Parallel Axis Theorem Calculator
Enter moment of inertia about the center of mass, mass and distance to compute the moment about a parallel axis I = I_cm + m·d².
Input Data
Results
At a glance:The parallel axis theorem relates the moment of inertia I of a body about any axis to that about a parallel axis through its center of mass: I = I_cm + m·d², where I_cm is the centroidal moment of inertia, m the total mass and d the perpendicular distance between the axes. It lets you build I for offset or composite shapes from known centroidal values without re-integrating. The theorem also implies I_cm is the minimum moment of inertia for any parallel axis (m·d² ≥ 0). Applications: pendulums, rolling bodies, machinery and structural dynamics. This tool returns I and the m·d² term.
Formula
I = I_cm + m·d²
$$I = I_{cm} + m \cdot d^2$$How to Use
- Enter the centroidal moment of inertia I_cm (kg·m²).
- Enter the mass m (kg) and axis distance d (to compute term) — enter d (m).
- The calculator returns m·d² and total I.
Case Studies
Rod about an end
Rod length L, I_cm = mL²/12 about center.
d = L/2 → m·d² = mL²/4.
I_end = mL²/12 + mL²/4 = mL²/3 (matches known result).
FAQ
Why add m·d²?
Shifting the rotation axis away from the CM adds rotational inertia because mass points are farther from the new axis on average; the exact extra amount is m·d².
Is I_cm always the smallest?
Yes, for any parallel axis, since m·d² ≥ 0. The centroidal axis gives the minimum moment of inertia.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.