Explanation Formula used: Unit tangent vector : T = v | v | . Principal unit normal vector: N = d T d t | d T d t | . Binomial vector: B = T × N Curvature: κ = | d T d s | = | v × a | | v | 3 . Torsion: τ = − d B d s ⋅ N = | x ˙ y ˙ z ˙ x ¨ y ¨ z ¨ x ⃛ y ⃛ z ⃛ | | v × a | 2 Tangential and normal scalar components of acceleration: a = a T T + a N N a T = d d t | v | a N = κ | v | 2 = | a | 2 − a T 2 Calculation: The torsion of the helix is given by as follows. r ( t ) = ( a cos t ) i + ( a sin t ) j + b t k . By using the formula of torsion determine the value of torsion. τ = − 1 | v | d B d t ⋅ N It is known that r ( t ) = ( a cos t ) i + ( a sin t ) j + b t k is a smooth curve for all t . v ( t ) = d r d t = d d t ( ( a cos t ) i + ( a sin t ) j + b t k ) = ( − a sin t ) i + ( a cos t ) j + b k Find the value of | v | as follows. | v | = | ( − a sin t ) i + ( a cos t ) j + b k | = ( − a sin t ) 2 + ( a cos t ) 2 + b 2 = a 2 ( sin 2 t + cos 2 t ) + b 2 = a 2 + b 2 Now find the value of Unit tangent vector. T = v | v | = ( − a sin t ) i + ( a cos t ) j + b k a 2 + b 2 = ( − a a 2 + b 2 sin t ) i + ( a a 2 + b 2 cos t ) j + ( b a 2 + b 2 ) k Now find the value of Principal unit normal vector as follows. N = d T d t | d T d t | = d d t ( ( − a a 2 + b 2 sin t ) i + ( a a 2 + b 2 cos t ) j + ( b a 2 + b 2 ) k ) | d d t ( ( − a a 2 + b 2 sin t ) i + ( a a 2 + b 2 cos t ) j + ( b a 2 + b 2 ) k ) | = ( a a 2 + b 2 cos t ) i + ( − a a 2 + b 2 sin t ) j + 0 k | ( a a 2 + b 2 cos t ) i + ( − a a 2 + b 2 sin t ) j + 0 k | = ( a a 2 + b 2 cos t ) i

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ISBN: 9780134606095

Author: Hass

Publisher: PEARSON

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Chapter 13.5, Problem 26E

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The torsion of the helix.

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Chapter 13 Solutions

EBK THOMAS'CALCULUS,EARLY TRANSCEND.

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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