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

A spring with an 8-kg mass is kept stretched 0.4 m beyond its natural length by a force of 32 N. The spring starts at its equilibrium position and is given an initial velocity of 1 m/s. Find the position of the mass at any time t.

To determine

To find: The position of the mass at any time t .

Explanation

Given data:

The spring is stretched beyond its natural length, so x=0.4 , Initial velocity x(t)=1ms ,

m=8kg , restoringforce=32N .

Formula used:

Write the expression for Hooke’s Law.

restoringforce=kx (1)

Here,

k is spring constant, and

x is difference between the natural length and length of due to force exerts.

Write the expression for Newton’s Second Law.

md2xdt2+kx=0

mx+kx=0 (2)

Write the expression for auxiliary equation.

mr2+k=0 (3)

Write the expression for the complex roots.

r=±iω (4)

Here,

ω=km

Write the expression for general solution of mr2+k=0 .

x(t)=c1cosωt+c2sinωt (5)

Substitute 0.4m for x and 32N for restoring force in equation (1),

32N=k(0.4m)k=32N0.4mk=80Nm

Substitute 80 for k and 8 for m in equation (2),

8x+80x=0

Find the auxiliary equation using equation (3).

8r2+80=0

Solve for r .

8r2=80r2=10r=10

Simplify r as follows.

r=±10i (6)

Compare equation (4) with equation (6).

ω=10

Substitute 10 for ω in equation (5),

x(t)=c1cos(10)t+c2sin(10)t (7)

Since, the spring is start at its equilibrium position the value of x(0)=0

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

Other Equations Find all real solutions of the equation. 112. x5x+6=0

Precalculus: Mathematics for Calculus (Standalone Book)

2. Define the terms population, sample, parameter and statistic.

Statistics for The Behavioral Sciences (MindTap Course List)

In Exercises 914, evaluate the expression. 9. (94)3/2

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

Rectangular coordinates of the point with polar coordinates are: (−1, 0) (0, 1) (0, −1) (1, 0)

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