Question 8 of 10 Sapling Learning You are designing the section of a roller coas

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Question 8 of 10 Sapling 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 13.7 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 that the nders' centripetal acceleration should be no more than n 1.81 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 decideto ignore the effects of friction and air resistance. s state 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 R, hump loop 35.5 m Roop figure not to scale Next Exi 99+

Explanation / Answer

given,

initial height = 35.5 m

initial speed = 13.7 m/s

by conservation of energy

initial energy = final energy

initial energy = mgh + 0.5 * mv^2

initial energy = m * 9.8 * 35.5 + 0.5 * m * 13.7^2

let the radius of the hump be r so,

centripetal acceleration at the top pf the hump = v^2 / r

v^2 / r = 1.81 g

v^2 = 1.81 * g * r

final energy = mgr + 0.5 * mv^2

final energy = m * 9.8 * r + 0.5 * m * 1.81 * g * r

m * 9.8 * 35.5 + 0.5 * m * 13.7^2 =  m * 9.8 * r + 0.5 * m * 1.81 * g * r

9.8 * 35.5 + 0.5 * 13.7^2 = 9.8 * r + 0.5 * 1.81 * 9.8 * r

r = 23.662 m

Rhump = 23.662 m

at bottom of loop

v^2 / r = 5.45 g

since height is 0 now so,

final energy = 0.5 * mv^2

9.8 * 35.5 + 0.5 * 13.7^2 = 0.5 * 5.45 * 9.8 * r

r = 16.54 m

Rloop = 16.54 m

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