   Chapter 12.4, Problem 31E

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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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