A uniform, horizontal beam is 5.00m long and has a weight of 200N. The beam is hinged at one end to a wall. A 600N weight is attached to the order end of the beam. The beam is supported by cable at the end opposite the wall. The cable attaches to the wall directly above the beam. If the tension in the cable is 1000N, what is the angle between the cable and the beam? What is the horizontal force of the hinge?
2.3
To find the angle between the cable and the beam, we can use trigonometry. Let's denote the angle as θ.
The horizontal force of the hinge can be found by taking the sum of the horizontal forces acting on the beam and setting it equal to zero.
First, let's determine the vertical forces acting on the beam. We have the weight of the beam (200N) acting downwards and the weight attached at the end (600N) acting upwards. The net vertical force is (600N - 200N) = 400N upwards.
Next, let's determine the horizontal forces acting on the beam. There are two horizontal forces: the tension in the cable and the horizontal force at the hinge.
Now, let's break down the tension in the cable into its vertical and horizontal components. The vertical component of the tension is equal to the force of gravity acting on the beam (200N), and the horizontal component is equal to the horizontal force at the hinge.
Since the tension in the cable is given as 1000N, we can use trigonometry to find the horizontal component. The horizontal component is equal to the tension multiplied by the cosine of the angle θ:
Horizontal component = Tension * cos(θ)
= 1000N * cos(θ)
Now, we can set up the equation for the sum of the horizontal forces:
Horizontal force at hinge - Horizontal component = 0
Solving for the horizontal force at the hinge:
Horizontal force at hinge = Horizontal component
= 1000N * cos(θ)
To find the angle θ, we need to use the vertical forces. We can use the vertical component of the tension to set up an equation:
Vertical component of tension + Weight of the beam + Weight attached at the end = 0
Substituting the values:
Tension * sin(θ) + 200N + 600N = 0
Simplifying:
Tension * sin(θ) = -800N
Now we can solve for sin(θ):
sin(θ) = -800N / Tension
Once we have the value of sin(θ), we can find the angle θ using inverse sine (arcsine) function:
θ = arcsin(-800N / Tension)
Substituting the given values:
θ = arcsin(-800N / 1000N)
Calculating the value of θ using a calculator or mathematical software, we get:
θ ≈ -53.13 degrees
Note: The negative sign indicates that the angle is measured below the horizontal line.
Finally, substituting the value of θ into the equation for the horizontal force at the hinge:
Horizontal force at hinge = 1000N * cos(-53.13)
≈ 573.2N
So, the angle between the cable and the beam is approximately -53.13 degrees, and the horizontal force at the hinge is approximately 573.2N.