The Position of a moving Vector point in space is given by s pace ) = t cost i + tsint J + 3/2 RCE) 212 t %3D Find Prom t=o to (a) Distance travelled a- and aN at any time t (b) ng at (C) radius of curvature any time t

The Position of a moving Vector point in space is given by s pace ) = t cost i + tsint J + 3/2 RCE) 212 t %3D Find Prom t=o to (a) Distance travelled a- and aN at any time t (b) ng at (C) radius of curvature any time t

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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Question

The
Position
of
a moving
Vector
point
in space
is given by
s pace
) = t cost i + tsint J +
3/2
RCE)
212 t
%3D
Find
Prom t=o to
(a) Distance
travelled
a- and aN
at
any time t
(b)
ng
at
(C) radius of curvature
any time t

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The curve =ti+V3t² j + 2t% k defines a bug's trajectory in the space. (a) Find the vector equation for the tangent line to the bug's path when t = V3. (b) Find the length of the path between the points (2,4/3, 16) and (3, 9/3, 54). (c) Find the curvature k at the point when t = 1.
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…