

For a parallel axis separated by d from the CM axis, I=I_cm+Md2 I=I_{cm}+Md^2 .

UnderstandInterpret and connect

The added term is never negative.

DeepenFormulation and conditions

The theorem requires parallel axes and known I_cm.

ExploreConnections for further study

The d=0 case returns the central axis.

Worked example

Rod about a displaced axis

A uniform 3.0 kg, 1.20 m rod rotates about an axis 0.30 m from its center.

Given
  • M=3.0 kg
  • L=1.20 m
  • d=0.30 m
Target

Find I.

  1. Central axis

    I_cm=(1/12)ML².

  2. Central value

    I_cm=0.36 kg·m².

  3. Parallel axes

    I=I_cm+Md2 I=I_{cm}+Md^2 =0.36+3(0.30²).

  4. Result

    I=0.63 kg·m².

Conclusion

I=0.63 kg·m².

Mathematical relation

Parallel-axis theorem

I=Icm+Md2 I=I_{cm}+Md^2
Represents

Moment about a displaced parallel axis.

Physical interpretation

Moving the axis adds Md²; it is never subtracted.

DeepenVariables, conditions, and checks

Variables

d
separation between parallel axes; usual unit: m

Conditions of application

  • The known axis crosses the CM and both axes are parallel.

Dimensional check

kg·m².

A body shows its center-of-mass axis and a parallel axis separated by d.bodydCM axisparallel axisCM

A parallel axis displaced by d has I=I_cm+Md2 I=I_{cm}+Md^2 .

Concept review

Common errors

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

Treating I as a function of mass alone.

It also depends on distribution and axis.

Using I=I_cm-Md².

For parallel axes, I=I_cm+Md2 I=I_{cm}+Md^2 .