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de lim 1₁-1₂ = -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)| (t₁) -l(t₂)| d'T de 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.

de lim 1₁-1₂ = -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)| (t₁) -l(t₂)| d'T de 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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 33E
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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 B = -TIN or, equivalently, by 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 = K₂ 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 x = 1/R where R is the radius of the circle.
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 B = -TIN or, equivalently, by 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 = K₂ 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 x = 1/R where R is the radius of the circle.
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