Geometric Representation
Mass Moment of Inertia
A measure of an object's resistance to changes in its rotational motion. Depends on the mass distribution relative to the axis of rotation. Key parameter in angular acceleration and torque calculations.
Formula
I = ∫ r² dm
Units
kg·m²
Common Shape Formulas
Rotational Dynamics| Shape | Formula | Axis |
|---|---|---|
| Solid Cylinder | I = ½MR² | Central axis |
| Hollow Cylinder | I = MR² | Central axis |
| Solid Sphere | I = ⅖MR² | Through center |
| Thin Rod | I = ¹⁄₁₂ML² | Through center |
Geometric Representation
Area Moment of Inertia
A geometric property of a cross-section that measures its resistance to bending and deflection. Also known as the second moment of area. Critical for beam design and structural analysis.
Formula
I = ∫ y² dA
Units
m⁴ or mm⁴
Common Shape Formulas
Structural Mechanics| Shape | Formula | Axis |
|---|---|---|
| Rectangle | I = bh³/12 | Centroidal axis |
| Circle | I = πr⁴/4 | Through center |
| Hollow Circle | I = π(r₀⁴ - rᵢ⁴)/4 | Through center |
| I-Beam | I = (BH³ - bh³)/12 | Major axis |
Geometric Representation
Polar Moment of Inertia
A measure of an object's resistance to torsion or twisting. For circular shafts, it's the sum of the area moments of inertia about two perpendicular axes in the cross-section.
Formula
J = ∫ r² dA
Units
m⁴ or mm⁴
Common Shape Formulas
Torsional Analysis| Shape | Formula | Axis |
|---|---|---|
| Solid Circle | J = πd⁴/32 | Central axis |
| Hollow Circle | J = π(d₀⁴ - dᵢ⁴)/32 | Central axis |
| Rectangle | J = bh(b² + h²)/12 | Centroidal axis |