The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. What is the speed of the roller coaster at the top of the loop (in m/s) if the radius of curvature there is 17.0 m and the downward acceleration of the car is 1.50 g? 15.00 Jx m/s
The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. What is the speed of the roller coaster at the top of the loop (in m/s) if the radius of curvature there is 17.0 m and the downward acceleration of the car is 1.50 g? 15.00 Jx m/s
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter7: Conservation Of Energy
Section: Chapter Questions
Problem 56P
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![**Modern Roller Coasters with Vertical Loops**
Modern roller coasters have vertical loops like the one shown in the figure below. The diagram illustrates a roller coaster loop with the points labeled A, B, C, and D, and two significant radii: \( r_{\text{minimum}} \) at the top of the loop and \( r_{\text{maximum}} \) on the sides of the loop.
![Roller Coaster Loop Diagram]
**Description of the Diagram:**
- **Loop Structure:** A vertical loop is depicted with the roller coaster track forming a circular path.
- **Points on the Track:**
- **A and D:** Points on the horizontal base level of the loop.
- **B and C:** Points towards the top of the loop, symmetrically opposite each other.
- **Radii:**
- **\( r_{\text{minimum}} \):** The smaller radius of curvature at the top of the loop.
- **\( r_{\text{maximum}} \):** The larger radius of curvature on the sides of the loop.
**Problem Statement:**
The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, ensuring passengers remain pressed firmly into their seats. Given the radius of curvature at the top is 17.0 meters and the downward acceleration of the car is 1.50 times the acceleration due to gravity (g), determine the speed of the roller coaster at the top of the loop in meters per second (m/s).
**Input Box:**
A text input box is provided to enter the calculated speed:
- Input value: 15.00 m/s (marked incorrect).
---
**Note:** Apply the physics concepts of centripetal force and acceleration to solve this problem. Use the formula for centripetal acceleration, \( a = \frac{v^2}{r} \), where \( v \) is the speed of the roller coaster, and \( r \) is the radius of curvature.
### Solution
Given:
\[ r = 17.0 \, \text{m} \]
\[ a = 1.50g \]
Where \( g \approx 9.81 \, \text{m/s}^2 \),
\[ a = 1.50 \times 9.81 \, \text{m/s}^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff65fb7b2-ba85-4e67-8fe1-01f97a564d1e%2F754fe7a4-60f4-46f1-bd84-c27f77fd4831%2Fsuunf9y_processed.png&w=3840&q=75)
Transcribed Image Text:**Modern Roller Coasters with Vertical Loops**
Modern roller coasters have vertical loops like the one shown in the figure below. The diagram illustrates a roller coaster loop with the points labeled A, B, C, and D, and two significant radii: \( r_{\text{minimum}} \) at the top of the loop and \( r_{\text{maximum}} \) on the sides of the loop.
![Roller Coaster Loop Diagram]
**Description of the Diagram:**
- **Loop Structure:** A vertical loop is depicted with the roller coaster track forming a circular path.
- **Points on the Track:**
- **A and D:** Points on the horizontal base level of the loop.
- **B and C:** Points towards the top of the loop, symmetrically opposite each other.
- **Radii:**
- **\( r_{\text{minimum}} \):** The smaller radius of curvature at the top of the loop.
- **\( r_{\text{maximum}} \):** The larger radius of curvature on the sides of the loop.
**Problem Statement:**
The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, ensuring passengers remain pressed firmly into their seats. Given the radius of curvature at the top is 17.0 meters and the downward acceleration of the car is 1.50 times the acceleration due to gravity (g), determine the speed of the roller coaster at the top of the loop in meters per second (m/s).
**Input Box:**
A text input box is provided to enter the calculated speed:
- Input value: 15.00 m/s (marked incorrect).
---
**Note:** Apply the physics concepts of centripetal force and acceleration to solve this problem. Use the formula for centripetal acceleration, \( a = \frac{v^2}{r} \), where \( v \) is the speed of the roller coaster, and \( r \) is the radius of curvature.
### Solution
Given:
\[ r = 17.0 \, \text{m} \]
\[ a = 1.50g \]
Where \( g \approx 9.81 \, \text{m/s}^2 \),
\[ a = 1.50 \times 9.81 \, \text{m/s}^
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