§10.2 — Moment of Inertia

A simple question: which is harder to spin? (1) A houseplant or (2) a steel wheel of the same mass? Obviously the wheel. But why? Both have the same mass; the difference is the distribution of that mass. In the wheel, mass is far from the axis. In the plant, mass is near the center. Moment of inertia measures this difference.

Definition of Moment of Inertia

Moment of inertia (\( I \)) is an object's resistance to starting or stopping rotation:

\[ I = \sum m_i r_i^2 \]

where:

• \( m_i \) = mass of a small piece
• \( r_i \) = distance of that piece from the rotation axis

For a continuous object:

\[ I = \int r^2 \, dm \]

Unit: kilogram–meter squared (kg·m²)

Key Point

Moment of inertia depends on the rotation axis. Change the axis, and \( I \) changes. For example, a rod rotating about its center has a smaller \( I \) than the same rod rotating about its end.

Why Moment of Inertia Matters

Angular momentum and rotational kinetic energy depend directly on \( I \):

\[ L = I\omega \]

\[ K_{\text{rot}} = \frac{1}{2}I\omega^2 \]

Just as mass resists changes in linear velocity, moment of inertia resists changes in angular velocity.

Moment of Inertia for Standard Shapes

For regular geometric shapes, the moment of inertia about various axes has been calculated:

Thin Rod, Axis Perpendicular Through Center

Mass \( M \), length \( L \), axis perpendicular through midpoint:

\[ I = \frac{1}{12}ML^2 \]

If the axis passes through the end of the rod:

\[ I = \frac{1}{3}ML^2 \]

Example: A rod of 2 kg and length 1 m. Rotating about its center: \[ I = \frac{1}{12} \times 2 \times 1^2 = 0.167 \text{ kg·m}^2 \]

Rotating about its end: \[ I = \frac{1}{3} \times 2 \times 1^2 = 0.667 \text{ kg·m}^2 \]

Four times larger! Because mass is farther from the axis.

Solid Cylinder or Disk, Central Axis

Mass \( M \), radius \( R \):

\[ I = \frac{1}{2}MR^2 \]

Solid Sphere, Axis Through Center

Mass \( M \), radius \( R \):

\[ I = \frac{2}{5}MR^2 \]

Thin Hoop, Axis Through Center

Mass \( M \), radius \( R \):

\[ I = MR^2 \]

All mass is at maximum distance, so \( I \) is largest!

Comparison Table

For objects of equal mass and radius (or length), which is hardest to spin?

Shape	Moment of Inertia	Relative
Hoop	\( I = MR^2 \)	1.0
Hollow cylinder	\( I \approx 0.9MR^2 \)	0.9
Solid cylinder	\( I = 0.5MR^2 \)	0.5
Solid sphere	\( I = 0.4MR^2 \)	0.4

Result: Hoop is hardest, sphere is easiest.

Parallel Axis Theorem

If \( I_{\text{cm}} \) is the moment of inertia about an axis through the center of mass, and you want the moment about a parallel axis at distance \( d \) away:

\[ I = I_{\text{cm}} + Md^2 \]

where \( M \) is the total mass.

Example

Rod of 2 kg, length 1 m. About center: \( I_{\text{cm}} = \frac{1}{12} \times 2 \times 1 = 0.167 \) kg·m²

About end (distance \( d = 0.5 \) m from center):

\[ I = 0.167 + 2 \times (0.5)^2 = 0.167 + 0.5 = 0.667 \text{ kg·m}^2 \]

This agrees with the direct formula \( \frac{1}{3}ML^2 = 0.667 \) kg·m² perfectly! ✓

Moment of Inertia and Scaling

If you enlarge an object (same material, just bigger):

\[ I \propto R^2 \]

Double the radius, and \( I \) quadruples!

Practical example: Basketball (radius 12 cm) versus tennis ball (radius 3 cm). If they had equal mass density, the basketball's moment would be 16 times larger!

Intrinsic Angular Momentum

Electrons inside atoms have intrinsic angular momentum (spin), like a sphere spinning inside itself. The moment of inertia of this spin is determined by the electron's mass, but the physics is fundamentally quantum, not classical.

What You Should Know

• Definition of moment: \( I = \sum m_i r_i^2 \)
• \( I \) depends on the axis
• Standard formulas: rod (\( \frac{1}{12}ML^2 \)), cylinder (\( \frac{1}{2}MR^2 \)), sphere (\( \frac{2}{5}MR^2 \)), hoop (\( MR^2 \))
• Parallel axis theorem: \( I = I_{\text{cm}} + Md^2 \)
• Links to momentum and energy: \( L = I\omega \) and \( K = \frac{1}{2}I\omega^2 \)

Preview of §10.3

Now that we know what moment of inertia is, we can reach Newton's second law for rotation: torque causes angular acceleration, but the amount depends on \( I \).

📚 See also: Halliday Vol 1, Ch 10, §10.2 — Moment of inertia. 🔗 Reference: §9.6 (Rigid body dynamics) — first introduction to moment of inertia in the context of angular momentum.

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