Solve Flows Homework: θ'=1-cosθ, Find T(A) & Behavior at A=0

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Homework Help Overview

The problem involves a differential equation θ' = 1 - cos(θ) and seeks to determine the time T(A) it takes for the flow to reach θ = π, starting from θ(0) = A. Additionally, it explores the behavior of the solution as A approaches 0.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between the given equation and the time required for the flow to complete a full cycle. There is uncertainty about how to apply the equation T = 2π/w in this context, with some suggesting a different interpretation based on the travel distance from A to π. Questions arise regarding the meaning of θ' and whether the differential equation can be solved.

Discussion Status

The discussion is ongoing, with participants clarifying terms and exploring different interpretations of the problem. Some guidance has been offered regarding the differential equation, but no consensus has been reached on the approach to finding T(A) or the behavior as A approaches 0.

Contextual Notes

There is a lack of clarity on the definitions and implications of the terms used, particularly concerning the velocity and the nature of the differential equation. Participants are navigating these uncertainties as they attempt to understand the problem better.

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Homework Statement



θ'=1-cosθ If we have the flow starting at θ(0)=A, determine the amount of time, T(A), that it takes to reach θ=pi and what is the leading behavior as A approaches 0.


Homework Equations





The Attempt at a Solution


Well, in my book, they give an equation T=2pi/w where w is the velocity of θ'. T means in this equatio the time the flow requires to go a full circle back to the initial point back to 0. Not sure how to use this equation in this case I think they got this equation by saying that θ(t) changes by 2pi meaning the travel distance while in this case the travel distance is pi-A so T=(pi-A)/w but not exactly sure about this. I really need help with this...
 
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1670frank said:
θ'=1-cosθ
By θ' I assume you mean dθ/dt, right? Can you solve that differential equation?
 
Yes, you can write it as you say
 
I also asked whether you can solve the DE. I don't understand what you mean by "the velocity of θ' "
 

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