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Assuming s is the same for all contacting surfaces, determine the minimum s such that equilibrium is maintained for the leaning 6.5 m ladder AB. The weight of the ladder is non- negligible. Hint: The equilibrium equations yield a quadratic equation for us, for which one solution is valid. B |--2.5m- 6 m

Assuming s is the same for all contacting surfaces, determine the minimum s such that equilibrium is maintained for the leaning 6.5 m ladder AB. The weight of the ladder is non- negligible. Hint: The equilibrium equations yield a quadratic equation for us, for which one solution is valid. B |--2.5m- 6 m

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
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### Minimum Coefficient of Static Friction for Ladder Stability **Problem Statement:** Assuming \(\mu_s\) is the same for all contacting surfaces, determine the minimum \(\mu_s\) such that equilibrium is maintained for the leaning 6.5 m ladder AB. The weight of the ladder is non-negligible. *Hint: The equilibrium equations yield a quadratic equation for \(\mu_s\), for which one solution is valid.* **Diagram Explanation:** The diagram depicts a ladder AB leaning against a vertical wall. The ladder is 6.5 meters long and forms a right-angled triangle with the wall and the ground. The point where the ladder touches the ground is labeled as A, and the point where it touches the wall is labeled as B. The horizontal distance from point A to the vertical wall is 2.5 meters, and the height at which the ladder touches the wall (point B) is 6 meters. This setup requires calculating the minimum coefficient of static friction (\(\mu_s\)) to maintain equilibrium and prevent the ladder from slipping. --- For solving the problem: 1. **Equilibrium Conditions:** - **Sum of Forces in the Horizontal Direction (ΣF_x = 0):** The horizontal component of the force exerted by the ground at A must balance the horizontal component of the force exerted by the wall at B. - **Sum of Forces in the Vertical Direction (ΣF_y = 0):** The vertical component of the force exerted by the ground at A must balance the weight of the ladder. - **Sum of Moments about Point A (ΣM_A = 0):** The moments caused by the forces acting on the ladder must sum to zero to ensure rotational equilibrium. 2. **Analyze the Forces:** - Normal reaction force at A (N_A) - Frictional force at A (f_A) - Normal reaction force at B (N_B) - Weight of the ladder (W) 3. **Formulate the Equations:** - Using trigonometry and the Pythagorean theorem to find the angles and distances for calculating the forces. - Applying the equilibrium conditions to form a system of equations. - Solving the quadratic equation to determine \(\mu_s\). The detailed solution will involve substituting known values into these equations and solving for the unknown coefficient of static friction \(\mu_s\
Transcribed Image Text:### Minimum Coefficient of Static Friction for Ladder Stability **Problem Statement:** Assuming \(\mu_s\) is the same for all contacting surfaces, determine the minimum \(\mu_s\) such that equilibrium is maintained for the leaning 6.5 m ladder AB. The weight of the ladder is non-negligible. *Hint: The equilibrium equations yield a quadratic equation for \(\mu_s\), for which one solution is valid.* **Diagram Explanation:** The diagram depicts a ladder AB leaning against a vertical wall. The ladder is 6.5 meters long and forms a right-angled triangle with the wall and the ground. The point where the ladder touches the ground is labeled as A, and the point where it touches the wall is labeled as B. The horizontal distance from point A to the vertical wall is 2.5 meters, and the height at which the ladder touches the wall (point B) is 6 meters. This setup requires calculating the minimum coefficient of static friction (\(\mu_s\)) to maintain equilibrium and prevent the ladder from slipping. --- For solving the problem: 1. **Equilibrium Conditions:** - **Sum of Forces in the Horizontal Direction (ΣF_x = 0):** The horizontal component of the force exerted by the ground at A must balance the horizontal component of the force exerted by the wall at B. - **Sum of Forces in the Vertical Direction (ΣF_y = 0):** The vertical component of the force exerted by the ground at A must balance the weight of the ladder. - **Sum of Moments about Point A (ΣM_A = 0):** The moments caused by the forces acting on the ladder must sum to zero to ensure rotational equilibrium. 2. **Analyze the Forces:** - Normal reaction force at A (N_A) - Frictional force at A (f_A) - Normal reaction force at B (N_B) - Weight of the ladder (W) 3. **Formulate the Equations:** - Using trigonometry and the Pythagorean theorem to find the angles and distances for calculating the forces. - Applying the equilibrium conditions to form a system of equations. - Solving the quadratic equation to determine \(\mu_s\). The detailed solution will involve substituting known values into these equations and solving for the unknown coefficient of static friction \(\mu_s\
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