Moment of Inertia Calculator

Calculate moment of inertia for solid sphere, disk, rod, ring, cylinder, and hollow sphere.

⚙️ Rotation📐 I = mr²🔄 Rotational Inertia

Select shape, enter mass and dimension(s).

Shape
Mass (m) kg
Radius (r) m
Length (L) m — rods only
⚠️ Enter valid positive numbers.

What Is Moment of Inertia?

Moment of inertia (I) is the rotational analogue of mass. While mass measures an object's resistance to linear acceleration (F = ma), moment of inertia measures resistance to rotational acceleration: τ = Iα, where τ is torque and α is angular acceleration. Higher I means harder to spin up or slow down.

Unlike mass (a single scalar), moment of inertia depends on the distribution of mass relative to the rotation axis. The formula is I = Σmᵢrᵢ² for point masses, where rᵢ is each mass's perpendicular distance from the axis. Mass concentrated farther from the axis contributes more to I (squared dependence on r).

Different shapes have different I formulas: solid sphere (2/5 mr²), hollow sphere (2/3 mr²), solid disk (1/2 mr²), ring/hoop (mr²), rod rotating about center (1/12 mL²), rod rotating about end (1/3 mL²). A hollow sphere always has higher I than a solid sphere of the same mass and radius because mass is farther from center.

The parallel axis theorem extends I calculations: I = I_cm + md², where I_cm is the moment about the center of mass and d is the distance to the new axis. This lets you calculate I for any axis from the simpler I_cm formula, without integrating from scratch.

Formula Reference Table

Solve ForFormulaNotes
General definitionI = Σmᵢrᵢ² = ∫r² dmMass × (distance from axis)²
Solid sphereI = (2/5)mr²Ball, uniform density
Hollow sphereI = (2/3)mr²Shell, all mass at surface
Solid disk / cylinderI = (1/2)mr²About central axis
Thin ring / hoopI = mr²All mass at radius r
Rod (center axis)I = (1/12)mL²About center, ⊥ to length
Rod (end axis)I = (1/3)mL²About one end, ⊥ to length
Parallel axis theoremI = I_cm + m·d²Shift axis by distance d

3 Worked Examples

Example 1
Solid Sphere — Basketball

A basketball (m = 0.62 kg, r = 0.12 m) modeled as solid sphere.

  • I = (2/5)mr² = 0.4 × 0.62 × 0.12²
  • I = 0.4 × 0.62 × 0.0144 = 0.00357 kg·m²
  • If hollow sphere: I = (2/3)×0.62×0.0144 = 0.00595 kg·m² (67% higher)
✓ I_solid = 0.00357 kg·m²; I_hollow = 0.00595 kg·m²
Example 2
Disk — Flywheel

A 50 kg steel flywheel disk, r = 0.4 m.

  • I = (1/2)mr² = 0.5 × 50 × 0.16 = 4.0 kg·m²
  • Rotational KE at ω = 100 rad/s: KE = ½Iω² = 0.5 × 4.0 × 10,000 = 20,000 J
  • Flywheels store energy — this one holds 20 kJ
✓ I = 4.0 kg·m²; stores 20 kJ at 100 rad/s
Example 3
Rod — Swinging Door

A door modeled as a rod about its hinge (end axis): m = 20 kg, L = 0.9 m.

  • I = (1/3)mL² = (1/3) × 20 × 0.81 = 5.4 kg·m²
  • Torque needed for α = 2 rad/s²: τ = Iα = 5.4 × 2 = 10.8 N·m
  • At arm length 0.45 m: force = 10.8/0.45 = 24 N
✓ I = 5.4 kg·m²; needs 24 N force at midpoint for α = 2 rad/s²

Real-World Applications

🎡
Flywheels & Energy Storage
Flywheel batteries store kinetic energy: KE = ½Iω². High I (from large r) and high ω maximize storage. Steel flywheels at 50,000 RPM store megajoules in compact systems for grid-scale storage.
⚙️
Engine Design
Crankshafts and pistons have carefully tuned I to smooth out the pulsed torque of combustion cycles. Higher I smooths rotation; lower I allows faster acceleration (sports cars vs. trucks).
🏋️
Gymnastics & Diving
Athletes manipulate I by changing body position. Tucked position: small I, fast rotation. Extended position: large I, slow rotation. Same angular momentum, different ω.
🛸
Spacecraft Attitude Control
Reaction wheels are spinning flywheels inside spacecraft. By spinning them faster or slower, angular momentum is exchanged to reorient the spacecraft without using propellant.
🌍
Planetary Formation
Earth's I = 8.04×10³⁷ kg·m² (about 0.3307 MR² — slightly less than solid sphere due to dense core). Angular momentum conservation during contraction from a nebula caused the solar system to spin.

