Polar Second Moment of Area, abbreviated as J, (English area unit \(in^4\), Metric area unit \(mm^4\)), also called the polar moment of inertia, describe the resistance of an objects cross-section to torsional (twisting) deformations. It is an extension of the concept of moment of inertia, which measures an object's resistance to rotational motion about a given axis. While the ordinary moment of inertia deals with bending or flexural deformations, the polar moment of inertia specifically quantifies an object's resistance to twisting.
Polar Second Moment of Area Formulas |
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\(\Large{ J \;=\; \int_A \; r^2 \; d \; A }\) (Polar Second Moment of Area) \(\Large{ J \;=\; I_x + \; I_y }\) (Relationship with the Second Moment of Area) \(\Large{ r^2 \;=\; x^2 + \; y^2 }\) (Radial Distance) |
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| Symbol | English | Metric |
| \( J \) = Polar Second Moment of Area | \(in^4\) | \(mm^4\) |
| \( r \) = Radial Distance from Polar Axis | \(in\) | \(mm\) |
| \( I_x \) = Second Moment of Area about the x-axis | \(in^4\) | \(mm^4\) |
| \( I_y \) = Second Moment of Area about the y-axis | \(in^4\) | \(mm^4\) |
| \( x \) = Horizontal Distance from the y-axis | \(in\) | \(mm\) |
| \( y \) = Vertical Distance from the x-axis | \(in\) | \(mm\) |
| \( d \) = Distance Between the Two Axes | \(in\) | \(mm\) |
| \( A \) = Area Cross-section | \(in^2\) | \(mm^2\) |
| \( dA \) = Differential Element of Area | \(in^2\) | \(mm^2\) |
| \( \int \) = Integral; Sums all Contributions Continuously | - | - |
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