You are designing the section of a roller coaster ride shown in the figure. Prev

Question

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 14.9 m/s at the top of the incline, which is 35.5 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 = 1.85 g at the top of the hump and no more than N = 5.13 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?

Explanation / Answer

at the top of hump :

v2hump = aRhump = 1.85 x 9.8 Rhump = 18.13 Rhump

using conservation of energy at top of incline and top of hump

mgH + (0.5) m v2 = mgRhump + (0.5) m v2hump

gH + (0.5) v2 = gRhump + (0.5) v2hump

(9.8) (35.5) + (0.5) (14.9)2 = (9.8) Rhump + (0.5) (18.13) Rhump

Rhump = 24.3 m

at the bottom of the loop :

v2loop = aRloop = 5.13 x 9.8 Rloop = 50.3 Rloop

using conservation of energy at top of incline and top of hump

mgH + (0.5) m v2 = (0.5) m v2loop

gH + (0.5) v2 = (0.5) v2loop

(9.8) (35.5) + (0.5) (14.9)2 = (0.5) (50.3) Rloop

Rloop = 18.25 m

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