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Calculate the curvature and torsion for the following trajectories given by x(t) a(t) x + y(t) ŷ + z(t) 2. Sketch each trajectory. (a) r(t) = Rsin(wt), y(t) = Rcos(wt), z(t) = 0, where R and w are constant. =

Calculate the curvature and torsion for the following trajectories given by x(t) a(t) x + y(t) ŷ + z(t) 2. Sketch each trajectory. (a) r(t) = Rsin(wt), y(t) = Rcos(wt), z(t) = 0, where R and w are constant. =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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We have shown in class that the acceleration of a particle can be decomposed into components that are tangential and normal to its trajector as a = kv²N+ÏT where T and N are the unit tangent vector and principal (unit) normal vector, respectively, is the length along the trajectory, v = is the speed, and is the curvature of the trajectory. The curvature is related to the radius of curvature p by k = 1/p. The plane formed by T and N is called the osculating plane, and can be thought of as the instantaneous plane of the trajectory. The unit vector B = TX N is called the binormal vector, and by definition is always perpendicular to the osculating plane. We have shown in class that both T and B are parallel to N. The torsion r(t) of the trajectory is defined by or, equivalently, by B = -TIN dB de = -T, N. (i) The concept of curvature and radius of curvature are defined by extending those con- cepts from circles to curves in general. The curvature & is defined as the rate (with respect to length along the curve) of rotation of the tangent vector. Show that the definition of curvature given in class, namely |T(t₁) - T(t₂)| lim t₁-t₂|l(t₁) -l(t₂)| d'T de = K, is in fact the rate of rotation of T (i.e., that it gives the rate of change of the angle T makes with a fixed direction). Show also that for a circle in the plane = 1/R where R is the radius of the circle. (ii) Prove the following Frenet-Serret formulae: T = KUN, N=-KUT + TUB.
Transcribed Image Text:We have shown in class that the acceleration of a particle can be decomposed into components that are tangential and normal to its trajector as a = kv²N+ÏT where T and N are the unit tangent vector and principal (unit) normal vector, respectively, is the length along the trajectory, v = is the speed, and is the curvature of the trajectory. The curvature is related to the radius of curvature p by k = 1/p. The plane formed by T and N is called the osculating plane, and can be thought of as the instantaneous plane of the trajectory. The unit vector B = TX N is called the binormal vector, and by definition is always perpendicular to the osculating plane. We have shown in class that both T and B are parallel to N. The torsion r(t) of the trajectory is defined by or, equivalently, by B = -TIN dB de = -T, N. (i) The concept of curvature and radius of curvature are defined by extending those con- cepts from circles to curves in general. The curvature & is defined as the rate (with respect to length along the curve) of rotation of the tangent vector. Show that the definition of curvature given in class, namely |T(t₁) - T(t₂)| lim t₁-t₂|l(t₁) -l(t₂)| d'T de = K, is in fact the rate of rotation of T (i.e., that it gives the rate of change of the angle T makes with a fixed direction). Show also that for a circle in the plane = 1/R where R is the radius of the circle. (ii) Prove the following Frenet-Serret formulae: T = KUN, N=-KUT + TUB.
(iii) Calculate the curvature and torsion for the following trajectories given by x(t) x(t) x + y(t) ŷ + z(t) 2. Sketch each trajectory. (a) (b) (c) x(t) = Rsin(wt), y(t) = R cos(wt), z(t) = 0, where R and w are constant. r(t) = Rsin(wt), y(t) = Rcos(wt), z(t) = vat, where R, w, and va are constant. = x(t) = R₁ sin(wt), y(t) = R₂ cos(wt), z(t) = 0, where R₁, R₂, and w are constant. Then set R₁ = 1, R₂ = 2, and w = 1 and plot k(t) for 0 ≤ t ≤ 2π.
Transcribed Image Text:(iii) Calculate the curvature and torsion for the following trajectories given by x(t) x(t) x + y(t) ŷ + z(t) 2. Sketch each trajectory. (a) (b) (c) x(t) = Rsin(wt), y(t) = R cos(wt), z(t) = 0, where R and w are constant. r(t) = Rsin(wt), y(t) = Rcos(wt), z(t) = vat, where R, w, and va are constant. = x(t) = R₁ sin(wt), y(t) = R₂ cos(wt), z(t) = 0, where R₁, R₂, and w are constant. Then set R₁ = 1, R₂ = 2, and w = 1 and plot k(t) for 0 ≤ t ≤ 2π.
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