BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

As in Exercise 9, consider a spring with mass m, spring constant k, and damping constant c = 0, and let ω = k / m . If an external force F(t) = F0 cos ωt is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to show that the motion of the mass is given by

x ( t ) = c 1 cos ω t + c 2 sin ω t + F 0 2 m ω t sin ω t

To determine

To show: If external force F(t)=F0cosωt is applied, then the motion of the mass is given by x(t)=c1cosωt+c2sinωt+F02mωtsinωt .

Explanation

Given data:

F(t)=F0cosωt , ω=km , c=0

Formula used:

Write the expression for Newton’s Second Law with external force.

md2xdt2+kx=F(t)

mx+kx=F(t) (1)

Write the expression for auxiliary equation.

mr2+k=0 (2)

Write the expression for general solution with complex roots.

x(t)=eαt[c1cos(βt)+c2sin(βt)] (3)

Write the expression for r .

r=α+βi (4)

Find the expression for Newton’s Second Law with external force using equation (1).

Substitute F0cosωt for F(t) in equation (1),

mx+kx=F0cosωt (5)

Find the roots using equation (2).

mr2=kr2=kmr=kmr=±kmi

Substitute ω for km ,

r=±ωi (6)

Compare equations (4) and (6).

α=0β=ω

Substitute 0 for α and ω for β in equation (3),

x(t)=e(0)t[c1cos(ωt)+c2sin(ωt)]

x(t)=c1cos(ωt)+c2sin(ωt) (7)

Modify equation (7) for complementary equation as follows.

xc(t)=c1cos(ωt)+c2sin(ωt)

The natural frequency of the system equals to the frequency of the external force.

Consider the value of xp(t) as follows.

xp(t)=t(Acosωt+Bsinωt) (8)

Differentiate equation (8) with respect to t .

xp(t)=t[ωAsinωt+ωBcosωt]+[Acosωt+Bsinωt]

Differentiate equation with respect to t .

xp(t)=t[ω2Acosωtω2Bsinωt]+[ωAsinωt+ωBcosωt]+[ωAsinωt+ωBcosωt]=t[ω2Acosωtω2Bsinωt]+[2ωAsinωt+2ωBcosωt]

Modify equation (5) as follows

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 17 Solutions

Show all chapter solutions add
Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Solve the equations in Exercises 126. 6x(x2+1)2(x2+2)48x(x2+1)3(x2+2)3(x2+2)8=0

Finite Mathematics and Applied Calculus (MindTap Course List)

Define the terms population and sample, and explain the role of each in a research study.

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

83-94 Evaluate the integral. ex1+exdx

Calculus (MindTap Course List)

In Exercises 4148, find the indicated limit given that limxaf(x)=3 and limxag(x)=4 41. limxa[f(x)g(x)]

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Exercises 43 to 46, list all subsets of the given set. ,

Mathematical Excursions (MindTap Course List)

In Problems 5-10, use integration to find the general solution to each differential equation.

Mathematical Applications for the Management, Life, and Social Sciences

f(x) dx = F(x) means a) f(x) = F(x) b) f(x) = F(x) c) f(x) = F(b) F(a) d) f(x) = F(x) + C

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False:

Study Guide for Stewart's Multivariable Calculus, 8th