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Little trees makes angle around turn

  1. Jun 26, 2014 #1
    1. The problem statement, all variables and given/known data
    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?


    2. Relevant equations
    radial acceleration= v2/R

    3. 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...

    ∑Fy=TcosΘ-mg=0
    ∑Fx=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!
     
  2. jcsd
  3. Jun 27, 2014 #2

    Simon Bridge

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    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?
     
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