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Answer :
The necessary values are: T ≈ 215 lb (directed in the positive x-direction)
θ ≈ -63.43°
To determine the necessary values of T and θ, we can analyze the forces acting on the eyebolt and set up equations based on the given information.
Let's consider the forces acting in the x and y directions:
In the x-direction:
T cosθ - F₂ = 0 (Equation 1)
In the y-direction:
T sinθ + F₁ - F₃ = 545 lb (Equation 2)
We have two equations and two unknowns (T and θ).
Now we can substitute the given values and solve for T and θ.
Given:
F₁ = 400 lb
F₂ = 215 lb
Net force in the y-direction = 545 lb
Substituting these values into Equation 2, we have:
T sinθ + 400 - F₃ = 545 lb
Since the net force in the y-direction is directly pulling in the positive y-direction, F₃ must be in the negative y-direction. Therefore, we have F₃ = -545 lb.
Substituting this into the equation, we get:
T sinθ + 400 - (-545) = 545 lb
Simplifying further:
T sinθ + 945 = 545 lb
T sinθ = -400 lb
Now let's substitute this into Equation 1:
T cosθ - 215 = 0
Solving this equation for T cosθ, we get:
T cosθ = 215 lb
Now we can divide the two equations to eliminate T:
(T sinθ) / (T cosθ) = (-400 lb) / (215 lb)
Simplifying, we have:
tanθ = -1.8605
Taking the arctan of both sides, we find:
θ ≈ -63.43°
Therefore, the necessary values are:
T ≈ 215 lb (directed in the positive x-direction)
θ ≈ -63.43°
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