Chapter 12.4, Problem 50E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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# Motion Along an Involute of a Circle The figure shows a particle moving along a path modeled by r ( t ) = âŒ© cos Ï€ t + Ï€ t sin Ï€ t , sin Ï€ t âˆ’ Ï€ t cos Ï€ t âŒª .The figure also shows the vectors v(t) and a(t) for t = 1 and t = 2 .(a) Find a T and a N at t = 1 and t = 2 .(b) Determine whether the speed of the particle is increasing or decreasing at each of the indicated values of t. Give reasons for your answers.

(a)

To determine

To Calculate: The value of the tangential and normal components of acceleration as, aT and aN at t=1 and t=2 if the position vector is,

r(t)= cosπt+πtsinπt, sinπtπtcosπt.

Explanation

Given:

The provided position vector is, r(t)= âŒ©cosÏ€t+Ï€tsinÏ€t, sinÏ€tâˆ’Ï€tcosÏ€tâŒª.

Formula Used:

The tangential velocity and tangential acceleration are:

v(t)=dr(t)dt and a(t)=dv(t)dt.

Calculation:

The velocity is calculated as shown below,

Now, find the magnitude of the velocity vector to find speed of the particle as shown below,

â€–v(t)â€–= Ï€2t â€¦â€¦ (1)

Now, find the acceleration vector of the particle as calculated below,

Now, find the tangential component of acceleration as shown below,

Since, cos2Ï€+sin2Ï€t=1

The value is found as shown below,

aT=Ï€2

At t=1 and t=2 the tangent vector is aT=Ï€2

(b)

To determine
Whether the speed of particle is increasing or decreasing at each of the indicated value of t from t=1 to t=2 if the position vector is,

r(t)= cosπt+πtsinπt, sinπtπtcosπt.

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