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2) Suppose you put an object on an inclined slope and let go. If there is enough friction between the object and the surface of the slope, it might not go anywhere. This can be important when figuring out what kind of tires you need so you don't slip on an icy hill! Whether there is friction or not, it is useful to be able to write the gravitational force acting on the object as the sum of two perpendicular vectors: one parallel to the surface of the slope, and one perpendicular to the surface of the slope. For this series of questions, suppose the inclined slope is fixed at an angle of 30° and the object has a mass of 3 kg, as shown in the diagram. 30° 3 kg a) Gravitational force has a magnitude equal to an object's mass times acceleration due to gravity, and a direction of "straight down." Since we are using kilograms, acceleration due to gravity should be given in meters per second squared, and the value is approximately 9.8 m/s². Use this information to write the components of vector F, the force of gravity acting on the object. Remember that "down" is usually associated with a negative direction. b) Use vector projection to find the part of vector F that is parallel to the inclined surface of the slope. Call this vector ū. (Hint: you may find it useful to first write any vector in the direction of the slope, call it vector v, and use that in projection calculations.) c) Use your answers to (a) and (c) to write a vector w which is perpendicular to the surface of the slope (that is, perpendicular to your vector v) and which, when added to ū, results in the gravitational force vector F. In other words, u + w should equal F. d) Draw a picture showing the slope, the object, the gravitational force F, and the vectors ū and w.

2) Suppose you put an object on an inclined slope and let go. If there is enough friction between the object and the surface of the slope, it might not go anywhere. This can be important when figuring out what kind of tires you need so you don't slip on an icy hill! Whether there is friction or not, it is useful to be able to write the gravitational force acting on the object as the sum of two perpendicular vectors: one parallel to the surface of the slope, and one perpendicular to the surface of the slope. For this series of questions, suppose the inclined slope is fixed at an angle of 30° and the object has a mass of 3 kg, as shown in the diagram. 30° 3 kg a) Gravitational force has a magnitude equal to an object's mass times acceleration due to gravity, and a direction of "straight down." Since we are using kilograms, acceleration due to gravity should be given in meters per second squared, and the value is approximately 9.8 m/s². Use this information to write the components of vector F, the force of gravity acting on the object. Remember that "down" is usually associated with a negative direction. b) Use vector projection to find the part of vector F that is parallel to the inclined surface of the slope. Call this vector ū. (Hint: you may find it useful to first write any vector in the direction of the slope, call it vector v, and use that in projection calculations.) c) Use your answers to (a) and (c) to write a vector w which is perpendicular to the surface of the slope (that is, perpendicular to your vector v) and which, when added to ū, results in the gravitational force vector F. In other words, u + w should equal F. d) Draw a picture showing the slope, the object, the gravitational force F, and the vectors ū and w.

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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**Inclined Plane Problem: Understanding Vectors and Forces** This educational content explores the concept of forces acting on an inclined plane. Here’s the scenario: 1. **Setup**: Imagine placing an object on an inclined slope, letting it go. The slope has a fixed angle, and we examine the components of gravitational force acting on the object. 2. **Diagram Explanation**: The diagram shows a slope inclined at 30 degrees with an object of mass 3 kg on it. This setup is used to understand force components on an inclined plane. 3. **Questions and Explorations**: a) **Calculate Gravitational Force Components (Vector \( \mathbf{F} \))**: - Gravitational force is calculated using the formula: mass (3 kg) * gravity (9.8 m/s²). - The gravitational force acts "straight down." Decompose this force into components parallel and perpendicular to the incline. b) **Find the Parallel Component (Vector \( \mathbf{u} \))**: - Use vector projection to determine the component of \( \mathbf{F} \) that is parallel to the incline. - Consider taking any vector in the direction of the slope (Vector \( \mathbf{v} \)). c) **Determine the Perpendicular Component (Vector \( \mathbf{w} \))**: - Utilize the components from parts (a) and (b) to find vector \( \mathbf{w} \), which is perpendicular to the slope. - Ensure that combining \( \mathbf{u} \) and \( \mathbf{w} \) results in the original gravitational force vector \( \mathbf{F} \). d) **Illustrate the Forces**: - Draw the slope, the object, and vectors \( \mathbf{F} \), \( \mathbf{u} \), and \( \mathbf{w} \). This exercise helps students understand how forces break down into components on an inclined plane, a fundamental concept in physics regarding motion and mechanics.
Transcribed Image Text:**Inclined Plane Problem: Understanding Vectors and Forces** This educational content explores the concept of forces acting on an inclined plane. Here’s the scenario: 1. **Setup**: Imagine placing an object on an inclined slope, letting it go. The slope has a fixed angle, and we examine the components of gravitational force acting on the object. 2. **Diagram Explanation**: The diagram shows a slope inclined at 30 degrees with an object of mass 3 kg on it. This setup is used to understand force components on an inclined plane. 3. **Questions and Explorations**: a) **Calculate Gravitational Force Components (Vector \( \mathbf{F} \))**: - Gravitational force is calculated using the formula: mass (3 kg) * gravity (9.8 m/s²). - The gravitational force acts "straight down." Decompose this force into components parallel and perpendicular to the incline. b) **Find the Parallel Component (Vector \( \mathbf{u} \))**: - Use vector projection to determine the component of \( \mathbf{F} \) that is parallel to the incline. - Consider taking any vector in the direction of the slope (Vector \( \mathbf{v} \)). c) **Determine the Perpendicular Component (Vector \( \mathbf{w} \))**: - Utilize the components from parts (a) and (b) to find vector \( \mathbf{w} \), which is perpendicular to the slope. - Ensure that combining \( \mathbf{u} \) and \( \mathbf{w} \) results in the original gravitational force vector \( \mathbf{F} \). d) **Illustrate the Forces**: - Draw the slope, the object, and vectors \( \mathbf{F} \), \( \mathbf{u} \), and \( \mathbf{w} \). This exercise helps students understand how forces break down into components on an inclined plane, a fundamental concept in physics regarding motion and mechanics.
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