Explanation Given data: r ( t ) = 2 t 2 i + ( 2 3 t 3 − 2 t ) j Formula used: Write the expression for tangential component. a T = r ′ ( t ) ⋅ r ″ ( t ) | r ′ ( t ) | (1) Write the expression for normal component. a N = | r ′ ( t ) × r ″ ( t ) | | r ′ ( t ) | (2) Find r ′ ( t ) r ′ ( t ) = d d t [ r ( t ) ] Substitute 2 t 2 i + ( 2 3 t 3 − 2 t ) j for r ( t ) , r ′ ( t ) = d d t [ 2 t 2 i + ( 2 3 t 3 − 2 t ) j ] = d d t [ ( 2 t 2 ) i ] + d d t [ ( 2 3 t 3 − 2 t ) j ] = 2 ( 2 t ) i + 2 3 ( 3 t 2 ) j − 2 j = 4 t i + 2 t 2 j − 2 j r ′ ( t ) = 4 t i + ( 2 t 2 − 2 ) j Find r ″ ( t ) . r ″ ( t ) = d d t [ r ′ ( t ) ] Substitute 4 t i + ( 2 t 2 − 2 ) j for r ′ ( t ) , r ″ ( t ) = d d t [ 4 t i + ( 2 t 2 − 2 ) j ] = d d t ( 4 t i ) + d d t [ ( 2 t 2 − 2 ) j ] = 4 i + 2 ( 2 t ) j − 0 j = 4 i + 4 t j Find | r ′ ( t ) | . | r ′ ( t ) | = ( 4 t ) 2 + ( 2 t 2 − 2 ) 2 = ( 4 t ) 2 + ( 2 t 2 ) 2 − 8 t 2 + 4 { ∵ ( a − b ) 2 = a 2 − 2 a b + b 2 } = 16 t 2 + 4 t 4 − 8 t 2 + 4 = 4 t 4 + 8 t 2 + 4 | r ′ ( t ) | = 4 ( t 2 + 1 ) 2 = 2 ( t 2 + 1

8th Edition

ISBN: 9780357300961

Author: Stewart

Publisher: CENGAGE C

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Question

Chapter 13.4, Problem 38E

To determine

To find: The tangential components of the acceleration vector and normal components of the acceleration vector.

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Chapter 13.4, Problem 37E

Chapter 13.4, Problem 39E

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Chapter 13 Solutions

CALC:EARLY TRANS.CUST W/WEBASSIGN>IC<

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The tangential components of the acceleration vector and normal components of the acceleration vector.

Solution Summary: The author explains how to find the tangential and normal components of the acceleration vector.

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To find: The tangential components of the acceleration vector and normal components of the acceleration vector.

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Chapter 13 Solutions