   Chapter 12.4, Problem 27E

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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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