Work Problems: Calculus Solutions Explained

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Then, use the definition of work as the integral of force to solve for the natural length of the spring. For the second problem, use the weight of the coal and the length of the cable to find the work done. For the third problem, use the given equations to solve for the work done by the gas as it expands from V1 to V2. In summary, work in calculus involves using integrals to find the total amount of force applied over a certain distance. This can be applied to problems involving springs, lifting objects, and expanding gases.
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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

 
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Start by writing the force as a function of distance.
 

FAQ: Work Problems: Calculus Solutions Explained

What is a work problem in calculus?

A work problem in calculus involves finding the work done by a variable force over a certain distance. This type of problem is typically solved using integration to find the area under a force-distance curve.

What are the steps to solving a work problem in calculus?

The steps to solving a work problem in calculus are as follows:

  • 1. Determine the force function, which represents the variable force acting on the object
  • 2. Determine the distance function, which represents the distance the object has moved
  • 3. Set up the integral by multiplying the force function by the distance function
  • 4. Integrate the resulting function over the given interval of distance
  • 5. Evaluate the integral to find the work done by the force

What are some common real-life applications of work problems in calculus?

Work problems in calculus are commonly used in physics and engineering to calculate the work done by varying forces, such as the force of gravity or the force of a spring. They are also used in economics to calculate the work done by a variable production rate.

What are some key concepts to understand when solving work problems in calculus?

Some key concepts to understand when solving work problems in calculus include the concept of a variable force and how it relates to the work done, the concept of integration and finding the area under a curve, and the concept of limits and how they are used in calculus.

Are there any tips for solving work problems in calculus more efficiently?

One tip for solving work problems in calculus more efficiently is to carefully choose the orientation of the coordinate system and the direction of the force and displacement vectors. This can simplify the integration process and make the problem easier to solve. Additionally, it can be helpful to break the problem down into smaller, more manageable parts and then sum up the total work done at the end.

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