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**Title: Dynamics of a Car on a Banked Circular Track** **Problem Statement:** A car of mass \( m \) drives on a banked, circular track with no friction. The track is banked at angle \( \beta \), and the car travels on the track with a radius \( R \) from the center of the track. **Tasks:** (a) **Free-Body Diagram:** Draw a free-body diagram of the car. Recall that the car’s acceleration will be radially inward. Therefore, you should use the coordinate system provided for determining force components. - **Diagram Description:** The diagram shows a car on a banked surface with an angle \( \beta \). There is a coordinate system with \( y \) pointing upwards and \( r \) pointing inward along the radius of the track. (b) **Calculate Car's Speed \( v_0 \):** What is the car’s speed \( v_0 \) in terms of \( g\), \( \beta\), and \( R\)? Why is your result independent of \( m \)? Explain why your result makes sense for \( \beta = 0^\circ \) and \( \beta = 90^\circ\). (c) **Speed for Safe Travel:** If \( R = 20.0 \, \text{m} \) and \( \beta = 20.0^\circ \), what speed \( v_0 \) will allow the car to safely travel around the track? (d) **Effect of Different Speeds:** What happens when the car has a speed \( v > v_0 \)? What happens when \( v < v_0 \)? --- This explanation focuses on understanding the dynamics of motion on a banked track and requires knowledge of forces and circular motion principles. Students are expected to apply their understanding of physics to solve the problem and explore the effects of different parameters on the car's motion.

**Title: Dynamics of a Car on a Banked Circular Track** **Problem Statement:** A car of mass \( m \) drives on a banked, circular track with no friction. The track is banked at angle \( \beta \), and the car travels on the track with a radius \( R \) from the center of the track. **Tasks:** (a) **Free-Body Diagram:** Draw a free-body diagram of the car. Recall that the car’s acceleration will be radially inward. Therefore, you should use the coordinate system provided for determining force components. - **Diagram Description:** The diagram shows a car on a banked surface with an angle \( \beta \). There is a coordinate system with \( y \) pointing upwards and \( r \) pointing inward along the radius of the track. (b) **Calculate Car's Speed \( v_0 \):** What is the car’s speed \( v_0 \) in terms of \( g\), \( \beta\), and \( R\)? Why is your result independent of \( m \)? Explain why your result makes sense for \( \beta = 0^\circ \) and \( \beta = 90^\circ\). (c) **Speed for Safe Travel:** If \( R = 20.0 \, \text{m} \) and \( \beta = 20.0^\circ \), what speed \( v_0 \) will allow the car to safely travel around the track? (d) **Effect of Different Speeds:** What happens when the car has a speed \( v > v_0 \)? What happens when \( v < v_0 \)? --- This explanation focuses on understanding the dynamics of motion on a banked track and requires knowledge of forces and circular motion principles. Students are expected to apply their understanding of physics to solve the problem and explore the effects of different parameters on the car's motion.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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**Title: Dynamics of a Car on a Banked Circular Track** **Problem Statement:** A car of mass \( m \) drives on a banked, circular track with no friction. The track is banked at angle \( \beta \), and the car travels on the track with a radius \( R \) from the center of the track. **Tasks:** (a) **Free-Body Diagram:** Draw a free-body diagram of the car. Recall that the car’s acceleration will be radially inward. Therefore, you should use the coordinate system provided for determining force components. - **Diagram Description:** The diagram shows a car on a banked surface with an angle \( \beta \). There is a coordinate system with \( y \) pointing upwards and \( r \) pointing inward along the radius of the track. (b) **Calculate Car's Speed \( v_0 \):** What is the car’s speed \( v_0 \) in terms of \( g\), \( \beta\), and \( R\)? Why is your result independent of \( m \)? Explain why your result makes sense for \( \beta = 0^\circ \) and \( \beta = 90^\circ\). (c) **Speed for Safe Travel:** If \( R = 20.0 \, \text{m} \) and \( \beta = 20.0^\circ \), what speed \( v_0 \) will allow the car to safely travel around the track? (d) **Effect of Different Speeds:** What happens when the car has a speed \( v > v_0 \)? What happens when \( v < v_0 \)? --- This explanation focuses on understanding the dynamics of motion on a banked track and requires knowledge of forces and circular motion principles. Students are expected to apply their understanding of physics to solve the problem and explore the effects of different parameters on the car's motion.
Transcribed Image Text:**Title: Dynamics of a Car on a Banked Circular Track** **Problem Statement:** A car of mass \( m \) drives on a banked, circular track with no friction. The track is banked at angle \( \beta \), and the car travels on the track with a radius \( R \) from the center of the track. **Tasks:** (a) **Free-Body Diagram:** Draw a free-body diagram of the car. Recall that the car’s acceleration will be radially inward. Therefore, you should use the coordinate system provided for determining force components. - **Diagram Description:** The diagram shows a car on a banked surface with an angle \( \beta \). There is a coordinate system with \( y \) pointing upwards and \( r \) pointing inward along the radius of the track. (b) **Calculate Car's Speed \( v_0 \):** What is the car’s speed \( v_0 \) in terms of \( g\), \( \beta\), and \( R\)? Why is your result independent of \( m \)? Explain why your result makes sense for \( \beta = 0^\circ \) and \( \beta = 90^\circ\). (c) **Speed for Safe Travel:** If \( R = 20.0 \, \text{m} \) and \( \beta = 20.0^\circ \), what speed \( v_0 \) will allow the car to safely travel around the track? (d) **Effect of Different Speeds:** What happens when the car has a speed \( v > v_0 \)? What happens when \( v < v_0 \)? --- This explanation focuses on understanding the dynamics of motion on a banked track and requires knowledge of forces and circular motion principles. Students are expected to apply their understanding of physics to solve the problem and explore the effects of different parameters on the car's motion.
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