Moment of Inertia Calculator
Choose an object, enter mass and dimensions in SI units, and calculate rotational inertia instantly.
What Is Moment of Inertia?
The moment of inertia is rotational inertia: a measure of how strongly an object resists changes in rotational motion. In linear motion, mass resists acceleration. In rotational motion, moment of inertia does that job. The farther mass is distributed from the axis of rotation, the larger the moment of inertia.
This integral says each tiny mass piece dm contributes according to the square of its distance r from the axis. That square term is why moving mass outward has a dramatic effect.
Why It Matters in Engineering and Physics
- Machine design: flywheels, shafts, gears, and motors depend on rotational dynamics.
- Robotics: joint control needs accurate inertia values for smooth motion planning.
- Vehicles: wheel inertia affects acceleration and braking response.
- Sports biomechanics: athletes change body configuration to control spin.
- Structural analysis: related area moments of inertia affect beam bending (a different but similarly named concept).
Common Formulas (Mass Moment of Inertia)
About standard centroidal axes
- Point mass at distance r: I = m r²
- Thin rod, center: I = (1/12)mL²
- Thin rod, end: I = (1/3)mL²
- Solid disk/cylinder, centerline axis: I = (1/2)mR²
- Thin hoop/ring, centerline axis: I = mR²
- Solid sphere, diameter axis: I = (2/5)mR²
- Thin spherical shell, diameter axis: I = (2/3)mR²
- Rectangular plate, center and perpendicular axis: I = (1/12)m(a²+b²)
How to Calculate Moment of Inertia Step by Step
1) Identify the axis of rotation
This is the most important step. The same object has different inertia values about different axes. Always define axis location and direction before selecting any formula.
2) Pick the matching geometry formula
Use idealized formulas for standard shapes. If geometry is complex, break it into simpler parts and sum contributions.
3) Use consistent SI units
Mass in kilograms (kg), dimensions in meters (m), and resulting inertia in kg·m². Unit mistakes are one of the most common errors in rotational calculations.
4) Apply axis-shift correction if needed
If your axis is shifted from the center of mass, apply the parallel-axis theorem:
where d is the distance between the centroidal axis and the new parallel axis.
Worked Example
Suppose a 4 kg solid disk has radius 0.30 m. About its center axis:
If the axis is moved 0.10 m away in parallel:
Mass Moment vs Area Moment of Inertia
These terms are often confused:
- Mass moment of inertia (kg·m²): used in rotational dynamics (torque, angular acceleration).
- Area moment of inertia (m⁴): used in beam bending and structural stiffness calculations.
Same phrase, different physical meaning, different units, and different applications.
Common Mistakes to Avoid
- Using the wrong axis formula.
- Mixing centimeters with meters.
- Forgetting to square the distance term.
- Skipping parallel-axis correction for off-center rotation.
- Confusing mass moment with area moment in structural problems.
Final Thoughts
Calculating moment of inertia is straightforward once axis definition, geometry, and units are clear. Use the calculator above for quick results, then verify critical designs with hand checks and engineering judgment. For advanced cases, integral methods and CAD tools can provide higher-fidelity inertia tensors for real-world components.