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 16.1 m/s at the top of the incline, which is 35.9 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 = 1.61 g at the top of the hump and no more than N = 5.77 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.

Explanation / Answer

Initially,
E = PE + KE
E = m*g*h + 1/2 * m*v^2
E = M * 9.8m/s² * 35.9 + 1/2 *M*(16.1m/s)² = M * 481.425

1.
At r = Rhump,
a = 1.61 * 9.8m/s² = v² / r
v² = 15.778 * r m/s²

E = M * 481.425 m²/s² = Mgr + ½Mv²
481.425  m²/s² = 9.8* r + 1/2* 15.778 * r
Solving for r
r = Rhump = 27.21 m

2. At r = Rloop,
a = 5.77 * 9.8m/s² = 56.546 m/s²
a = v² / r
v² = 56.546 m/s² * r

E = 481.425 m²/s² * M =1/2 * Mv²
481.425 m²/s² = 1/2 * 56.546 m/s² * r

Solving for r
r = Rloop = 17.03 m

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