![]() ![]() You can add Ixx and Iyy or I₁₁ and I₂₂, the result (the polar moment of inertia) should be the same in both cases. The polar moment of inertia of a shape describes its ability to withstand torsional deformation (twist). This sum of area moments is called the ' polar moment of inertia' of the shape. The second moment of area of any shape about any axis plus the second moment of area at right-angles to it will be equal to the sum of any other two second moments of area at right-angles to each other in the same plane. Two point masses m₁ and m₂, with reduced mass μ, separated by a distance r with axis of rotation going through the center of mass and perpendicular to the line joining the two particles.The second moment of area ( moment of inertia) and radius of gyration (also called second area moments) of any shape are properties that define its structural rigidity (ability to withstand deformation) about a given axis. Torus with minor radius a, major radius b and mass m with axes of rotating going through its center: perpendicular to the major diameter and parallel to the major diameter. Solid and hollow, regular tetrahedron (four flat faces) of side s and mass m with axis of rotation going through its center and one of vertices. Spherical shell of inner radius r₁, outer radius r₂ and mass m with axis of rotation going through its center. Hollow sphere of radius r and mass m with axis of rotation going through its center. Rod of length L and mass m with two axes of rotation: about its center and one end. Solid right circular cone of radius r, height h and mass m with three axes of rotation passing trough its center: parallel to the x, y or z axes. Hollow right circular cone of radius r, height h and mass m with three axes of rotation passing trough its center: parallel to the x, y or z axes. Plane regular polygon with n vertices, radius of the circumscribed circle R and mass m with axis of rotation passing through its center, perpendicular to the plane. Thin rectangular plate of length l, width w and mass m with axis of rotation going through its center, perpendicular to the plane. Point mass m at a distance r from the axis of rotation. Solid and hollow, regular octahedron (eight flat faces) of side s and mass m with axis of rotation going through its center and one of vertices. An isosceles triangle of mass m, vertex angle 2β and common-side length L with axis of rotation through tip, perpendicular to plane. Solid and hollow, regular icosahedron (twenty flat faces) of side s and mass m with axis of rotation going through its center and one of vertices. Solid ellipsoid of semiaxes a, b, c and mass m with three axes of rotation going through its center: parallel to the a, b or c semiaxes. Solid and hollow, regular dodecahedron (twelve flat faces) of side s and mass m with axis of rotation going through its center and one of vertices. Thin solid disk of radius r and mass m with three axes of rotation going through its center: parallel to the x, y or z axes. Cylindrical shell of radius r and mass m with axis of rotation going through its center, parallel to the height. ![]() Cylindrical tube of inner radius r₁, outer radius r₂, height h and mass m with three axes of rotation going through its center: parallel to x, y and z axes. Solid cylinder of radius r, height h and mass m with three axes of rotation going through its center: parallel to x, y and z axes. Solid cuboid of length l, width w, height h and mass m with four axes of rotation going through its center: parallel to the length l, width w, height h or to the longest diagonal d. Thin circular hoop of radius r and mass m with three axes of rotation going through its center: parallel to the x, y or z axes. Solid ball of radius r and mass m with axis of rotation going through its center.
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