The total curvature of the portion of a smooth curve that runs from s=So to s₁ > So can be found by integrating K from so to s₁. If the curve hast₁some other parameter, say t, then the total curvature is K=S₁jx ds= }₁KThe total curvature isdsKời dt =dt50toa. Find the total curvature of the portion of the helix r(t) = (8 cos t)i + (8 sin t)j +tk, 0≤t≤ 4.b. Find the total curvature of the parabola y = 4x², -∞0

Given equation of helixr→(t)=(8cos⁡t)i→+(8sin⁡t)j→+tk→ Where…

Answered: The total curvature of the portion of a…

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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Find the unit tangent vector T and the curvature k for the following parameterized curve:
A particle which moves in two-dimensional curvilinear motion has coordinates in millimeters which vary with time t in seconds according to x = 8.9t² + 6.5 and y = 6t³+ 4.4. For time t = 4.6 s, determine the radius of curvature p of the particle path and the magnitudes of the normal and tangential accelerations. Answers: At t = 4.6 s, p= an = at = i i i mm mm/s² mm/s²
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…
6. Find the total length, curvature and torsion of a curve: [cos² t, sin² t,0] 0≤t≤n?
Suppose C is the curve given by r(t) = (3 cos(t), -3 sin(t), 4t). Show that both the curvature as well as the torsion of C are constant, and find their values.
Find Consider the curve dy dx in terms of and y: Y 2 y And at the point (1,11), the tangent line to this curve is
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…
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…

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