Explanation Given: The position vector r ( t ) = a cos ω t i + a sin ω t j Formula Used: Differentiation formula is: d d t x n = n x n − 1 a T = v . a ‖ v ‖ if, r ( t ) = a i+b j+c k then ||r(t)||= a 2 + b 2 + c 2 and a N = ‖ v × a ‖ ‖ v ‖ T ( t ) = r ' ( t ) ‖ r ' ( t ) ‖ and N ( t ) = T ' ( t ) ‖ T ' ( t ) ‖ Calculation: Consider the given position vector, r ( t ) = a cos ω t i + a sin ω t j Differentiate r ( t ) to get r ' ( t ) = − a w sin ω t i + a w cos ω t j Now magnitude of the velocity vector, ‖ r ' ( t ) ‖ = ‖ − a ω sin ω t i + a ω cos ω t j ‖ = a 2 ω 2 sin 2 ω t + a 2 ω 2 cos 2 ω t = a 2 ω 2 ( sin 2 ω t + cos 2 ω t ) = a 2 ω 2 = a ω Therefore, the unit tangent vector, T ( t ) = r ' ( t ) ‖ r ' ( t ) ‖ = − a ω sin ω t i + a ω cos ω t j a ω = − sin ω t i + cos ω t j Differentiate T ( t ) with respect to t , T ′ ( t ) = − ω cos ω t i − ω sin ω t j Now magnitude of the T ′ ( t ) vector, ‖ T ′ ( t ) ‖ = ‖ − ω cos ω t i − ω sin ω t j ‖ = ω 2 sin 2 ω t + ω 2 cos 2 ω t = ω 2 ( sin 2 ω t + cos 2 ω t ) = ω 2 = ω The principal unit normal vector N ( t ) = T ' ( t ) ‖ T ' ( t ) ‖

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 12.4, Problem 31E

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To calculate: The T(t), N(t), aT and aN for an object moving according to position vector r(t)=acosωti+asin ωtj

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Chapter 12.4, Problem 30E

Chapter 12.4, Problem 32E

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The T ( t ) , N ( t ) , a T and a N for an object moving according to position vector r ( t ) = a cos ω t i + a sin ω t j

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To calculate: The T(t), N(t), aT and aN for an object moving according to position vector r(t)=acosωti+asin ωtj

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Chapter 12 Solutions