![]() 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. To begin, we will assume that the plate has mass (M) and length sides (L).#1 - Ball. I x = (⅓) a 4 Moment of Inertia of a Square PlateĪ few factors must be considered when calculating the moment of inertia of a square plate. (2) The following derivation is for a square when the centre of mass is moved a certain distance (d).īy using the parallel axis theorem we can now state (1) When we look at the square with its centre of mass passing through the x-axis, we see that it is made up of two equal-sized rectangles. Remember that the moment of inertia of a rectangle is given as In addition to integration, we will use a rectangle as a reference to find the M.O.I. ![]() The parallel axis theorem states that the moment of inertia can be easily calculated.Įven so, in this lesson, we will replace mass (M) with the area (A). I = a 4 / 3 Moment of Inertia of a Square Derivation If indeed the centre of mass (cm) is moved to a certain distance (d) from the x-axis, we will use a different expression to calculate the moment of inertia of the same square. This would be the equation for a solid square with its centre of mass along the x-axis.Ī square’s diagonal moment of inertia can also be calculated as In this case, a = the square section’s sides. ![]()
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