What Is the Tension in the Top Cables of the Sculpture?

[mandi182]

Homework Statement

An abstract sculpture is constructed by suspending a sphere of polished rock along the vertical with four cables connected to a square wooden frame 1.7 m. If the stone has mass m = 5.2 kg, and the bottom two cables are under a tension of 25 N each, what is the tension in each top cable?

The picture just shows a square with a ball in the middle and 4 cables on each side.

Homework Equations

I don't know what equation to use. I know that it is not accelerating but that is all the I know.

The Attempt at a Solution

I thought that it may be the tension given because the top would hold more. I also tried half that and quadruple that. None of the other people in my class were able to figure it out.

I really just need someone to explain it to me.

Thanks!

 

[PhanthomJay]

Without a picture, I'm envisioning a square open frame standing vertically, with the sphere suspended in the center, supported by 4 cables, each of which is attached to the corners of the frame, making an X pattern Is this correct?

 

[mandi182]

Yes, it is basically an X with a ball in the middle.

 

[PhanthomJay]

OK. Draw a free body diagram of the sphere (isolate it to draw the forces acting on it). It's weight will act veritically down, acting away from the sphere, and the 2 lower tension forces as given (25 N each) pull away from the sphere, each at a 45 degree angle, left or right of the vertical. The 2 unknown top tension forces also pull away from the sphere, at a 45 degree angle, left or right of the vertical. Since, as you have noted, the sphere is in equilibrium, use Newton's 1st law in both the x and y directions to solve for the value of the tensions in the upper cords (you can make use of the symmetry of the problem to make it even easier).

 

[rwisz]

I can provide you with a solution to this problem using the principles of static equilibrium. The first step is to draw a free body diagram of the sphere, with all the forces acting on it. In this case, we have the weight of the sphere acting downwards, and four tension forces acting upwards from the cables.

Next, we can use the principle of static equilibrium, which states that the sum of all the forces acting on an object must be equal to zero if the object is not accelerating. In this case, we can write the following equation:

ΣF = 0

This means that the sum of all the forces acting in the vertical direction must be equal to zero. Since the sphere is not accelerating, we can assume that it is in equilibrium, meaning that the forces acting on it are balanced.

Now, we can calculate the tension in the top cables by using the fact that the bottom two cables are under a tension of 25 N each. Since the wooden frame is a square, we can assume that the tension in each top cable is the same. Therefore, we can write the following equation:

2(25 N) + 2(T) = mg

Where T is the tension in each top cable, and mg is the weight of the sphere. Solving for T, we get:

T = (mg - 50 N) / 2

Substituting in the values given in the problem, we get:

T = (5.2 kg x 9.8 m/s^2 - 50 N) / 2 = 24.6 N

Therefore, the tension in each top cable is 24.6 N. I hope this explanation helps you understand how to approach this type of problem and solve it using the principles of static equilibrium. If you have any further questions, please feel free to ask.