   Chapter 13.4, Problem 42E

Chapter
Section
Textbook Problem

# Find the tangential and normal components of the acceleration vector at the given point.42. r ( t ) = 1 t i + 1 t 2 j + 1 t 3 k ,  (1,1,1)

To determine

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

Explanation

Given data:

r(t)=1ti+1t2j+1t3k and (1,1,1) .

Formula used:

Write the expression for tangential component.

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

Write the expression for normal component.

aN=|r(t)×r(t)||r(t)| (2)

Equate the given vector for point (1,1,1) .

1t=1t=11t=1

Similarly,

1t2=1t2=11t=1t=1

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

Find r(t) .

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

Substitute 1ti+1t2j+1t3k for r(t) ,

r(t)=ddt[1ti+1t2j+1t3k]=ddt(1ti)+ddt(1t2j)+ddt(1t3k)=t2i2t3j3t4k

Substitute 1 for t ,

r(1)=12i2(13)j3(14)k=(1)i2(1)j3(1)k=i2j3k

Find r(t) .

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

Substitute t2i2t3j3t4k for r(t) ,

r(t)=ddt[t2i2t3j3t4k]=ddt(t2i)+ddt(2t3j)+ddt(3t4k)=2t3i+6t4j+12t5k

Substitute 1 for t ,

r(1)=2(13)i+6(14)j+12(15)k=2(1)i+6(1)j+12(1)k=2i+6j+12k

Find |r(1)|

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