![]() ![]() The red r’s in this image show the distance that is being measured when adding up each little infinitesimal dm. Due to the symmetry, rotation about the x axis and z axis looks identical. In this image, rotation about the y axis and x axis produce different types of rotation. You can imagine sticking your pencil into an object and twisting along that axis. As shown in the following figure, rotating about the different axes will produce different types of rotation. A skill that you can develop is your visualization of the rotation about each axis. If there is more mass closer to the axis of rotation, the inertia is smaller. Inertia is always positive and has units of kgm 2 or slugft 2.įor an infinitesimal unit of mass, the inertia depends on how far it is from the axis of rotation.Īs shown in this image, each little dm at r distance from the axis of rotation (y) is added up (through integration). The bigger the inertia, the slower the rotation. Mass moment of inertia, or inertia as it will be referred to from here on, is resistance to rotation. 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.7.4 Mass Moment of Inertia 7.4.1 Intro to Mass Moment of Inertia 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 through 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 through 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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