Explanation Acceleration vector a ( t ) can be decompose into tangential component (tangent to the path of motion) and normal component (perpendicular to the path of motion) as a ( t ) = a T T ( t ) + a N N ( t ) ------------------------(1) where unit tangent vector, T ( t ) = r ′ ( t ) ∥ r ′ ( t ) ∥ = v ∥ v ∥ and unit normal vector, N ( t ) = T ′ ( t ) ∥ T ′ ( t ) ∥ a T = a · T = a · v ∥ v ∥ a N = a · N = ∥ a × v ∥ ∥ v ∥ Given: Position vector r ( t ) = 〈 t , cos t , t sin t 〉 Formula used: Decomposition of a ( t ) into tangential component T ( t ) is, a T = a · T = a · v ∥ v ∥ ----------------(2) Decomposition of a ( t ) into normal component N ( t ) is, a N = a · N = ∥ a × v ∥ ∥ v ∥ ---------------(3) Calculation: Since position vector r ( t ) = 〈 t , cos t , t sin t 〉 ----------------(4) Therefore, velocity, v ( t ) = r ′ ( t ) = 〈 1 , − sin t , sin t + t cos t 〉 ------------(5) and acceleration, a ( t ) = v ′ ( t ) = r ″ ( t ) = 〈 0 , − cos t , 2 cos t − t sin t 〉 ---------(6) At point t = π / 2 , we get From equation (5), v ( π 2 ) = r ′ ( π 2 ) = 〈 1 , − 1 , 1 〉 --------------------------(7) From equation (6), a ( π 2 ) = v ′ ( π 2 )

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Author: Rogawski

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Chapter 13.5, Problem 43E

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Decomposition of acceleration vector $a (t)$ of $r (t) = 〈 t, \cos t, t \sin t 〉$ into tangential and normal components at the point $t = π / 2$ .

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Chapter 13.5, Problem 42E

Chapter 13.5, Problem 44E

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Chapter 13 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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Decomposition of acceleration vector a ( t ) of r ( t ) = 〈 t , cos t , t sin t 〉 into tangential and normal components at the point t = π / 2 .

Decomposition of acceleration vector a(t) of r(t)=〈t,cost,tsint〉 into tangential and normal components at the point t=π/2.

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Chapter 13 Solutions

Decomposition of acceleration vector $a (t)$ of $r (t) = 〈 t, \cos t, t \sin t 〉$ into tangential and normal components at the point $t = π / 2$ .