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A Circular Motion Problem

  1. Jun 11, 2009 #1
    A skier starts from rest at the top of a frictionless slope of ice in the shape of a hemispherical dome with radius R and slides down the slope. At a certain hight h, teh normal force becomes zero and the skier leaves the surface of teh ice. What is h in term of R?

    I know the answer is 2R/3, but couldn't figure out how to get there.

    Thanks.
     
  2. jcsd
  3. Jun 11, 2009 #2

    Doc Al

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    Show what you've done so far and where you are stuck.

    Hint: Consider the forces acting on the skier at any point and apply Newton's 2nd law.
     
  4. Jun 11, 2009 #3

    tiny-tim

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    Welcome to PF!

    Hi Almond ! Welcome to PF! :smile:
    Hint: centripetal acceleration :wink:
     
  5. Jun 12, 2009 #4
    Well, I tried to use conservation of energy,
    mgR=mgh+1/2mv2
    Then I used centripetal acceleration, a=V2 /R
    gR=gh+1/2aR,

    And I'm stuck, not sure how to get a.
     
  6. Jun 12, 2009 #5

    tiny-tim

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    ok, now draw a diagram (for yourself) and call the angle θ …

    what is the normal force when the skier is at angle θ from the vertical? :smile:
     
  7. Jun 12, 2009 #6
    another hint:

    Draw a free body diagram of the skier as he leaves the slope. What is the normal force at that point? what are the sum of the forces in the direction of the normal force? The skier "drops" from R to h, is there an equation of motion that relates velocity as a function of position? The geometry of the skier as he leaves the slope in terms of R and h will be helpful.
     
  8. Jun 14, 2009 #7
    I have managed to get the answer somehow by using these two equations:

    h=Rsin[tex]\theta[/tex] and
    a=gsin[tex]\theta[/tex]

    But I don't understand the second equation, shouldn't it be g=asin[tex]\theta[/tex], since mg points downwards and is the component of centripetal force, I suppose? I am totally confused with the free body diagram.:frown:

    Thanks for all the help so far!:smile:
     
  9. Jun 14, 2009 #8

    Doc Al

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    If mg points down, what is its radial component?

    Draw a diagram at the moment that the skier just begins to lose contact with the surface. What forces act? What are their components? What net force acts toward the center of the sphere?
     
    Last edited: Jun 14, 2009
  10. Jun 14, 2009 #9
    An approach that helped me solve this problem was drawing a free body diagram for the skier, from the point of view of his accelerated system and look at all the forces he has acting upon him. From there, just look at the specific instant where N=0 and find the correlating angle Θ from the vertical.
     
  11. Jun 14, 2009 #10

    ideasrule

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    I agree with RoyalCat. Whenever I come across a problem involving centripetal acceleration, I always pretend there's a centrifugal force acting on the accelerated body. It makes so much more sense for me.
     
  12. Jun 14, 2009 #11
    and I could never work my way with centrifugal force.. always wondered, "where is this coming from!?"
     
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