Common Mistakes to Avoid

⚠️
Confusing moment of inertia with mass

I (kg·m²) is not the same as mass (kg). Mass measures linear inertia; I measures rotational inertia. I depends on both mass AND how it is distributed relative to the rotation axis.

⚠️
Using wrong formula for shape

I = (1/2)mr² for a solid disk ≠ mr² for a hoop. Always check which shape formula applies. A filled cylinder uses disk formula; a bicycle wheel approximates a hoop.

⚠️
Using diameter instead of radius

r in I formulas is the radius, not diameter. r = d/2. Using diameter gives I that is 4× too large.

⚠️
Forgetting the axis of rotation matters

I = (1/12)mL² for a rod rotating about its center; I = (1/3)mL² rotating about its end. The same object has different I for different rotation axes.

⚠️
Ignoring parallel axis theorem

To find I about a non-center axis, use I = I_cm + md², not just the standard formula. Applying the center-of-mass formula for an off-center axis gives wrong answers.

Frequently Asked Questions

Why does mass distribution matter more than total mass? ▾
I = Σmᵢrᵢ² — each mass element contributes in proportion to the SQUARE of its distance from the axis. Doubling the radius of a point mass quadruples its contribution to I. This is why a ring (all mass at r) has higher I than a disk (mass spread from 0 to r) of equal mass and radius.
What is the parallel axis theorem? ▾
I = I_cm + md². To find I about any axis parallel to the center-of-mass axis, add md² where d is the separation. A rod's I about center = mL²/12; about end (d = L/2): I = mL²/12 + m(L/2)² = mL²/12 + mL²/4 = mL²/3 ✓
Why do figure skaters spin faster when pulling in arms? ▾
Conservation of angular momentum: L = Iω = constant. Pulling arms inward reduces I (arms closer to axis). Since L = Iω = const, ω must increase. A skater reducing I by 60% spins 2.5× faster — no external torque needed.
What is radius of gyration? ▾
k = √(I/m) — the effective radius at which all mass could be concentrated to give the same I. For solid sphere: k = r√(2/5) = 0.632r. For rod (center): k = L/√12 = 0.289L. Engineers use k to compare rotational inertia of complex cross-sections.
How does I relate to kinetic energy? ▾
Rotational KE = ½Iω². For a rolling object (both translating and rotating): total KE = ½mv² + ½Iω² = ½mv²(1 + I/mr²). A solid sphere rolling: KE = ½mv²(1 + 2/5) = 7/10 mv². A ring rolling: KE = ½mv²(1 + 1) = mv². Rings roll slower down ramps than spheres for equal mass.
What is the inertia tensor? ▾
For 3D rotation, I is not a single number but a 3×3 tensor with nine components (six unique due to symmetry). The principal moments of inertia describe rotation about three orthogonal axes. For symmetric objects, the principal axes align with symmetry axes, simplifying analysis.
How do engines use moment of inertia? ▾
Reciprocating engines have pulsating torque. A high-I crankshaft/flywheel system averages out these pulses, maintaining smooth RPM. Formula: α = τ_net/I. Higher I → smaller α for the same torque fluctuation → smoother rotation. Large diesel engines use massive flywheels (I up to 100 kg·m²).
What is the moment of inertia of a human body? ▾
Varies enormously with posture. Standing straight (rotation about vertical axis): I ≈ 10–15 kg·m² (arms away). Arms pulled in: I ≈ 0.8–1.0 kg·m². This 10–15× difference explains spin speed changes in figure skating, gymnastics, and divers. The body distributes mass at average r ≈ 0.1–0.15 m when tucked.

Related Physics Calculators

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Formula Explorer connections

Interpretation: This formula is the rotational counterpart of linear mechanics, relating angle, angular motion, torque, inertia or rotational energy. Assumption: Define the rotation axis and sign convention. Rigid-body behavior, no slipping, steady rotation or negligible bearing losses may be assumed.

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