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a) The speed meter on my car reads a steady 60 mph, it is impossible that I am accelerating. b) The torsion of a smooth plane curve is zero because B is a constant vector. c) Curvature is the magnitude of dT/ds, which describes the rate at which T turns per unit of lenoth along the curve

a) The speed meter on my car reads a steady 60 mph, it is impossible that I am accelerating. b) The torsion of a smooth plane curve is zero because B is a constant vector. c) Curvature is the magnitude of dT/ds, which describes the rate at which T turns per unit of lenoth along the curve

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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### Problem Statements for Discussion: a) The speed meter on my car reads a steady 60 mph, it is impossible that I am accelerating. b) The torsion of a smooth plane curve is zero because B is a constant vector. c) Curvature is the magnitude of dT/ds, which describes the rate at which T turns per unit of length along the curve. ### Explanation: 1. **Steady Speed and Acceleration:** - When the speedometer of a car shows a constant speed of 60 mph, can acceleration be truly absent? - Consider that acceleration can occur not only due to changes in speed but also due to changes in direction (i.e., centripetal acceleration). 2. **Torsion of Plane Curves:** - In the context of 3D space, torsion measures how much a curve twists out of the plane. - A smooth plane curve having zero torsion implies that it lies entirely within a single plane, causing vector B (binormal vector) to remain constant. 3. **Curvature and Rate of Turning:** - Curvature represents how sharply a curve bends at a given point. - \(\frac{dT}{ds}\) defines how the tangent vector T changes with respect to the arc length \(s\), which is a measure of the curve’s rate of rotation in space.
Transcribed Image Text:### Problem Statements for Discussion: a) The speed meter on my car reads a steady 60 mph, it is impossible that I am accelerating. b) The torsion of a smooth plane curve is zero because B is a constant vector. c) Curvature is the magnitude of dT/ds, which describes the rate at which T turns per unit of length along the curve. ### Explanation: 1. **Steady Speed and Acceleration:** - When the speedometer of a car shows a constant speed of 60 mph, can acceleration be truly absent? - Consider that acceleration can occur not only due to changes in speed but also due to changes in direction (i.e., centripetal acceleration). 2. **Torsion of Plane Curves:** - In the context of 3D space, torsion measures how much a curve twists out of the plane. - A smooth plane curve having zero torsion implies that it lies entirely within a single plane, causing vector B (binormal vector) to remain constant. 3. **Curvature and Rate of Turning:** - Curvature represents how sharply a curve bends at a given point. - \(\frac{dT}{ds}\) defines how the tangent vector T changes with respect to the arc length \(s\), which is a measure of the curve’s rate of rotation in space.
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