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Roller Coaster and Centripetal Acceleration

  1. Jul 11, 2011 #1
    1. The problem statement, all variables and given/known data

    Part of a roller-coaster ride involves coasting down an incline and entering a loop 8.00 m in diameter. For safety considerations, the roller coaster speed at the top of the loop must be such that the force of the seat on a rider is equal in magnitude to the rider's weigt. From what height above the bottom of the loop must the roller coaster descend to satisfy this requirement?

    2. Relevant equations

    a_c = v^2/r
    Conservation of Energy


    3. The attempt at a solution
    m* a_c = mg

    m* v^2/r = mg
    v^2/r = g
    v^2 = rg

    mgh = 1/2 mv^2 + mg * 8
    9.8 * h = 1/2 * rg + 8 * 9.8

    9.8 * h = 1/2 * 4 * 9.8 + 8 * 9.8
    h = 2 + 8
    h = 10

    However, the answer is 12.

    Please guide me to the solution.

    Thanks!
     
  2. jcsd
  3. Jul 11, 2011 #2

    ehild

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    There act two forces on the rider, one is gravity (mg) and the other is the normal force N from the seat. The problem says that the seat must exert a force on the rider equal to mg. The resultant force is equal to the centripetal force.


    ehild
     
  4. Jul 11, 2011 #3
    So does this mean that n = -mg, so

    [itex] m * a_c = 2mg ? [/itex]

    [itex]a_c = 2g[/itex]


    [itex]a_c = 19.6[/itex]

    [itex]v^2/r = 19.6[/itex]
    [itex]v^2 = 19.6 * 4 = 78.4 [/itex]
    [itex]v = 8.85[/itex]

    [itex]9.8 * h = 1/2 * 9.85^2 + 8 * 9.8[/itex]

    [itex]x = 12.95[/itex]

    Hmm.. this is closer, but still not correct.
     
    Last edited: Jul 11, 2011
  5. Jul 11, 2011 #4

    ehild

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    You used v = 9.85 instead of 8.85. It is much better if you do not evaluate the equations, but express the height h in terms of R.

    ehild
     
  6. Jul 11, 2011 #5
    Ah thank you! I plugged in 8.85 now and it came out to 11.996, which is close enough.
    Thanks!
     
  7. Jul 11, 2011 #6

    ehild

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    I show the other way: mv^2/R=2mg
    hmg=mg*2R+1/2 mv^2 -->

    v^2=2gR,
    hg=2gR+1/2v^2--->hg = 3gR ---> h=3R.

    ehild
     
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