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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 ) = 4 cos 3 t i + 4 sin 3 t j , t = π

To determine

To Calculate: The tangential and normal components of Acceleration at the given time t for the plane curve r(t).

Explanation

Given: r(t)=4cos3t i+4sin3t j, t=π

Formula Used:

The tangential and normal component of the acceleration are shown below:

If,  r(t)=a i+b j+c k then ||r(t)||=a2+b2+c2 and aT=v.av, aN=v×av

Calculation:

The velocity vector is v(t)= r'(t)=12sin3t i+12cos3t j

The Speed of the velocity vector can be calculated as:

v(t)=144 sin23t+144 cos2t = 12 [as sin23t + cos23t i]

The acceleration vector is,

a(t)=36cos3t i36sin3t j

Now, the tangential component of acceleration can be calculated as shown below,

aT=v

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