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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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Circular Motion In Exercises 31-34, consider an object moving according to the position vector

r ( t ) = a cos ω t i + a sin ω t j .

Determine the speed of the object at any time t and explain its value relative to the value of a T .

To determine

To calculate: The speed of the object with position vector r(t)=acosωti+asin ωtj at any time

t and explain its value relative to the value of aT.

Explanation

Formula Used:

The magnitude of the vector and tangential 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

Consider the provided position vector is,

r(t)=acosωti+asin ωtj

Now, differentiate the provided position vector r(t) to find the velocity vector as shown below,

r'(t)=awsinωt i+awcosωt j

since, the Speed of the object is equal to the magnitude of the velocity vector,

Therefore, the speed of the object is,

r'(t)=aωsinωt i+aωcosωt j=a2ω2sin2ωt+a2ω2cos2ωt=a2ω2(sin2ωt+cos2ωt)=a2ω2= aω

Now, find the second derivative of the position vector to find the acceleration vector

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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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