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

[1670frank]

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...

 

[haruspex]

θ'=1-cosθ

By θ' I assume you mean dθ/dt, right? Can you solve that differential equation?

 

[1670frank]

Yes, you can write it as you say

 

[haruspex]

I also asked whether you can solve the DE. I don't understand what you mean by "the velocity of θ' "