Need Timely Help with Unique Work by a Variable Force Problem

In summary, the conversation is about a problem involving the compression of gas in a circular cylinder. The work done in compressing the gas is given by a definite integral equation. The question asks to find the work done in a specific scenario, but the individual is confused about how to approach the problem and asks for help. The solution involves using the definition of work, the relationship between force and pressure, and the gas law equation to solve for the unknown variable.
  • #1
IFailedAlgebra
1
0
Hi, I'm taking a university extension program, so I can't talk with my professor. The Calc teacher at my high school doesn't understand the problem either.

The Problem:

Suppose that the gas in a circular cylinder of cross section area A is being compressed by a piston. If p is the pressure of the gas in psi, and V is the volume in cubic inches, the work done in compressing the gas from state (P1,V1) **(P sub 1, V sub 1)** to state (P2,V2) is given by the equation:

Work= [definite integral from (P2,V2) to (P1,V1)] p*dV


Find the work done in compressing the gas from V1=243 cubic inches to V2=32 cubic inches if P1=50 lb/cubic inch AND p and V obey the gas law pV^(1.4)=constant.


I cannot make heads or tails of this problem. I have never seen a coordinate pair used as a limit of integration.
I first thought that I should use the gas constant to make those limits into usable numbers, but P2 is not given.
I then thought that maybe I should use the gas constant (using P1 and V1) to find P2 (by setting P2(V2)^1.4 equal to the number given by plugging in P1 and V1). This creates a problem, making the upper and lower limit the same (109,350 is the exact number). This would defeat the purpose of integrating the equation, because the result would obviously be 0.
Any thoughts? Any help is greatly appreciated. My email is M_Daun@msn.com
 
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  • #2
It is a matter of putting one variable in terms of the other.

Starting with the definition of work, which is the integral of force applied over distance

[tex]W\,=\,\int_{x_1}^{x_2}\,F\,dx[/tex], but F = PA, where P is the pressure and A is area. Remember that F, P and x are vectors (x is the displacement vector.

Now the Work equation becomes

[tex]W\,=\,\int_{x_1}^{x_2}\,PA\,dx[/tex]

Now Volume V = Ax, and dV = A dx, and Vi corresponds to xi

[tex]W\,=\,\int_{V_1}^{V_2}\,P\,dV[/tex]

Now one is given that

[tex]PV^\gamma\,=\,k[/tex] or

[tex]P\,=\,kV^{-\gamma}[/tex], and substituting into the Work equation

[tex]W\,=\,\int_{V_1}^{V_2}\,kV^{-\gamma}\,dV[/tex]

k is known since one knows one of the state points P1, V1. Knowing either P2 or V2, one can solve for the other using the relationship between P and V.
 
  • #3


Hi there,

I understand that you are struggling with a problem involving a variable force and need timely help with unique work. I would suggest reaching out to your university extension program for assistance. They may have resources available such as tutoring or study groups that can help you with this specific problem.

Additionally, you can also try searching online for similar problems or reaching out to online communities or forums for assistance. There are many knowledgeable individuals who may be able to provide helpful insights and explanations.

In regards to the problem itself, it seems like you are on the right track by using the gas law to find the missing variable. Have you tried using the ideal gas law (PV = nRT) to find P2? This may help you in solving the problem and using the definite integral to find the work done.

I hope this helps and that you are able to find a solution soon. Don't hesitate to reach out for further assistance or clarification. Best of luck!
 

1. What is a variable force problem?

A variable force problem is a physics or engineering problem in which the force acting on an object changes over time or distance. This can make it more challenging to solve compared to problems with a constant force.

2. Why is timely help important for solving variable force problems?

Timely help is important for solving variable force problems because these types of problems often require a deep understanding of the underlying principles and mathematical concepts. Without timely help, it can be difficult to fully grasp the problem and find the correct solution.

3. What unique challenges do variable force problems present?

Variable force problems can present several unique challenges, such as the need to use calculus to calculate the changing force, the potential for multiple forces acting on an object, and the need to consider the direction of the force as well as its magnitude.

4. What steps can I take to solve a variable force problem?

To solve a variable force problem, it is important to carefully read and analyze the problem, draw a diagram to visualize the situation, identify the known and unknown variables, and apply the relevant equations and principles to find the solution.

5. How can I improve my skills in solving variable force problems?

To improve your skills in solving variable force problems, you can practice solving various types of problems, seek help from a tutor or teacher, and actively engage in class discussions and problem-solving sessions. It can also be helpful to review and understand the underlying principles and concepts involved in these types of problems.

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