   Chapter 12.4, Problem 30E

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# Circular MotionIn Exercises 31–34, consider an object moving according to the porition vector r ( t ) = a cos ω t i + a sin ω t j . Determine the direction of T and N relative to the position vector r.

To determine

To calculate: The directions of T and N relative to the position vector r(t)=acosωti+asin ωtj

Explanation

Given:

The position vector js, r(t)=acosωti+asin ωtj.

Formula used:

The magnitude of the vector is:

If,  r(t)=a i+b j+c k then ||r(t)||=a2+b2+c2

The principle unit tangent and normal vectors are:

T(t)=r'(t)r'(t) and N(t)=T'(t)T'(t)

Calculation:

Consider the provided position vector is,

r(t)=acosωti+asin ωtj

Now, differentiate provided position r(t) with respect to t therefore,

r'(t)=awsinωt i+awcosωt j

Now find the magnitude of the velocity vector.

Then the magnitude of velocity vector is,

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

Now, find, the unit tangent vector,

T(t)=r'(t)r'(t) = aωsinωt i+aω cosωt jaω=sinωt i +cosωt j

Now, differentiate unite tangent vector T(t) with respect to t. then,

T(t)=ωcosωt iωsinωtj

Now, find the magnitude of the T(t) vector

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