A block with some mass m is connected to a string that is attached to the ceiling. The block on the end of the string is going around a circular path with a constant radius r and constant speed. Applying Newton's second law to the x and y components of forces seperately in order to find the expressions for the tension of the string in terms of mass m, angle θ, and constant g.
A block with some mass m is connected to a string that is attached to the ceiling. The block on the end of the string is going around a circular path with a constant radius r and constant speed. Applying Newton's second law to the x and y components of forces seperately in order to find the expressions for the tension of the string in terms of mass m, angle θ, and constant g.
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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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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A block with some mass m is connected to a string that is attached to the ceiling. The block on the end of the string is going around a circular path with a constant radius r and constant speed. Applying Newton's second law to the x and y components of forces seperately in order to find the expressions for the tension of the string in terms of mass m, angle θ, and constant g.
Transcribed Image Text:**Understanding Circular Motion**
The diagram illustrates the essential components involved in circular motion.
1. **Circular Path:**
The larger circle denotes the circular path that an object follows as it moves in a circular motion.
2. **Radius (r):**
Represented by the letter "r", the radius is the distance from the center of the circle to the object in motion.
3. **Angle (θ):**
The angle θ indicates the angle formed between a reference line (like a vertical axis) and the radius line extending to the moving object.
4. **Object in Motion:**
The object in motion, marked with the letter "m", indicates the position of the moving object on the circular path.
Understanding these components is crucial for analyzing various aspects of circular motion, such as angular displacement, angular velocity, and centripetal force.
![**Understanding Vector Components in Physics**
The diagram presented here illustrates the breakdown of a force vector \( \vec{F} \) into its constituent components along the \( x \) and \( y \) axes. Understanding these components is fundamental in vector analysis and physics problems involving forces.
### Diagram Explanation
#### Left Side Diagram:
- **Vector \( \vec{F} \)**: This is the resultant force vector, positioned at an angle \( \theta \) from the vertical axis.
- **Angle \( \theta \)**: The angle between \( \vec{F} \) and the vertical component \( \vec{F}^y \).
- **\( \vec{F}^x \)**: The horizontal component of \( \vec{F} \). It represents the projection of \( \vec{F} \) on the \( x \)-axis.
- **\( \vec{F}^y \)**: The vertical component of \( \vec{F} \). It represents the projection of \( \vec{F} \) on the \( y \)-axis.
- **\( \vec{F}^g \)**: The gravitational force vector acting downwards (usually equal to \( mg \), where \( m \) is mass and \( g \) is the acceleration due to gravity).
The vector components are situated such that:
\[ \vec{F}^x = \vec{F} \cos(\theta) \]
\[ \vec{F}^y = \vec{F} \sin(\theta) \]
#### Right Side Diagram:
- **Coordinate Axes**: Traditional \( x, y \)-coordinate system. The horizontal axis represents the \( x \)-direction and the vertical axis represents the \( y \)-direction, which are typically used to reference the components of vectors in two dimensions.
### Key Points:
- Any vector in a plane can be decomposed into its horizontal and vertical components.
- The components can be calculated using trigonometric functions sine and cosine relative to the angle \( \theta \).
- This decomposition is useful for solving physics problems that involve forces, motion, and equilibrium.
By understanding and using these components, we can analyze various physical scenarios with greater ease and accuracy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6376748b-2546-4783-941d-656c42a0492c%2Fa87c3c28-fb2a-4803-8d56-85612afaebe0%2Fkabdlls_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Vector Components in Physics**
The diagram presented here illustrates the breakdown of a force vector \( \vec{F} \) into its constituent components along the \( x \) and \( y \) axes. Understanding these components is fundamental in vector analysis and physics problems involving forces.
### Diagram Explanation
#### Left Side Diagram:
- **Vector \( \vec{F} \)**: This is the resultant force vector, positioned at an angle \( \theta \) from the vertical axis.
- **Angle \( \theta \)**: The angle between \( \vec{F} \) and the vertical component \( \vec{F}^y \).
- **\( \vec{F}^x \)**: The horizontal component of \( \vec{F} \). It represents the projection of \( \vec{F} \) on the \( x \)-axis.
- **\( \vec{F}^y \)**: The vertical component of \( \vec{F} \). It represents the projection of \( \vec{F} \) on the \( y \)-axis.
- **\( \vec{F}^g \)**: The gravitational force vector acting downwards (usually equal to \( mg \), where \( m \) is mass and \( g \) is the acceleration due to gravity).
The vector components are situated such that:
\[ \vec{F}^x = \vec{F} \cos(\theta) \]
\[ \vec{F}^y = \vec{F} \sin(\theta) \]
#### Right Side Diagram:
- **Coordinate Axes**: Traditional \( x, y \)-coordinate system. The horizontal axis represents the \( x \)-direction and the vertical axis represents the \( y \)-direction, which are typically used to reference the components of vectors in two dimensions.
### Key Points:
- Any vector in a plane can be decomposed into its horizontal and vertical components.
- The components can be calculated using trigonometric functions sine and cosine relative to the angle \( \theta \).
- This decomposition is useful for solving physics problems that involve forces, motion, and equilibrium.
By understanding and using these components, we can analyze various physical scenarios with greater ease and accuracy.
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