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Four forces are exerted on the eyebolt as shown. If the net effect on the bolt is a direct pull of 545 lb in the y-direction, determine the necessary values of [tex]T[/tex] and [tex]\theta[/tex]. Assume [tex]F_1 = 400[/tex] lb, [tex]F_2 = 215[/tex] lb.

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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