Moment of Inertia Examples

Why This Matters

In AP Physics C: Mechanics, you're not just being tested on whether you can plug numbers into formulas—you're being tested on whether you understand why mass distribution affects rotational motion. Moment of inertia is the rotational analog of mass: it tells you how resistant an object is to angular acceleration. The key insight? Where the mass is located matters just as much as how much mass there is. Mass farther from the rotation axis contributes more to moment of inertia, which is why a hollow sphere is harder to spin up than a solid one of the same mass and radius.

These examples demonstrate core principles you'll encounter repeatedly: mass distribution effects, axis location dependence, and the parallel axis theorem. When you see a rolling motion problem or a rotating system on the exam, you need to instantly recognize which formula applies and—more importantly—why. Don't just memorize $I = \frac{1}{2}mr^2$ for a disk; understand that this coefficient reflects mass being distributed from the center outward. That conceptual understanding will save you on FRQs where you need to justify your approach or compare different objects.

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Solid Objects: Mass Distributed Throughout the Volume

When mass fills the entire volume of an object, more of it sits closer to the rotation axis. This concentrates mass near the center, resulting in smaller moment of inertia coefficients compared to hollow objects of the same shape.

Solid Sphere

• Formula: $I = \frac{2}{5}mr^2$—the coefficient $\frac{2}{5}$ reflects mass distributed throughout the volume, with more mass near the center
• Rolling motion favorite—solid spheres appear constantly in energy conservation problems where you must account for both translational and rotational kinetic energy
• Smallest coefficient among spheres—compare to hollow sphere's $\frac{2}{3}$; this difference determines which object reaches the bottom of a ramp first

Solid Cylinder (Axis Through Center)

• Formula: $I = \frac{1}{2}mr^2$—identical to a circular disk because both have the same radial mass distribution
• Wheels and axles—this is your go-to formula for any problem involving rolling cylinders, pulleys, or rotating shafts
• Intermediate coefficient—falls between the solid sphere ($\frac{2}{5}$) and hollow cylinder ($1$), making it useful for comparison questions

Circular Disk (Axis Through Center)

• Formula: $I = \frac{1}{2}mr^2$—mathematically identical to solid cylinder; the thickness doesn't affect the coefficient, only the radial distribution matters
• Rotating machinery standard—flywheels, turntables, and grinding wheels all use this formula
• Two-dimensional simplification—treat any flat circular object rotating about its center with this expression

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Hollow Objects: Mass Concentrated at the Surface

When mass is pushed to the outer edge or surface, every bit of it sits at maximum distance from the rotation axis. This maximizes moment of inertia for a given mass and radius.

Hollow Sphere (Thin Spherical Shell)

• Formula: $I = \frac{2}{3}mr^2$—larger coefficient than solid sphere because all mass sits at radius $r$
• Surface concentration—think of a basketball or tennis ball; the shell structure means no mass near the center
• Rolling comparison essential—hollow spheres accelerate more slowly down ramps than solid spheres of equal mass and radius

Hollow Cylinder (Thin Cylindrical Shell)

• Formula: $I = mr^2$—coefficient of 1 is the maximum possible; all mass sits at the outer radius
• Pipes and rings—any thin-walled tube rotating about its central axis uses this formula
• Hoop equivalent—a thin hoop rotating about its central axis perpendicular to the plane has the same formula

Thin Hoop (Axis Through Diameter)

• Formula: $I = \frac{1}{2}mr^2$—note this is for rotation about a diameter, not the central axis perpendicular to the hoop
• Axis orientation matters—the same hoop has $I = mr^2$ about its central axis but $I = \frac{1}{2}mr^2$ about a diameter
• Symmetry application—mass distribution relative to the chosen axis determines the coefficient, not the object's shape alone

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Rods: Axis Location Changes Everything

For elongated objects, where you place the rotation axis dramatically affects moment of inertia. Moving the axis from center to end increases $I$ because more mass is now farther from the axis.

Thin Rod (Axis Through Center)

• Formula: $I = \frac{1}{12}mL^2$—the smallest moment of inertia for a rod because mass is symmetrically distributed on both sides
• Balanced rotation—equal amounts of mass at equal distances on either side of the axis
• Pendulum reference—physical pendulum problems often start here before applying the parallel axis theorem

Thin Rod (Axis Through End)

• Formula: $I = \frac{1}{3}mL^2$—exactly four times larger than the center-axis case ($\frac{1}{3} = 4 \times \frac{1}{12}$)
• Parallel axis verification—you can derive this from the center formula: $I = \frac{1}{12}mL^2 + m\left(\frac{L}{2}\right)^2 = \frac{1}{3}mL^2$
• Lever and door problems—any rod pivoting about one end, like a swinging door or baseball bat, uses this formula

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Extended Shapes and the Parallel Axis Theorem

Some objects have more complex geometry, and many problems require finding moment of inertia about an axis that doesn't pass through the center of mass. The parallel axis theorem bridges these situations.

Rectangular Plate (Axis Through Center, Perpendicular to Plate)

• Formula: $I = \frac{1}{12}m(a^2 + b^2)$—where $a$ and $b$ are the plate's length and width
• Pythagorean structure—the sum of squares reflects contributions from both dimensions to the radial distance
• Engineering applications—rotating panels, doors about central pivots, and flat machinery components

Parallel Axis Theorem

• Formula: $I = I_{cm} + md^2$—where $d$ is the perpendicular distance between the center-of-mass axis and the new parallel axis
• Always increases $I$—the $md^2$ term is always positive, so moment of inertia about any axis other than the center of mass is larger
• Problem-solving essential—this theorem lets you find $I$ for any parallel axis once you know $I_{cm}$, appearing in nearly every complex rotation problem

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Quick Reference Table

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Self-Check Questions

1. Two objects—a solid sphere and a hollow sphere—have the same mass and radius. Which reaches the bottom of a frictionless ramp first, and why does the moment of inertia formula explain this?

2. A thin rod can rotate about its center or about one end. Using the parallel axis theorem, show mathematically why $I_{end} = 4 \times I_{center}$ for this specific geometry.

3. Which two objects in this guide share the same moment of inertia formula, and what physical principle explains why they're equivalent?

4. Compare and contrast a solid cylinder and hollow cylinder rolling down an incline. If both have mass $m$ and radius $r$, which has greater translational speed at the bottom? Explain using energy conservation.

5. An FRQ gives you a disk rotating about its center and asks you to find its moment of inertia about an axis at the edge of the disk, parallel to the original axis. Write the expression for this new moment of inertia and identify which theorem you applied.