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The angular twist of a shaft with given cross section is given by(1)(Roark 1954, p. 174), where is the twisting moment (commonly measured in units of inch-pounds-force), is the length (inches), is the modulus of rigidity (pounds-force per square inch), and (sometimes also denoted ) is the torsional rigidity multiplier for a given geometric cross section (inches to the fourth power). Note that the quantity is sometimes denoted (e.g., Timoshenko and Goodier 1951, p. 264).Values of are known exactly only for a small number of cross sections, and in closed form for even fewer. The following table lists approximate values for some common shapes (Timoshenko and Goodier 1951, pp. 258-280; Roark 1954, pp. 174-179).cross section approxOEIScircle1.570796...A019669equilateral triangle0.021650...A180317half-disk0.297556...A180310isosceles right triangle0.026089...A180314quarter-disk0.0825...sliced..

The area moment of inertia is a property of a two-dimensional plane shape which characterizes its deflection under loading. It is also known as the second moment of area or second moment of inertia. The area moment of inertia has dimensions of length to the fourth power. Unfortunately, in engineering contexts, the area moment of inertia is often called simply "the" moment of inertia even though it is not equivalent to the usual moment of inertia (which has dimensions of mass times length squared and characterizes the angular acceleration undergone by a solids when subjected to a torque).The second moment of area about the about the -axis is defined by(1)while more generally, the "product" moment of area is defined by(2)Here, the positive sign convention is used (e.g., Pilkey 2002, p. 15).More generally, the area moment of inertia tensor is given by(3)(4)by analogy with the moment of inertia tensor, which has negative..

A positive number such that a lamina or solid body with moment of inertia about an axis and mass is given byPickover (1995) defines a generalization of as a function quantifying the spatial extent of the structure of a curve and given bywhere is the length distribution function. Small compact patterns have small .

The centroid is center of mass of a two-dimensional planar lamina or a three-dimensional solid. The mass of a lamina with surface density function is(1)and the coordinates of the centroid (also called the center of gravity) are(2)(3)The centroid of a lamina is the point on which it would balance when placed on a needle. The centroid of a solid is the point on which the solid would "balance."The geometric centroid of a region can be computed in the WolframLanguage using Centroid[reg].For a closed lamina of uniform density with boundary specified by for and the lamina on the left as the curve is traversed, Green's theorem can be used to compute the centroid as(4)(5)The centroid of a set of point masses located at positions is(6)which, if all masses are equal, simplifies to(7)The centroid of the vertices of a quadrilateral occurs at the point of intersection of the bimedians (i.e., the lines and joining pairs of opposite midpoints) (Honsberger..

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