The polar moment of inertia describes the distribution of the area of a body with respect to a point in the plane of the body. Alternately, the point can be considered to be where a perpendicular axis crosses the plane of the body. The subscript on the symbol \(j\) indicates the point or axis.
There is a particularly simple relationship between the polar moment of inertia and the rectangular moments of inertia. Referring to the figure, apply the Pythagorean theorem \(r^2 = x^2 +y^2\) to the definition of polar moment of inertia to get
The polar moment of inertia is an important factor in the design of drive shafts. When a shaft is subjected to torsion, it experiences internal distributed shearing forces throughout its cross-section which counteract the external torsional load.
This distributed shearing force is called shear stress, and is usually given the symbol tau, \(\tau\text{.}\) Shear stress is zero at the shaft centerline, called the neutral axis, and increases linearly with \(r\) to a maximum value, \(\tau_\text{max}\) at the outside surface where \(r=c\text{,}\) so
The force at any point is \(dF = \tau\ dA\text{,}\) and the moment \(dM\) exerted at any point is \(dF\) times the moment arm, \(r\text{.}\) The total moment is the integral of this quantity over the area of the cross section, and is proportional to the polar moment of inertia.
This is the relationship between the maximum stress in a circular shaft, the applied torque \(T\text{,}\) and the geometric properties of the shaft \(J_O\) and \(c\text{.}\)