Work 

Spring of Unknown Length 

Copyright Maplesoft, a division of Waterloo Maple Inc., 2007 

Introduction 

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos. 

The steps in the document can be repeated to solve similar problems. 

 

Problem Statement 

A spring obeying Hooke's law is extended to a total length of .  To extend it to a total length of requires .  To extend it further to a total length of requires an additional .  Find the natural length of the spring. 

Solution 

For a spring obeying Hooke's law, the extension is a linear function of the applied force, or , where is the spring constant (with units )  and is the force exerted by the spring. A joule () is a unit of mechanical energy generated when a force of one Newton is applied over a distance of one meter. The work done as the spring is extended to a total length of from a total length is given by the integral 

 

 

 

where is the natural (unstretched) length of the spring. 

 

Step	Result
Input the equation for the work done when the spring is extended from to . is the natural length of the spring.    Use the definite integral template in the Expression palette to construct the equation. Select the entry from the Units(SI) palette and enter the appropriate units in place of "units." Right-click, Simplify.	(3.1)     (3.2)
Input the equation for the work done when the spring is extended from to .	(3.3)     (3.4)
Simultaneously solve the two equations generated above.     Using the label references ([Ctrl+L] and the label number) of the two equations of work, place the two equations on one line with a comma between them. Right-click, select Solve.	(3.5)

 

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