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The negative sign is simply a matter of convention: it is a byproduct of the historical development of the subject. Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In ...
Torsion of a square section bar Example of torsion mechanics. In the field of solid mechanics, torsion is the twisting of an object due to an applied torque [1] [2].Torsion could be defined as strain [3] [4] or angular deformation [5], and is measured by the angle a chosen section is rotated from its equilibrium position [6].
Here the vectors N, B and the torsion are not well defined. If the torsion is always zero then the curve will lie in a plane. A curve may have nonzero curvature and zero torsion. For example, the circle of radius R given by r(t) = (R cos t, R sin t, 0) in the z = 0 plane has zero torsion and curvature equal to 1/R. The converse, however, is false.
The torsion tensor thus is related to, although distinct from, the torsion of a curve, as it appears in the Frenet–Serret formulas: the torsion of a connection measures a dislocation of a developed curve out of its plane, while the torsion of a curve is also a dislocation out of its osculating plane.
Examples of circular motion include: special satellite orbits around the Earth (circular orbits), a ceiling fan's blades rotating around a hub, a stone that is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a ...
The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness.
The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative. The directions in the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k 1 does not equal k 2 , a result of Euler (1760), and are called principal directions .
The envelope of lines perpendicular to the pedal is then the envelope of reflected rays or the catacaustic of C′. This proves that the catacaustic of a curve is the evolute of its orthotomic. As noted earlier, the circle with diameter PR is tangent to the pedal. The center of this circle is R′ which follows the curve C′.