To determine the magnitude and coordinate direction angles (α, β, γ) of the resultant force acting on the tower, you can use vector addition. Given that the tower is held in place by three cables, you'll need to sum up the forces from each cable.
Let's represent the forces from each cable as vectors:
F1 = 30 N (tension in cable 1)
F2 = 40 N (tension in cable 2)
F3 = 50 N (tension in cable 3)
Now, you can find the resultant force (R) by summing these vectors:
R = F1 + F2 + F3
Find the x-component of R:
Rx = ΣFx = F1x + F2x + F3x
Rx = 0 + (-40)cos(30°) + (-50)cos(45°)
Rx = -20√3 - 50√2 N
Find the y-component of R:
Ry = ΣFy = F1y + F2y + F3y
Ry = 30 + (-40)sin(30°) + (-50)sin(45°)
Ry = 30 - 20 - 35.36
Ry = -25.36 N
Now, you can find the magnitude of R using the Pythagorean theorem:
|R| = √(Rx² + Ry²)
|R| = √((-20√3 - 50√2)² + (-25.36)²)
|R| ≈ 81.53 N (rounded to two decimal places)
Now, let's find the direction angles:
α = arctan(Ry / Rx) = arctan(-25.36 / (-20√3 - 50√2))
α ≈ 70.18°
β = arctan(Rx / Ry) = arctan((-20√3 - 50√2) / -25.36)
β ≈ 293.82°
γ = 180° - (α + β)
γ ≈ 16°
So, the magnitude of the resultant force is approximately 81.53 N, and the coordinate direction angles are:
α ≈ 70.18°
β ≈ 293.82°
γ ≈ 16°