Work Problems: Calculus Solutions Explained

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SUMMARY

This discussion focuses on solving calculus problems related to work, specifically through the application of integrals. Key examples include calculating work done on a spring and lifting a cable with a defined weight. The fundamental equation highlighted is that work equals the integral of force, with specific scenarios provided to illustrate its application. The discussion emphasizes the importance of expressing force as a function of distance to facilitate problem-solving.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with the concept of force and work
  • Knowledge of spring constants and Hooke's Law
  • Basic principles of pressure and volume in gas laws
NEXT STEPS
  • Study the application of Hooke's Law in spring problems
  • Learn how to derive work done from force functions in calculus
  • Explore the relationship between pressure, volume, and work in thermodynamics
  • Practice solving problems involving integrals of force over distance
USEFUL FOR

Students in calculus courses, educators teaching physics and mathematics, and anyone looking to deepen their understanding of work-related problems in calculus.

Gauss177
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We recently covered work in my calculus class, and I'm confused on how to approach these problems. I know work is the integral of force, but I just don't know how to start these problems. Thanks for the help.

Homework Statement


1. If 6 J of work are needed to stretch a spring from 10 cm to 12 cm and another 10 J are needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?

2. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Find the work done.

3. When gas expands in a cylinder with radius r, the pressure at any given time is a function of the volume: P = P(V). The force exerted by the gas on the a piston (in the cylinder) is the product of the pressure and the area: [tex]F = \pi r^2P[/tex]. Show that the work done by the gas when the volume expands from volume V1 to volume V2 is: W = Integral from V1 to V2 of P dV.

Homework Equations


Work = integral of Force

The Attempt at a Solution

 
Last edited:
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Start by writing the force as a function of distance.
 

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