# Polar moment of inertia

The English used in this article may not be easy for everybody to understand. |

*The*

**Note:***polar moment of inertia*must not be confused with the

*moment of inertia*, which characterizes an object's angular acceleration due to an applied torque.

The **polar moment of inertia** is a measure of an object's ability to resist torsion as a function of its shape. It is one aspect of the **area moment of inertia**, or second moment of area, linked through the perpendicular axis theorem. Where the **planar** second moment of area uses a beam's cross-sectional shape to describe its resistance to deformation (bending) when subjected to a force applied in a plane parallel to its neutral axis, the **polar** second moment of area uses a beam's cross-sectional shape to describe its resistance to deformation (torsion) when a moment (torque) is applied in a plane perpendicular to the beam's neutral axis. While the **planar** second moment of area is most often denoted by the letter, <math>I</math>, the **polar** second moment of area is most often denoted by either, <math>I_z</math>, or the letter, <math>J</math>, in engineering textbooks.

The calculated values for the polar moment of inertia are most often used describe a solid or hollow cylindrical shaft's resistance to torsion, as in a vehicle's axle or drive shaft. When applied to non-cylindrical beams or shafts, the calculations for the polar moment of inertia becomes erroneous due to warping of the shaft/beam. In these instances, a torsional constant should be used, where a correctional constant is added to the value's calculation.

The polar moment of inertia carries the units of length to the fourth power (<math>L^4</math>); meters to the fourth power (<math>m^4</math>) in the metric unit system, and inches to the fourth power (<math>in^4</math>) in the imperial unit system. The mathematical formula for direct calculation is given as a multiple integral over a shape's area, <math>R </math>, at a distance <math>\rho </math> from an arbitrary axis <math>O </math>.<math>J_{O} = \iint\limits_R \rho^2 dA </math>.

In the most simple form, the **polar** moment of inertia is a summation of the two **planar** second moments of area, <math>I_x</math> and <math>I_y</math>. Using the Pythagorean theorem, the distance from axis <math>O </math>, <math>\rho </math>, can be broken into its <math>x </math> and <math>y </math> components, and the change in area, <math>dA </math>, broken into its <math>x </math> and <math>y </math> components, <math>dx </math> and <math>dy </math>.

Given the two formulas for the **planar** second moments of area:

<math>I_x = \iint\limits_R x^2 dxdy </math>, and <math>I_y = \iint\limits_R y^2 dxdy </math>

The relation to the **polar** second moment of area can be shown as:

<math>J_{O} = \iint\limits_R \rho^2 dA </math>

<math>J_{O} = \iint\limits_R (x^2+y^2) dx dy </math>

<math>J_{O} = \iint\limits_R x^2 dxdy + \iint\limits_R y^2 dxdy </math>

<math>\therefore J = I_x + I_y </math>

In essence, as the magnitude of the polar moment of inertia increases (i.e. large object cross-sectional shape), more torque will be required to cause a torsional deflection of the object. However, it must be noted that this does not have any bearing on the torsional rigidity provided to an object by its constituent materials; the polar moment of inertia is simply rigidity provided to an object by its shape alone. Torsional rigidity provided by material characteristics is known as the shear modulus, <math>G </math>. Linking these two components of rigidity, one can calculate the angle of twist of a beam, <math>\theta </math>, using:

<math>\theta = \frac{Tl}{JG} </math>

Where <math>T </math> is the applied moment (torque) and <math>l </math> is the length of the beam. As shown, higher torques and beam lengths lead to higher angular deflections, where higher values for the polar moment of inertia, <math>J </math>, and material shear modulus, <math>G </math>, reduces the potential for angular deflections.

## Related pages

- Moment (physics)
- Second moment of area
- List of second moments of area for standard shapes
- Shear modulus

## Other websites

- Torsion of Shafts - engineeringtoolbox.com
- Elastic Properties and Young Modulus for some Materials - engineeringtoolbox.com
- Material Properties Database - matweb.com