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Determine the magnitude of the force that you need to apply to keep the block at rest

Determine the magnitude of the force that you need to apply to keep the block at rest

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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## Physics Mechanics: Force Analysis on an Inclined Plane ### Problem Statement: A block of mass \(m = 10 \, \text{kg}\) sits on a frictionless ramp that has an inclination angle \(\theta = 20\) degrees. A rope connects the block to the ceiling. The rope hangs straight down and exerts a tension \(T = 50 \, \text{N}\) on the block. In order to keep the block stationary, you need to apply a force \( \vec{F} \) up along the ramp as shown in the diagram. ### Diagram Explanation: The diagram depicts the physical setup of the problem: - An inclined plane is shown with an angle of inclination \(\theta\). - A block of mass \( m \) is placed on the inclined plane. The block is represented by a blue rectangular object labeled with \( m \). - A rope connects the block to the ceiling, preventing it from moving down the incline. The tension in the rope is indicated as \( T \). - A force \( \vec{F} \) is applied along the incline, directed upwards, which is represented by a pink arrow pointing up the ramp. ### Key Points: - Inclination angle \( \theta = 20^\circ \) - Block mass \( m = 10 \, \text{kg} \) - Rope tension \( T = 50 \, \text{N} \) - Applied force \( \vec{F} \) needs to be determined to keep the block stationary. ### Solution Strategy: To solve for the applied force \( \vec{F} \): 1. **Resolve Forces Along the Incline:** - Gravitational force component parallel to the incline: \( mg \sin(\theta) \) - Tension force component along the incline: This needs to be adjusted according to the geometry - The applied force \( \vec{F} \) directly counteracts the combined effect of the gravitational component and tension along the incline. 2. **Newton's Second Law Along the Incline:** - Since the block is stationary, the net force along the incline must be zero. - Set up the equilibrium condition: \( \vec{F} = mg \sin(\theta) - T \cdot \cos(\theta) \) 3. **Calculate the Required Force:** - Substitute the given values into the equation to
Transcribed Image Text:## Physics Mechanics: Force Analysis on an Inclined Plane ### Problem Statement: A block of mass \(m = 10 \, \text{kg}\) sits on a frictionless ramp that has an inclination angle \(\theta = 20\) degrees. A rope connects the block to the ceiling. The rope hangs straight down and exerts a tension \(T = 50 \, \text{N}\) on the block. In order to keep the block stationary, you need to apply a force \( \vec{F} \) up along the ramp as shown in the diagram. ### Diagram Explanation: The diagram depicts the physical setup of the problem: - An inclined plane is shown with an angle of inclination \(\theta\). - A block of mass \( m \) is placed on the inclined plane. The block is represented by a blue rectangular object labeled with \( m \). - A rope connects the block to the ceiling, preventing it from moving down the incline. The tension in the rope is indicated as \( T \). - A force \( \vec{F} \) is applied along the incline, directed upwards, which is represented by a pink arrow pointing up the ramp. ### Key Points: - Inclination angle \( \theta = 20^\circ \) - Block mass \( m = 10 \, \text{kg} \) - Rope tension \( T = 50 \, \text{N} \) - Applied force \( \vec{F} \) needs to be determined to keep the block stationary. ### Solution Strategy: To solve for the applied force \( \vec{F} \): 1. **Resolve Forces Along the Incline:** - Gravitational force component parallel to the incline: \( mg \sin(\theta) \) - Tension force component along the incline: This needs to be adjusted according to the geometry - The applied force \( \vec{F} \) directly counteracts the combined effect of the gravitational component and tension along the incline. 2. **Newton's Second Law Along the Incline:** - Since the block is stationary, the net force along the incline must be zero. - Set up the equilibrium condition: \( \vec{F} = mg \sin(\theta) - T \cdot \cos(\theta) \) 3. **Calculate the Required Force:** - Substitute the given values into the equation to
### Physics Problem: Determining the Magnitude of Force \( F \) **Objective:** To determine the magnitude of the force \( F \) that needs to be applied to keep a block at rest. The given problem requires you to apply concepts of equilibrium and force analysis. Use your knowledge of Newton's Laws and static equilibrium to solve for the force. **Equation to Consider:** \[ \sum F = 0 \] **Steps for Solving the Problem:** 1. **Identify all forces acting on the block:** - Gravitational force (\( F_g \)) acting downward. - Normal force (\( N \)) acting perpendicular to the surface. - Applied force (\( F \)) which needs to be determined. - Frictional force (\( F_f \)) if any, acting opposite to the direction of potential movement. 2. **Set up the equilibrium equations:** - For vertical forces: \( \sum F_y = 0 \) - For horizontal forces: \( \sum F_x = 0 \) 3. **Solve the equations for the unknown force \( F \).** **Example Analysis:** - Assume there is no friction for simplicity. - The block is resting, so the forces in the vertical direction must balance. - If there are no other forces acting horizontally, \( F \) will need to balance any horizontal components. Use these steps as a guideline to derive the precise value of the force \( F \) in your specific situation. --- **Note:** There are no graphs or diagrams provided in the given text. For visual learners, it would be beneficial to include a free-body diagram illustrating the block and the forces acting on it to aid in understanding. Draw a rectangle (representing the block) with vectors showing the forces \( F_g \) (downward), \( N \) (upward), and \( F \) in the required direction.
Transcribed Image Text:### Physics Problem: Determining the Magnitude of Force \( F \) **Objective:** To determine the magnitude of the force \( F \) that needs to be applied to keep a block at rest. The given problem requires you to apply concepts of equilibrium and force analysis. Use your knowledge of Newton's Laws and static equilibrium to solve for the force. **Equation to Consider:** \[ \sum F = 0 \] **Steps for Solving the Problem:** 1. **Identify all forces acting on the block:** - Gravitational force (\( F_g \)) acting downward. - Normal force (\( N \)) acting perpendicular to the surface. - Applied force (\( F \)) which needs to be determined. - Frictional force (\( F_f \)) if any, acting opposite to the direction of potential movement. 2. **Set up the equilibrium equations:** - For vertical forces: \( \sum F_y = 0 \) - For horizontal forces: \( \sum F_x = 0 \) 3. **Solve the equations for the unknown force \( F \).** **Example Analysis:** - Assume there is no friction for simplicity. - The block is resting, so the forces in the vertical direction must balance. - If there are no other forces acting horizontally, \( F \) will need to balance any horizontal components. Use these steps as a guideline to derive the precise value of the force \( F \) in your specific situation. --- **Note:** There are no graphs or diagrams provided in the given text. For visual learners, it would be beneficial to include a free-body diagram illustrating the block and the forces acting on it to aid in understanding. Draw a rectangle (representing the block) with vectors showing the forces \( F_g \) (downward), \( N \) (upward), and \( F \) in the required direction.
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