Polar Moment (solid shaft)⚠ unverified

Understanding the Parameters

• Diameter $d$ — the only geometric input for a solid round shaft. Because area grows as $d^2$ and each area element's contribution is weighted by its distance-squared from the axis (another $d^2$), $J$ scales as $d^4$. This steep dependence is why a small increase in shaft diameter buys a large gain in torsional capacity — and why hollow shafts (which remove low-leverage material near the axis) are so weight-efficient (see Polar Moment Hollow).

• Units of m⁴ — the same as bending's $I$, reflecting area (length²) times distance-squared (length²). For a circle, $J = I_x + I_y = 2I$ exactly, because the polar moment is the sum of the two rectangular moments (perpendicular axis theorem).

Derivation (Approaching a Proof)

By definition, the polar second moment of area is the integral of distance-squared over the cross-section:

$$J = \int_A r^2\, \mathrm{d}A.$$

For a solid circle, use an annular ring element of radius $r$ and thickness $\mathrm{d}r$, whose area is $\mathrm{d}A = 2\pi r\, \mathrm{d}r$. Integrating from the axis to the outer radius $R = d/2$:

$$J = \int_0^{R} r^2 (2\pi r\, \mathrm{d}r) = 2\pi \int_0^{R} r^3\, \mathrm{d}r = 2\pi\,\frac{R^4}{4} = \frac{\pi R^4}{2}.$$

Substituting $R = d/2$:

$$J = \frac{\pi (d/2)^4}{2} = \frac{\pi d^4}{32}.$$

The ring-element choice exploits the circular symmetry: every point on a ring is at the same radius $r$, so the $r^2$ weighting is constant over each ring and the area integral collapses to a single integration in $r$. This is also why $J = 2I$ for a circle — the perpendicular-axis theorem $J = I_x + I_y$ with $I_x = I_y$ by symmetry.

Dimensional check. $[J] = \text{m}^4$. ✓

History and Development

The polar moment and the torsion theory built on it descend from Coulomb (1784, torsion balance) and Saint-Venant (1850s, general theory of torsion). For circular sections the elementary $J = \pi d^4/32$ result is exact because plane sections remain plane under torsion — a special property of the circle that makes round shafts the default for transmitting torque. It is a staple of every strength-of-materials text (Timoshenko, Shigley).

Related Concepts: Torsional Shear Stress, Polar Moment Hollow, Shaft Diameter torsion, Rectangular Moment of Inertia, Critical Speed Shaft

Notes: Solid round section only; use Polar Moment Hollow for tubes. For circles $J = 2I$. The elementary formula is exact for circular sections (non-circular sections warp and need Saint-Venant's torsion constant instead).