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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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FindingTangentialand NormalComponents of Acceleration In Exercises 25-30, Find the tangential and normal components of acceleration at the given time t for the plane curve r(t).

r ( t ) = e t cos t i + e t sin t j , t = π 2

To determine

To calculate:

Tangential and normal components of acceleration at the time t=π2 for the plane curve

r(t)=etcosti+etsintj

Explanation

Given:

r(t)=etcosti+etsintj,t=π2

Formula used:

Tangential component of acceleration, aT=va|v|

Normal component of acceleration, aN=|v×a||v|

Calculation:

According to question:

The given plane curve is r(t)=etcosti+etsintj

The velocity, speed and acceleration for the given curve are given as:

v(t)=r'(t)=et(costsint)i+et(sint+cost)j|v(t)|=(et(costsint))2+(et(sint+cost))2=e2t(2)=2e2ta(t)=r"(t)=v'(t)=et(2sint)i+et(2cost)j=2etsinti+2etcostj

Now calculate dot and cross product of velocity and acceleration:

va=(et(costsint)i+et(sint+cost)j)(2etsinti+2etcostj)=2e2t(costsintsin2t)+2e2t(costsint+cos2t)=2

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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 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