Moment of Inertia Formulas

Why This Matters

Moment of inertia isn't just another formula to memorize—it's the rotational equivalent of mass, and it determines how objects respond to torques in everything from spinning flywheels to swinging crane arms. You're being tested on your ability to select the correct formula for a given geometry, apply axis theorems to shift reference points, and understand why mass distribution matters more than total mass when it comes to rotation.

The key insight is this: mass farther from the rotation axis contributes more to rotational inertia. That's why a hollow cylinder resists angular acceleration more than a solid one of the same mass, and why the parallel axis theorem always adds inertia when you move away from the centroid. Don't just memorize the fractions in front of each formula—understand what geometric principle each one represents and when to apply the powerful shortcut theorems.

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Fundamental Axis Theorems

These two theorems are your problem-solving workhorses. They let you calculate moment of inertia about any axis once you know the centroidal value, saving you from integrating every time.

Parallel Axis Theorem

• $I = I_{cm} + md^2$—shifts the rotation axis parallel to the centroidal axis by distance $d$
• Always adds inertia because $md^2$ is always positive; moving away from the centroid increases rotational resistance
• Essential for composite bodies where you must reference all parts to a common axis for summation

Perpendicular Axis Theorem

• $I_z = I_x + I_y$—relates the out-of-plane moment of inertia to two in-plane axes
• Only valid for planar (2D) bodies like thin plates, discs, and laminae
• Powerful shortcut when one in-plane inertia is known or easily calculated by symmetry

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Cylindrical and Circular Geometries

Rotating machinery, shafts, wheels, and pipes all fall into this category. The key variable is how mass is distributed radially from the central axis.

Solid Cylinder About Central Axis

• $I = \frac{1}{2}mr^2$—the baseline formula for shafts, rollers, and solid wheels
• Coefficient of 1/2 reflects that mass is distributed from center to edge, averaging the $r^2$ contribution
• Same formula applies to solid discs rotating about the axis perpendicular to their face

Hollow Cylinder About Central Axis

• $I = \frac{1}{2}m(R_{outer}^2 + R_{inner}^2)$—accounts for the missing inner mass
• Higher inertia than solid cylinder of same mass because remaining mass sits farther from the axis
• Critical for pipes, tubes, and flywheels where weight savings and high inertia are both desirable

Circular Disc About Diameter

• $I = \frac{1}{4}mr^2$—rotation about an axis in the plane of the disc
• Half the central-axis value because mass is distributed closer to this axis on average
• Verify with perpendicular axis theorem: $I_z = I_x + I_y$ gives $\frac{1}{2}mr^2 = 2 \times \frac{1}{4}mr^2$ ✓

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Slender Rods and Beams

Thin rods are idealized as having all mass along a line. The axis location relative to the rod's length dramatically changes the result.

Thin Rod About Center

• $I = \frac{1}{12}mL^2$—rotation about the centroidal axis perpendicular to the rod's length
• Smallest possible value for a rod because mass is symmetrically distributed about the axis
• Starting point for parallel axis calculations when the pivot is elsewhere

Thin Rod About End

• $I = \frac{1}{3}mL^2$—rotation about an axis through one end of the rod
• Four times the centroidal value because you've shifted the axis by $d = L/2$
• Derived via parallel axis theorem: $\frac{1}{12}mL^2 + m(L/2)^2 = \frac{1}{3}mL^2$

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Rectangular Plates

Flat rectangular shapes appear in structural analysis, doors, and panels. Pay attention to which dimension defines the rotation axis.

Rectangular Plate About Centroidal Axis

• $I = \frac{1}{12}m(b^2 + h^2)$—for rotation about the axis perpendicular to the plate's face
• Both dimensions contribute because mass extends in both the $b$ and $h$ directions from the central axis
• For rotation about an in-plane centroidal axis, use $I = \frac{1}{12}mh^2$ (about the axis parallel to width $b$)

Rectangular Plate About Edge

• $I = \frac{1}{3}mh^2$—rotation about an edge parallel to the width
• Same factor of 1/3 as the rod about its end—the geometry is analogous in that direction
• Apply parallel axis theorem from centroidal: $\frac{1}{12}mh^2 + m(h/2)^2 = \frac{1}{3}mh^2$

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Spheres and Triangular Shapes

These geometries appear less frequently but are still fair game, especially for conceptual questions about mass distribution.

Solid Sphere About Diameter

• $I = \frac{2}{5}mr^2$—valid for rotation about any diameter due to symmetry
• Coefficient of 2/5 is smaller than a solid cylinder's 1/2 because more mass concentrates near the center
• Important for rolling motion problems where both translation and rotation occur simultaneously

Triangular Plate About Base

• $I = \frac{1}{6}mh^2$—rotation about the base of a triangular lamina (note: original formula had an error)
• Mass distribution is non-uniform along the height, with more area near the base
• Less common on exams but tests your understanding of how shape affects inertia

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Quick Reference Table

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Self-Check Questions

1. A solid cylinder and a hollow cylinder have the same mass and outer radius. Which has the greater moment of inertia about the central axis, and why?

2. You know the moment of inertia of a thin rod about its center is $\frac{1}{12}mL^2$. Using the parallel axis theorem, derive the moment of inertia about one end.

3. The perpendicular axis theorem applies to which type of bodies—3D solids, planar laminae, or both? What's the key limitation?

4. Compare and contrast: A circular disc rotating about its central axis versus rotating about a diameter. Which has greater inertia, and by what factor?

5. An FRQ shows a composite body made of a rectangular plate with a rod attached at its edge. What strategy would you use to find the total moment of inertia about the plate's opposite edge?