Explanation Given: The acceleration of the motion r ( t ) = ( cosh t ) i − ( sinh t ) j + t k . 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: From Exercise 9 of section 13.4 the value of T = ( 1 2 tanh t ) i − 1 2 j + ( 1 2 sech t ) k and the value of N = ( sech t ) i − ( tanh t ) k . So, the value of B can be calculated as follows. B = T × N = | i j k 1 2 tanh t − 1 2 1 2 sech t sech t 0 − tanh t | = i ( − 1 2 ⋅ ( − tanh t ) − 0 ( 1 2 sech t ) ) − j ( 1 2 tanh t ⋅ ( − tanh t ) − ( sech t ) 1 2 sech t ) + k ( 1 2 tanh t ⋅ 0 − sech t ⋅ − 1 2 ) = ( 1 2 tanh t ) i + 1 2 j + ( 1 2 sech t ) k The value of v is v = ( sinh t ) i − ( cosh t ) j + k

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ISBN: 9780135430903

Author: Hass

Publisher: PEARSON

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Chapter 13.5, Problem 16E

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The value of B and τ for space curves.

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Chapter 13.5, Problem 15E

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Chapter 13 Solutions

THOMAS' CALC. EARLY TRANS.W/ACCESS

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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