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he movers are trying to get a heavy crate (mass m) up a ramp into the loading dock of a uilding. One of the movers is pushing on the crate with a force of constant magnitude F₁ arallel with the surface of the ramp. The other mover is pulling a rope attached to the crate, xerting a force of constant magnitude F2, oriented at an angle above the surface of the amp. m F₁₂ 20 b

he movers are trying to get a heavy crate (mass m) up a ramp into the loading dock of a uilding. One of the movers is pushing on the crate with a force of constant magnitude F₁ arallel with the surface of the ramp. The other mover is pulling a rope attached to the crate, xerting a force of constant magnitude F2, oriented at an angle above the surface of the amp. m F₁₂ 20 b

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 Problem: Motion on a Frictionless Ramp **Problem Statement:** There is no friction between the crate and the surface of the ramp. The movers applied their forces continuously (i.e., constant forces). Given this information, and the fact that the crate started at rest at the bottom of the ramp, determine how fast the crate moves when it reaches the top of the ramp. --- **Solution Outline:** To solve this problem, consider the principles of classical mechanics, specifically Newton's Laws of Motion. The key points to consider are: 1. **No Friction**: Since there is no friction, the force applied by the movers is the only force causing the crate to move. 2. **Constant Force**: The constant force implies a uniform acceleration which can be determined using Newton’s second law (F = ma). 3. **Initial Conditions**: The crate started at rest, meaning its initial velocity (v₀) is zero. 4. **Kinematic Equations**: Utilize kinematic equations to relate the displacement of the crate up the ramp to its final velocity. --- **Detailed Steps and Formulas:** 1. **Determine the Net Force (F) Applied on the Crate**: - Since the ramp is frictionless, the entire magnitude of the force is contributing to the crate's motion. 2. **Calculate the Acceleration (a)**: - Use Newton’s second law: \( F = m \cdot a \) - Rearrange to find \( a = \frac{F}{m} \) 3. **Use Kinematic Equations**: - Given that the initial velocity \( v_0 = 0 \), - and knowing the displacement (d) of the crate up the ramp, - We can use the kinematic equation: \( v^2 = v_0^2 + 2ad \) 4. **Solving for the Final Velocity (v)**: - Simplify to find \( v = \sqrt{2ad} \) By defining the force applied, mass of the crate, and displacement, we can calculate the exact speed of the crate at the top of the ramp using the derived formulas.
Transcribed Image Text:#### Physics Problem: Motion on a Frictionless Ramp **Problem Statement:** There is no friction between the crate and the surface of the ramp. The movers applied their forces continuously (i.e., constant forces). Given this information, and the fact that the crate started at rest at the bottom of the ramp, determine how fast the crate moves when it reaches the top of the ramp. --- **Solution Outline:** To solve this problem, consider the principles of classical mechanics, specifically Newton's Laws of Motion. The key points to consider are: 1. **No Friction**: Since there is no friction, the force applied by the movers is the only force causing the crate to move. 2. **Constant Force**: The constant force implies a uniform acceleration which can be determined using Newton’s second law (F = ma). 3. **Initial Conditions**: The crate started at rest, meaning its initial velocity (v₀) is zero. 4. **Kinematic Equations**: Utilize kinematic equations to relate the displacement of the crate up the ramp to its final velocity. --- **Detailed Steps and Formulas:** 1. **Determine the Net Force (F) Applied on the Crate**: - Since the ramp is frictionless, the entire magnitude of the force is contributing to the crate's motion. 2. **Calculate the Acceleration (a)**: - Use Newton’s second law: \( F = m \cdot a \) - Rearrange to find \( a = \frac{F}{m} \) 3. **Use Kinematic Equations**: - Given that the initial velocity \( v_0 = 0 \), - and knowing the displacement (d) of the crate up the ramp, - We can use the kinematic equation: \( v^2 = v_0^2 + 2ad \) 4. **Solving for the Final Velocity (v)**: - Simplify to find \( v = \sqrt{2ad} \) By defining the force applied, mass of the crate, and displacement, we can calculate the exact speed of the crate at the top of the ramp using the derived formulas.
### Moving a Heavy Crate Up a Ramp: An Educational Example The movers are trying to get a heavy crate (mass \( m \)) up a ramp into the loading dock of a building. One of the movers is pushing on the crate with a force of constant magnitude \( F_1 \) parallel with the surface of the ramp. The other mover is pulling a rope attached to the crate, exerting a force of constant magnitude \( F_2 \), oriented at an angle \( \theta \) above the surface of the ramp. #### Diagram Explanation The diagram below visually represents the scenario described: 1. **Crate and Forces**: - A crate, labeled with mass \( m \), is positioned on a ramp. - A mover on the left is shown pushing the crate up the ramp. This force is depicted as an arrow parallel to the surface of the ramp, labeled \( \vec{F_1} \). - Another mover on the right is pulling the crate using a rope. The force exerted is represented by an upward diagonal arrow, labeled \( \vec{F_2} \), which is at an angle \( \theta \) relative to the ramp surface. 2. **Ramp Dimensions**: - The ramp extends horizontally for a distance \( d \) from the base to the loading dock. - The height of the ramp, from the bottom to the top, where the loading dock is located, is \( h \). 3. **Angles and Directions**: - The angle \( \theta \) indicates the incline of the force \( \vec{F_2} \) relative to the ramp's surface. - Arrows indicating directions of forces and dimensions are clearly marked to show the direction and point of application of these forces. This setup illustrates the physics concepts of forces acting on an object on an incline, considering components parallel and perpendicular to the surface, as well as understanding the resultant motion due to combined forces.
Transcribed Image Text:### Moving a Heavy Crate Up a Ramp: An Educational Example The movers are trying to get a heavy crate (mass \( m \)) up a ramp into the loading dock of a building. One of the movers is pushing on the crate with a force of constant magnitude \( F_1 \) parallel with the surface of the ramp. The other mover is pulling a rope attached to the crate, exerting a force of constant magnitude \( F_2 \), oriented at an angle \( \theta \) above the surface of the ramp. #### Diagram Explanation The diagram below visually represents the scenario described: 1. **Crate and Forces**: - A crate, labeled with mass \( m \), is positioned on a ramp. - A mover on the left is shown pushing the crate up the ramp. This force is depicted as an arrow parallel to the surface of the ramp, labeled \( \vec{F_1} \). - Another mover on the right is pulling the crate using a rope. The force exerted is represented by an upward diagonal arrow, labeled \( \vec{F_2} \), which is at an angle \( \theta \) relative to the ramp surface. 2. **Ramp Dimensions**: - The ramp extends horizontally for a distance \( d \) from the base to the loading dock. - The height of the ramp, from the bottom to the top, where the loading dock is located, is \( h \). 3. **Angles and Directions**: - The angle \( \theta \) indicates the incline of the force \( \vec{F_2} \) relative to the ramp's surface. - Arrows indicating directions of forces and dimensions are clearly marked to show the direction and point of application of these forces. This setup illustrates the physics concepts of forces acting on an object on an incline, considering components parallel and perpendicular to the surface, as well as understanding the resultant motion due to combined forces.
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