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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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Motion Along an Involute of a Circle The figure shows a particle moving along a path modeled by

r ( t ) = cos π t + π t sin π t , sin π t π t cos π t .

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.

Chapter 12.4, Problem 50E, Motion Along an Involute of a Circle The figure shows a particle moving along a path modeled by

(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)= cosπt+πtsinπt, sinπtπtcosπt.

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,

v(t)=r(t)=πsinπt+πsinπt+π2tcosπt, πcosπtπcosπt+π2sinπt= π2tcosπt,π2tsinπt

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

v(t)= π2t …… (1)

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

a(t)= r(t)=π2cosπtπ3tsinπt, π2sinπt+π3t cosπt

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

aT=v·av=π2tcosπt,π2tsinπt·π2cosπtπ3tsinπt, π2sinπt+π3t cosπtπ2t=π4tcos2πtπ5tcosπtsinπt+π4tsin2πt+π5t2sinπtcosπtπ2t=π4t(cos2πt+sin2πt)π2t

Since, cos2π+sin2πt=1

The value is found as shown below,

aT=π2

At t=1 and t=2 the tangent vector is aT=π2

(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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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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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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