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 h = 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 = 1.85 g at the top of the hump and no more than N = 5.53 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. (Figure not to scale) h Ryoop

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What is the minimum radius \( R_{\text{hump}} \) you can use for the semi-circular hump?

\( R_{\text{hump}} = \)  [text box] m

What is the minimum radius \( R_{\text{loop}} \) you can use for the vertical loop?

\( R_{\text{loop}} = \)  [text box] m
Transcribed Image Text:What is the minimum radius \( R_{\text{hump}} \) you can use for the semi-circular hump? \( R_{\text{hump}} = \) [text box] m What is the minimum radius \( R_{\text{loop}} \) you can use for the vertical loop? \( R_{\text{loop}} = \) [text box] m
### Roller Coaster Design Considerations

You are designing a section of a roller coaster ride as shown in the illustration. Previous sections of the ride give the train a speed of 10.7 m/s at the top of the incline, which is \( h = 36.7 \) 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 \( n = 1.85 \, g \) at the top of the hump and no more than \( N = 5.53 \, g \) at the bottom of the loop. For this initial phase of your design, you decide to ignore the effects of friction and air resistance. (Figure not to scale)

#### Diagram Explanation

- **Incline and Hump:**
  - The train starts from a height \( h \) and gains speed as it moves downward.
  - It then ascends a hump where centripetal acceleration is limited to \( 1.85 \, g \).

- **Loop:**
  - The figure shows a circular loop with a radius labeled \( R_{\text{loop}} \).
  - The centripetal acceleration at the bottom of the loop must not exceed \( 5.53 \, g \).

The goal of this design is to ensure safe and thrilling experiences while adhering to government safety standards by carefully calculating speeds and accelerations at various points.
Transcribed Image Text:### Roller Coaster Design Considerations You are designing a section of a roller coaster ride as shown in the illustration. Previous sections of the ride give the train a speed of 10.7 m/s at the top of the incline, which is \( h = 36.7 \) 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 \( n = 1.85 \, g \) at the top of the hump and no more than \( N = 5.53 \, g \) at the bottom of the loop. For this initial phase of your design, you decide to ignore the effects of friction and air resistance. (Figure not to scale) #### Diagram Explanation - **Incline and Hump:** - The train starts from a height \( h \) and gains speed as it moves downward. - It then ascends a hump where centripetal acceleration is limited to \( 1.85 \, g \). - **Loop:** - The figure shows a circular loop with a radius labeled \( R_{\text{loop}} \). - The centripetal acceleration at the bottom of the loop must not exceed \( 5.53 \, g \). The goal of this design is to ensure safe and thrilling experiences while adhering to government safety standards by carefully calculating speeds and accelerations at various points.
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