This moment of inertia calculator computes I — rotational inertia — for eight standard shapes: point mass, rod (center and end axes), solid disk/cylinder, hoop, solid sphere, hollow sphere and rectangular plate. It also applies the parallel-axis theorem, because real rotation rarely happens about the center of mass.

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Mass, Weighted by Distance Squared

Moment of inertia is the rotational analog of mass, with a twist that decides everything: I = Σ mᵢrᵢ² — each piece of mass counts not for what it weighs but for its distance from the axis, squared. Hence the coefficient zoo: a hoop (all mass at radius R) takes the full MR²; a solid sphere, mass tucked close to the axis, takes only ⅖MR²; the rod quadruples its I when you move the axis from center (ML²/12) to end (ML²/3). The defaults run the flywheel-relevant case: a 4 kg solid disk of radius 0.5 m → I = ½ × 4 × 0.25 = 0.50 kg·m². The parallel-axis theorem (I = I_cm + Md²) then prices any off-center mounting — and its d² means sloppy axis placement is punished quadratically, a fact every unbalanced machine demonstrates loudly.

Where I earns its keep: angular acceleration (α = τ_net/I — the same torque spins a figure skater faster than a flywheel because their I’s differ 100-fold), rotational kinetic energy (½Iω²) and angular momentum (L = Iω) — the next two calculators in this cluster consume I directly. Engineering cousins include section moment of inertia in beam theory (a different, area-based quantity with the same name and squared-distance soul — don’t mix the units).

Worked Examples (Verified)

  • Solid disk (defaults): I = ½(4)(0.5²) = 0.50 kg·m². As a hoop the same mass/radius gives 1.00 — rim concentration doubles it.
  • Rod two ways: M = 3 kg, L = 2 m → center axis (1/12)(3)(4) = 1.0 kg·m²; end axis (1/3)(3)(4) = 4.0 kg·m² — check the parallel-axis route: 1.0 + 3 × (1)² = 4.0 ✔ (the theorem reproducing the table entry, a good sign for both).
  • Solid sphere: M = 5 kg, R = 0.3 m → ⅖ × 5 × 0.09 = 0.18 kg·m²; hollow sphere same data: ⅔ × 5 × 0.09 = 0.30 — hollowness pushes mass outward and I follows.

Moment-of-Inertia Discipline

  • The axis is part of the answer: “the I of a rod” is meaningless until the axis is named — every formula in the selector carries its axis with it.
  • Radii vs diameters: tables use R; drawings often give diameter. Halving errors square into 4× mistakes — the input label says radius deliberately.
  • Compound bodies add: I_total = Σ of parts about the SAME axis (shift each with parallel-axis first if needed). A disk plus a rim is just addition once axes agree.
  • Thin-shell and solid versions differ by factors that matter (sphere ⅔ vs ⅖) — real objects sit between; when in doubt, the shell value is the conservative bound for spin-up torque.
  • Units kg·m², always: a mass in grams or a radius in cm contaminates by powers of ten squared — convert before, not after.

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Frequently Asked Questions

How do you calculate moment of inertia?

For standard shapes, from the table this calculator implements: hoop MR², solid disk ½MR², solid sphere ⅖MR², hollow sphere ⅔MR², rod about center ML²/12, rod about end ML²/3, point mass Mr², plate M(a²+b²)/12 — then shift axes with I = I_cm + Md² when needed. Defaults: a 4 kg, 0.5 m solid disk gives 0.50 kg·m². Non-standard shapes are decomposed into standard parts and summed about a common axis.

Why does distance count squared?

Because both rotational energy and angular momentum weight outer mass more: a mass at radius r moves at speed ωr, so its kinetic energy is ½m(ωr)² — the r² enters through the velocity. Physically: mass far from the axis travels farther per revolution and resists changes in spin correspondingly more. It’s why tightrope walkers carry long poles and why flywheel rims are heavy and hubs are light.

What is the parallel axis theorem?

I = I_cm + Md²: the moment of inertia about any axis equals the center-of-mass value about a parallel axis plus total mass times the squared separation. Check it on the rod: center 1.0 + 3 kg × (1 m)² = 4.0 = the end-axis table value — the theorem and the table agree, as they must. The calculator applies it whenever you enter a nonzero shift d.

What is the radius of gyration?

k = √(I/M) — the radius at which the body’s entire mass, concentrated in a thin ring, would produce the same I. It compresses shape into one length: disk k = R/√2 ≈ 0.707R, hoop k = R, sphere k ≈ 0.632R. Engineers use it for quick comparisons and for column/buckling work in its area form. This calculator prints it with every result.

Which shape has the largest moment of inertia for its mass and size?

The hoop/ring: every gram sits at the full radius, so no shape of the same M and R can exceed MR² (mass can’t be farther than R). It’s the upper bound the other coefficients are fractions of — ½ for the uniform disk, ⅖ for the sphere. Racing engineers chase the opposite end (mass near the hub); flywheel designers chase the hoop.

How does moment of inertia affect acceleration?

Through α = τ_net / I — Newton’s second law in rotation. Double I at the same torque and angular acceleration halves; that’s why a potter’s wheel with a heavy flywheel smooths speed fluctuations (stored I resists changes) and why a car’s wheel/brake package affects acceleration feel beyond mere weight: unsprung rotating mass charges its toll through I, not just M.

Is mass moment of inertia the same as area moment of inertia?

No — same name, different quantity. Mass MOI (this page, kg·m²) governs rotational dynamics. Area MOI (m⁴) governs beam bending stiffness and appears in structural formulas (σ = My/I). They share the ∫r² structure and nothing else; confusing their units is the tell. Our Construction cluster cares about the area kind; this Physics page is the mass kind.

Is this moment of inertia calculator free?

Yes — free, instant, no sign-up. Pick a shape, enter mass and dimensions (plus optional axis shift) for I and the radius of gyration.