   # Motion of a SpringIn Exercise 35-36, a 64-pound weight stretched a spring 4 3 foot from its natural length. Use the given information to find a formula for the position of the weight as a function of time. The weight is pulled 3 4 foot below equilibrium and released. The motion takes place in a medium that furnishes a damping force of magnitude 1 8 | v | at all times. ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 36RE
Textbook Problem
1 views

## Motion of a SpringIn Exercise 35-36, a 64-pound weight stretched a spring 4 3 foot from its natural length. Use the given information to find a formula for the position of the weight as a function of time.The weight is pulled 3 4 foot below equilibrium and released. The motion takes place in a medium that furnishes a damping force of magnitude 1 8 | v | at all times.

To determine

The formula for the position of weight as a function of time when a 64-pounds weight stretches a spring 43 foot from its natural length. The weight is pulled 34 foot below the equilibrium and released. The motion takes place in a medium that furnishes a damping force of magnitude 18|v| at all times.

### Explanation of Solution

Given information: The weight is pulled 34 foot below the equilibrium and released. The motion takes place in a medium that furnishes a damping force of magnitude 18|v| at all times.

The 64-pounds weight stretches a spring 43 foot from its natural length, so by Hooke’s Law,

64=k(43)64×32=k48=k

As the weight w is given by mg, we have

m=wg=6432=2

So, the resulting differential equation for the motion is

d2ydt2+pmdydt+kmy=0d2ydt2+(18)2dydt+481y=0d2ydt2+116dydt+24y=0

The characteristic equation for the differential equation d2ydt2+116dydt+24y=0 is m2+116m+24=0.

The roots of the characteristic equation m2+116m+24=0 are:

m2+116m+24=016m2+m+384=0m=1±14×16×3842×16m=1±12457632m=1±2457532m=1±24575i32

If m1=α+iβ and m2=αiβ are complex zeros of the characteristics equation, then the general solution is y=C1eαtcosβt+C2eαtsinβt.

Thus, the general solution of the equation d2ydt2+116dydt+24y=0 is y=C1e132tcos2457532t+C2e132tsin2457532t.

Now, when t=0, y=34, so

y=C1e132tcos2457532t+C2e132tsin2457532t34=C1(1)cos(0)+C2(1)sin(0)34=C1(1)(1)34=C1

Now, differentiate y=C1e132tcos2457532t+C2e132tsin2457532t with respect to t

### 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

Find more solutions based on key concepts
The gradient vector field for is:

Study Guide for Stewart's Multivariable Calculus, 8th

In Exercises 16, find the points of intersection of the graphs of the functions. Express your answer accurate t...

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

The slope of the tangent line to r = cos θ at is:

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

Convert the expressions in Exercises 6584 to power form. 23x1.2x2.13

Finite Mathematics and Applied Calculus (MindTap Course List)

In Problems 25-38, find y'.

Mathematical Applications for the Management, Life, and Social Sciences

Solve each equation and check: 17=3+w

Elementary Technical Mathematics

In Exercise 9-16, find the steady state vector for the transition matrix. 

Finite Mathematics for the Managerial, Life, and Social Sciences

Describe how replication protects against fraud being committed in research.

Research Methods for the Behavioral Sciences (MindTap Course List)

Finding Inverse Function In problems A-5 through A-10, find a formula for the inverse of the given function. f(...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

For many years businesses have struggled with the rising cost of health care. But recently, the increases have ...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Use a Maclaurin series to show that a power series solution of the initial-value problem d2dt2+glsin=0,(0)=6,(0...

A First Course in Differential Equations with Modeling Applications (MindTap Course List) 