A crane suspends a 500-lb steel beam horizontally by support cables (with negligible weight) attached from a hook to each end of the beam. The support cables each make an angle of 60° with the beam. Find the tension vector in each support cable and the magnitude of each tension.
How does you get 250-lb?
the magnitude of the vertical component is 250-lb
tension = 250-lb / sin(60º)
the beam weighs 500-lb
it is symmetrically supported by a cable at each end ... 250-lb per end
To find the tension vector in each support cable and the magnitude of each tension, we can use trigonometry.
Let's start by drawing a diagram to visualize the situation. The steel beam is suspended horizontally, and two support cables are attached to each end of the beam, making a 60° angle with the beam.
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We'll call the tension in the left cable T₁ and the tension in the right cable T₂.
Now let's split the problem into two vertical forces acting on the steel beam: the weight of the beam (500 lbs) acting downward and the vertical component of the tension force in each cable acting upward.
Since the beam is in equilibrium, the sum of the vertical forces must be zero. So the upward force in each cable must equal the downward force due to the weight of the beam.
Now let's find the vertical component of the tension force for each cable. This will be the value we need to balance the weight of the beam.
To find the vertical component, we can use trigonometry. Since the angle between each cable and the beam is 60°, we can use the sine function.
The vertical component of T₁ is given by:
Vertical component of T₁ = T₁ * sin(60°)
Similarly, the vertical component of T₂ is given by:
Vertical component of T₂ = T₂ * sin(60°)
Since the vertical component of each tension force must equal the weight of the beam (500 lbs), we have:
T₁ * sin(60°) = 500 lbs (Equation 1)
T₂ * sin(60°) = 500 lbs (Equation 2)
Now let's solve these equations simultaneously to find the tensions in each cable.
From Equation 1, we have:
T₁ = 500 lbs / sin(60°)
Similarly, from Equation 2, we have:
T₂ = 500 lbs / sin(60°)
Now we can calculate the tensions in each cable.
T₁ = 500 lbs / sin(60°) ≈ 577.35 lbs
T₂ = 500 lbs / sin(60°) ≈ 577.35 lbs
So the tension vector in each support cable is approximately 577.35 lbs, and the magnitude of each tension is also approximately 577.35 lbs.