The Ferris Wheel

  1. 1. The problem statement, all variables and given/known data


    The figure below shows a Ferris wheel that rotates three times each minute. It carries each car around a circle of diameter 19.0 m.

    a) What is the centripetal acceleration of a rider?
    Answer in m/s^2

    (b) What force does the seat exert on a 40.0 kg rider at the lowest point of the ride?
    Answer in N

    (c) What force does the seat exert on the rider at the highest point of the ride?
    Answer in N

    (d) What force (magnitude and direction) does the seat exert on a rider when the rider is halfway between top and bottom, going up?
    Answer in N
    Answer in ° (measured inward from the vertical)

    2. Relevant equations



    3. The attempt at a solution

    a) Angular velocity w = 3 rev/min = 3 * 2pi/60 rad/s
    = pi/10 rad/s
    Or w = 0.314 rad/s
    Radius r = 19.0 m/2 = 9.5 m
    Centripetal acceleration a = w^2 * r
    = 0.314^2 * 9.5
    = 0.937 m/s^2

    b) Mass M = 40.0 kg
    Let force = F
    F is upward and weight Mg is downward. Centripetal force is upward.
    F - Mg = Ma
    Or F = M(a+g) = 40 * (9.8 + 0.937) = 429.5 N

    c) Mass M = 40.0 kg
    Let force = F
    F is upward and Mg is downward. Centripetal force is downward.
    Mg - F = Ma
    Or F = M(g - a) = 40 * (9.8 - 0.937) = 354 N

    d) Let force = F at angle theta below vertical.
    Vertical component of F = F cos(theta) upward.
    Horizontal component of F = F sin(theta)

    There is no acceleration in vertical direction.
    Therefore F cos(theta) = Mg--------------------------(1)
    Centripetal force is horizontal.
    Therefore F sin(theta) = Ma---------------------------(2)
    Dividing (2) by (1),
    tan(theta) = a/g
    Or theta = atan(a/g) = atan(0.937/9.8) = 5.46 deg
    From (1),
    F = Mg/cos(theta) = 40 * 9.8/cos(5.46 deg) = 394 N
    Therefore, force by the seat is 394 N at 5.46 deg below vertical.
    (Note: I do not know what exactly is meant by inward from vertical. I have calculated the angle from vertically upward. If you want the angle from vertically downward, then it is 180 - 5.46 = 174.54 deg



    I only have one chance to enter this, please check to see if the way i did it is correct.
     
  2. jcsd
  3. Delphi51

    Delphi51 3,410
    Homework Helper

    All calcs look good. For (d) it would be the 5.44 degree answer because the Fg is much larger than the Fc, so only a little off vertical.
     
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