Little trees makes angle around turn

[Calculon]

Homework Statement

Your car is smelly so you've placed a Little Trees air freshener in your rear view mirror. You begin to turn around a corner of radius R and the little tree starts to make an angle θ from the vertical. What is the angle θ as a function of the speed v and and radius R?

Homework Equations

radial acceleration= v²/R

The Attempt at a Solution

So I began by drawing a triangle with the hypotenuse representing the tension vector, T, at an angle of Θ from the vertical. I represented the opposite side TsinΘ and the adjacent side as TcosΘ. I then calculated the net forces on the little tree...

∑F_y=TcosΘ-mg=0
∑F_x=TsinΘ=ma

I figured that a here represents the radial acceleration, so I let a= v²/R. Because there is no y movement, I ignored y-axis forces.

So I changed the equation to be:
TsinΘ=mv²/R

This is sort of where I ran into trouble. So I can't have T in the equation so I substituted in T=mg/cosΘ from the y equation. When I plug this into the primary equation, I end up having after cancellation:
gtanΘ=v²/R. I further simplified this to give me the desired function:

arctan(v²/Rg)=Θ

I'm not sure if I did it correctly because I have g left in the final equation, and I'm not sure how to get it out.

This is my first post on this website, I am teaching myself intro mechanics over the summer and I found this off of a university's old exam and there is no solutions guide, so any confirmation or correction of my solution would be much appreciated. Thanks!

 

[Simon Bridge]

Well done - why do you need to get rid of the g in the equation - if gravity were stronger, what do you think would happen to the angle the tree hangs for the same centripetal acceleration?