Math Problem Statement

Compute the forces in the two angled rods in Figure P3–9 for an applied force, F = 1500 lb, if the angle u is 45

Solution

To compute the forces in the two angled rods for an applied force of F=1500lbF = 1500 \, \text{lb} and an angle θ=45\theta = 45^\circ, we will follow the steps of resolving the applied force into components along the directions of the rods.

Assumptions:

  • The system is in static equilibrium, meaning the sum of the forces in both horizontal and vertical directions is zero.
  • There are two rods, each at a 4545^\circ angle to the horizontal. Let's assume the forces in the rods are F1F_1 and F2F_2, and they act along the direction of the rods.

Steps:

  1. Resolve forces into horizontal and vertical components: The applied force F=1500lbF = 1500 \, \text{lb} acts vertically downward. Each rod force has horizontal and vertical components due to the angle of 4545^\circ.

  2. Equilibrium in the horizontal direction: The horizontal components of the forces in the two rods must balance each other. If F1F_1 and F2F_2 are the forces in the rods: F1cos(45)=F2cos(45)F_1 \cos(45^\circ) = F_2 \cos(45^\circ) Therefore, F1=F2F_1 = F_2.

  3. Equilibrium in the vertical direction: The sum of the vertical components of the forces in the rods must equal the applied force FF: F1sin(45)+F2sin(45)=1500lbF_1 \sin(45^\circ) + F_2 \sin(45^\circ) = 1500 \, \text{lb} Since F1=F2F_1 = F_2, we can simplify this to: 2F1sin(45)=1500lb2F_1 \sin(45^\circ) = 1500 \, \text{lb} F1=15002sin(45)F_1 = \frac{1500}{2 \sin(45^\circ)}

  4. Calculate F1F_1: F1=15002×0.70711060.66lbF_1 = \frac{1500}{2 \times 0.7071} \approx 1060.66 \, \text{lb} Since F1=F2F_1 = F_2, the force in each rod is: F1=F2=1060.66lbF_1 = F_2 = 1060.66 \, \text{lb}

Conclusion:

The force in each rod is approximately 1060.66lb1060.66 \, \text{lb}.

Do you need more details or have any questions?

Here are 5 related questions:

  1. How does changing the angle affect the forces in the rods?
  2. What would happen if the applied force FF increased?
  3. How do we compute the forces if the rods have different angles?
  4. How would the forces change if there was friction at the supports?
  5. What is the impact of adding a horizontal force component in this setup?

Tip: In static equilibrium problems, always check that the sum of forces and moments in all directions equals zero.

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Math Problem Analysis

Mathematical Concepts

Statics
Trigonometry
Force Resolution
Equilibrium

Formulas

F_1 = F_2
2F_1 \sin(\theta) = F
F_1 = \frac{F}{2 \sin(\theta)}

Theorems

Equilibrium of Forces

Suitable Grade Level

Undergraduate Engineering or Physics