We are analyzing a static force diagram where two cables are tied together at point C. Point C is subjected to a horizontal load of 660 Newtons to the right. Point A is located 3 meters above the horizontal level of point C, and point B is 1.4 meters below. The horizontal distance from the vertical support to point C is 2.25 meters. Our goal is to determine the tension in cables AC and BC.
To solve for the tensions in cables AC and BC, we apply the principles of static equilibrium. At point C, the sum of all forces in both the horizontal and vertical directions must equal zero. We'll first calculate the angles alpha and beta that the cables make with the horizontal. Then, we'll set up equations based on the force components in each direction.
We calculate the angles using the given dimensions. For cable AC, the vertical distance is 3 meters and the horizontal distance is 2.25 meters. So the tangent of angle alpha is 3 divided by 2.25, which gives us approximately 53.13 degrees. For cable BC, the vertical distance is 1.4 meters and the horizontal distance is 2.25 meters. So the tangent of angle beta is 1.4 divided by 2.25, which gives us approximately 31.89 degrees.
Now we set up our force equations at point C. In the horizontal direction, the component of tension in cable AC times cosine alpha minus the component of tension in cable BC times cosine beta must equal the applied load of 660 Newtons. In the vertical direction, the component of tension in cable AC times sine alpha minus the component of tension in cable BC times sine beta must equal zero. These two equations form a system that we can solve for the two unknown tensions.
Using the calculated angles, we find that cosine alpha is approximately 0.6 and sine alpha is 0.8. For beta, cosine is approximately 0.84 and sine is 0.54. Substituting these values into our force equations and solving the system, we determine that the tension in cable AC is approximately 350 Newtons and the tension in cable BC is approximately 530 Newtons. This matches the second option in our multiple choice question.