If the resultant force acting on the support is to be 1200 lb directed horizontally to the right, determine the force F in rope A and the corresponding angle
The correct answer and explanation is :
To determine the force ( F ) in rope A and the corresponding angle ( \theta ), we need to apply the principles of static equilibrium. In a static system, the sum of forces in each direction (horizontal and vertical) must equal zero. Additionally, the sum of moments around any point must also equal zero.
Step 1: Define the forces and the situation
Let:
- ( F ) be the force in rope A,
- ( \theta ) be the angle between rope A and the horizontal,
- ( 1200 \, \text{lb} ) be the resultant force acting horizontally to the right.
Assume that the force in the rope A has both horizontal and vertical components. The horizontal component of the force in rope A is ( F \cos(\theta) ), and the vertical component is ( F \sin(\theta) ).
Step 2: Apply the equilibrium conditions
Horizontal equilibrium:
The horizontal forces must balance. If the only horizontal force is ( F \cos(\theta) ), and the resultant force is 1200 lb, we have:
[
F \cos(\theta) = 1200
]
Vertical equilibrium:
The vertical forces must balance. If there are no vertical external forces given (assuming there’s no additional load or forces vertically), the vertical component of the rope A’s force must be zero:
[
F \sin(\theta) = 0
]
This implies ( \theta = 0^\circ ), meaning rope A is horizontal.
Step 3: Solve for ( F )
From the horizontal equilibrium equation:
[
F \cos(0^\circ) = 1200
]
Since ( \cos(0^\circ) = 1 ), the equation simplifies to:
[
F = 1200 \, \text{lb}
]
Step 4: Conclusion
Thus, the force ( F ) in rope A is 1200 lb, and the corresponding angle ( \theta ) is 0°, meaning rope A is aligned horizontally.
Image Representation:
The diagram would show rope A being pulled horizontally to the right with a force of 1200 lb, and the angle ( \theta ) being 0°, indicating that rope A is horizontal.
Here is the diagram illustrating the situation, where the rope A is horizontal, and the force in the rope is 1200 lb acting horizontally to the right with an angle ( \theta = 0^\circ ).
