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inimum distance x from point A at which an additional weight 2w (twice the weight of the rod) can be hung without causing the rod to

inimum distance x from point A at which an additional weight 2w (twice the weight of the rod) can be hung without causing the rod to

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As shown in the figure below, a uniform beam is supported by a cable at one end and the force of friction at the other end. The cable makes an angle of 

? = 30°,

 the length of the beam is 

L = 3.25 m,

 the coefficient of static friction between the wall and the beam is 

?s = 0.560,

 and the weight of the beam is represented by w. Determine the minimum distance x from point A at which an additional weight 2w (twice the weight of the rod) can be hung without causing the rod to slip at point A.

### Analysis of Beam with a Suspended Weight #### Diagram Description: The diagram presented shows a horizontal beam, labeled \(AB\), attached perpendicularly to a vertical wall at point \(A\). - A rope or cable is connected from point \(B\) of the beam to the wall at a higher point above \(A\), creating a right triangle with the beam and the wall. - The angle between the rope and the beam is labeled as \(\theta\). - A weight labeled \(2w\) is suspended from a point on the beam, a distance \(x\) from point \(A\) (the point where the beam is attached to the wall). #### Detailed Analysis: **Beam:** The horizontal beam extends from point \(A\) on the wall to point \(B\). The attachment at \(B\) is supported by a tensioned rope creating an angle \(\theta\) with the beam, offering vertical and possibly horizontal support. **Rope/Cable Angle:** The angle \(\theta\) between the rope and the beam is crucial for determining the components of tension in the rope. This will influence both vertical and horizontal forces acting on the beam. **Suspended Weight:** A weight of \(2w\) is shown hanging a distance \(x\) from point \(A\). This uniform load contributes to the bending moment and shear force on the beam. #### Mechanical Analysis Considerations: - **Tension in the Support Rope:** Needs to be calculated considering the angle \(\theta\) and distributing the forces along the beam. - **Reaction Forces at \(A\):** Will have both vertical and horizontal components due to the tension in the rope and the weight. - **Shear Force and Bending Moment:** At any section of the beam will vary as a function of the distance from the wall, \(x\), and can be derived using equilibrium equations. This setup exemplifies fundamental principles in statics involved in analyzing forces and moments in a beam with multiple supports and loads.
Transcribed Image Text:### Analysis of Beam with a Suspended Weight #### Diagram Description: The diagram presented shows a horizontal beam, labeled \(AB\), attached perpendicularly to a vertical wall at point \(A\). - A rope or cable is connected from point \(B\) of the beam to the wall at a higher point above \(A\), creating a right triangle with the beam and the wall. - The angle between the rope and the beam is labeled as \(\theta\). - A weight labeled \(2w\) is suspended from a point on the beam, a distance \(x\) from point \(A\) (the point where the beam is attached to the wall). #### Detailed Analysis: **Beam:** The horizontal beam extends from point \(A\) on the wall to point \(B\). The attachment at \(B\) is supported by a tensioned rope creating an angle \(\theta\) with the beam, offering vertical and possibly horizontal support. **Rope/Cable Angle:** The angle \(\theta\) between the rope and the beam is crucial for determining the components of tension in the rope. This will influence both vertical and horizontal forces acting on the beam. **Suspended Weight:** A weight of \(2w\) is shown hanging a distance \(x\) from point \(A\). This uniform load contributes to the bending moment and shear force on the beam. #### Mechanical Analysis Considerations: - **Tension in the Support Rope:** Needs to be calculated considering the angle \(\theta\) and distributing the forces along the beam. - **Reaction Forces at \(A\):** Will have both vertical and horizontal components due to the tension in the rope and the weight. - **Shear Force and Bending Moment:** At any section of the beam will vary as a function of the distance from the wall, \(x\), and can be derived using equilibrium equations. This setup exemplifies fundamental principles in statics involved in analyzing forces and moments in a beam with multiple supports and loads.
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