Map Sapling Learning macmillan learning You are designing the section of a rolle
ID: 1790936 • Letter: M
Question
Map Sapling Learning macmillan learning You are designing the section of a roller coaster ride shown in the figure. Previous sections of the ride give the train a speed of 10.7 m/s at the top of the incline, which is 36.7 m above the ground. As any good engineer would, you begin your design with safety in mind. Your local government's safety regulations state that the riders' centripetal acceleration should be no more than n = 1.77 g at the top of the hump and no more than N = 5.45 g at the bottom of the loop. For this initial phase of your design, you decide to ignore the effects of friction and air resistance. What is the minimum radius you can use for the semi-circular hump? What is the minimum radius you can use for the vertical loop? Number Number R. = loop hump 36.7 m Rloop figure not to scaleExplanation / Answer
given speed of train at the top of the incline, u = 10.7 m/s
so speed of train on the top of the bump, = v
speed of train at the bottom of the vertical loop = v'
hence
0.5mv^2 + mgR = 0.5mu^2 + mgH
where R is radius of buump and H is initial maximum height
0.6v^2 + g*R = 0.5*10.7^2 + g*36.7
hence
0.6v^2 + 9.81R = 417.272
also,
for minimum radius of hump
v^2/R = 1.77g
0.6*1.77gR + gR = 417.272
R = 20.62 m is the lowest value of R that may be used
similiarly
for the vertical loop, speed at bottom = v'
0.5mv'^2 = 0.5mu^2 + mgH = 417.272*m
0.5v'^2 = 417.272
v' = 28.888 m/s
also, v'^2/Rloop = 5.45*9.81
Rloop = 15.60931 m