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J o = π 64 π 64 d 4 + π 64 π 64 d 4. This is the equation for finding the polar moment of inertia for the circular shaft. The above figure shows the cross-section profile of a hollow circular shaft with an outer diameter (do) and inner diameter (di).
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 j indicates the point or axis.
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.
The " Polar Moment of Inertia" is defined with respect to an axis perpendicular to the area considered. It is analogous to the " Area Moment of Inertia " - which characterizes a beam's ability to resist bending - required to predict deflection and stress in a beam.
The polar section modulus (also called section modulus of torsion), Z p, for circular sections may be found by dividing the polar moment of inertia, J, by the distance c from the center of gravity to the most remote fiber.
In following sections we will use the integral definitions of moment of inertia (10.1.3) to find the moments of inertia of five common shapes: rectangle, triangle, circle, semi-circle, and quarter-circle with respect to a specified axis.
The moment of inertia about an axis perpendicular to the plane of the ellipse and passing through its centre is \(c_3ma^2 \), where, of course (by the perpendicular axes theorem), \( c_3 = c_1 + c_2 \).
