Is my use of ##e_x## in my solution for Fluids Angular Momentum correct?

[member 428835]

I don't typically do this, but attached is a figure with a problem statement. This is not homework because I have already solved it! My question is a subtle one.
Please see the attachments, one is the question and the other is my solution.

I know my solution is correct, but at the end of equation (2) I wrote $n = e_x$ where $e_x$ is the unit vector in the $x$ direction. At the time I am not sure why I wrote $e_x$ this way, because I now think it should be $\cos \theta e_x + \sin \theta e_y$. Any help here would be awesome in understanding why what I had was correct.

Thanks so much.

 

[haruspex]

I am not going to try to follow your intricate algebra, but I will suggest an easier way. Take moments about the sprinkler's axis and apply conservation of angular momentum. What is the tangential velocity of the jet in the ground frame?

 

[member 428835]

Tangential velocity is $-r \omega + Q \cos \theta / 3 A$.

I think I know how to solve the problem assuming velocity out of the sprinkler pipe is constant (or rather averaged), but I want to make sure I can do it both ways (the way you mention above and using integrals).

Thanks for the reply

 

[member 428835]

Never mind, I figured it out. For others, in case they want to know, I'll attach the solution.