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

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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 13.1 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 = 1.81 g at the top of the hump and no more than N = 5.21 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.

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 13.1 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 = 1.81 g at the top of the hump and no morethan N = 521 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. efecs ol ricion and air resislatnoef the loop.For this initial phase of your design. you decide to ignore the 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 Rhump loop 35.9 m Rloop hump figure not to scale

Explanation / Answer

Here ,

initial speed , u = 13.1 m/s

Height , H = 35.9 m

N = 5.21 g at the bottom

n = 1.81 g at the top of hump

Here , for the hump

Using conservation of energy

0.5 * m * v^2 - 0.5 * m * u^2 = m * g * (H - Rhump)

v^2 = 2 * g* (H - Rhump) + 13.1^2

Now , at the top of hump

m * v^2/Rhump = m * 1.81 * g

(2 * g* (H - Rhump) + 13.1^2)/Rhump = 1.81 * g

(2 * 9.8* (H - Rhump) + 13.1^2)/Rhump = 1.81 * 9.8

Rhump = 23.44 m

the radius of hump is 23.44 m

for the loop ,

at the bottom

v^2 = 13.1^2 + 2 * 9.8 * 35.9

N = v^2/Rloop

5.21 * 9.8 = (13.1^2 + 2 * 9.8 * 35.9 )/Rloop

Rloop = 17.14 m

the radius of loop is 17.14 m

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