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Circular Motion in a Roller Coaster

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data
    In a certain loop-the-loop roller coaster ride, the coaster is given an intial speed of v ms-1. The coaster then rolls freely under the influence of gravity. The coaster travels down a 10m high slope before arriving at the start of the loop. It then travels the loop which has a diameter of 20m. Calculate the minimum value of v required for the coaster to make the loop.

    2. Relevant equations
    mv^2/r = mg
    E(kinetic) = 0.5mv^2
    E(potential) = mgh

    3. The attempt at a solution
    I found the increase in speed due to gravity using motion formulae
    Assuming up is positive and down is negative,
    v^2 = u^2 + 2as
    v^2 = 0^2 + 2(-9.8)(10)
    v = 14 ms-1 down

    When the train reaches the loop it has completely horizontal velocity, therefore its velocity at the bottom of the loop is v+14 ms-1 horizontally

    To successfully travel the loop, F(centripetal) must equal F(gravity)
    Fc = Fg
    mv^2/r = mg
    v^2/r = g
    v^2 = 10 x 9.8
    v = 9.9 ms-1 at the top of the loop

    v+14 ms-1 must be able to travel the 20m and still achieve a speed of 9.9ms-1 at the top
    E(potential) = mgh
    E(potential) = m x 9.8 x 20
    E(potential) = 196m

    Since Ep at top equals Ek at bottom,
    Ep(top) = Ek(bottom)
    196m = 0.5mv^2
    196/0.5 = v^2
    v = 19.8 ms-1

    19.8ms-1 is the speed required to reach the top, 9.9ms-1 is the speed it must have at the top, therefore the total speed required is 19.8+9.9 =29.69ms-1

    Since 14ms-1 is provided by the travel down the slope,
    v = 29.69-14 = 15.7ms-1

    Can someone tell me if i am right, and correct me if im not? Your help is much appreciated =)
     
  2. jcsd
  3. Apr 9, 2012 #2

    What about Ek at the top?
     
  4. Apr 9, 2012 #3
    That was added on after that calculation, 9.9ms-1 is the Ek at the top
     
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