   Chapter 13.4, Problem 41E

Chapter
Section
Textbook Problem

# Find the tangential and normal components of the acceleration vector at the given point.41. r ( t ) = ln t i + ( t 2 + 3 t ) j + 4 t k ,  (0,4,4)

To determine

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

Explanation

Given:

r(t)=lnti+(t2+3t)j+4tk and (0,4,4) .

Formula used:

Write the expression for tangential component.

aT=r(t)r(t)|r(t)| (1)

Equate the given vector function for point (0,4,4) .

lnt=0t=e0t=1

Similarly,

4t=4t=44t=1t=12

t=1

Therefore the point (0,4,4) corresponds to t=1 .

Find r(t)

r(t)=ddt[r(t)]

Substitute Inti+(t2+3t)j+4tk for r(t) ,

r(t)=ddt[Inti+(t2+3t)j+4tk]=ddt(Inti)+ddt[(t2+3t)j]+ddt(4tk)=(1t)i+(2t+3)j+(2t)k

Substitute 1 for t ,

r(1)=(11)i+[2(1)+3]j+(21)k=1i+(2+3)j+(21)k=i+5j+2k

Find r(t) .

r(t)=ddt[r(t)]

Substitute (1t)i+(2t+3)j+(2t)k for r(t) ,

r(t)=ddt[(1t)i+(2t+3)j+(2t)k]=ddt[(1t)i]+ddt[(2t+3)j]+ddt[(2t)k]=(1t2)i+2j(1t32)k

Substitute 1 for t ,

r(1)=(112)i+2j(1132)k=(11)i+2j(11)k=i+2jk

Find |r(1)|

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