Calculating Cable Length and Stretch for Uniform Iron Beam in Static Equilibrium

In summary: So, summing torques about the center of mass (which is the same as the end of the bar, because it's uniform): \Sigma M_o = T_y\cdot L - \frac{WL}{2} = 0 You can use this to find T_y. Then use σ = εE to solve for the strain ε. Then use the equation for elongation under tension: \Delta L = \frac{L\sigma}{AE} to solve for the stretch, ΔL.In summary, the picture attached shows a uniform iron beam with a mass of 254 kg and a length of 3 m. The cable holding the beam in place can withstand a
  • #1
Juntao
45
0
Picture is attached.

The above figure shows a uniform iron beam of mass 254 kg and length L = 3 m. The cable holding the beam in place can take a tension of 1300 N before it breaks. (You may ignore the small mass of the cable in this calculation.)

======
a)What minimum length of the cable?
b) Assume the cable is made of steel and has a diameter of 1". How much will it stretch before it breaks?

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Okay. I am stuck on part a so far. I know that this is a static equilibrium, therefore, the summation of all the forces in x and y direction equal zero.
Also, the summation of torques must also equal zero.

But I'm not sure how to write out the equations. Can someone give me a lending hand?
 

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  • #2
Let me try to label some forces without drawing a picture:

The weight of the bar, W, acts at the center of mass of the bar (1.5 m from the left end (set your origin at the left end)).

The tension in the cable, T, has components Tx (clearly pointing to the left) and Ty (pointing up).

The mounting of the bar (pin joint) can support a vertical force, Ry (pointing up) and a horizontal force, Rx (pointing to the right).

Some equations for static equilibrium:

[tex] \Sigma F_y = R_y + T_y - W = 0 [/tex]

[tex] \Sigma F_x = R_x - T_x = 0 [/tex]

[tex] \Sigma M_o = T_y\cdot L - \frac{WL}{2} = 0 [/tex]

You can use these to find the y-component of the tension (actually, you only need the moment equation), but not much else.

Say you've solved for Ty and you want to find the minimum length of cable. You know that for the minimum length of cable, the cable tension will be at its maximum (not a great plan from an engineering standpoint, but nonetheless...). So:

[tex] T = \sqrt{T_x^2+T_y^2} = 1300 [/tex]

use that to solve for Tx.

The length of the cable is given by:

[tex] L_c = \frac{L}{\cos \theta} [/tex]

where Lc is the length of the cable, L is the length of the bar (3 m), and θ is the angle between the cable and the bar, so that:

[tex] \tan\theta = \frac{T_y}{T_x} [/tex]

That should allow you to find Lc

For part b, you should be able to use Hooke's law, σ = εE, to find the length it stretches (you have the tension, the cross sectional area, the unstretched length, and you can look up the Young's modulus).
 
  • #3
I'm not sure as I haven't actually worked it through, but because both θ and Lc are unknown, you might also have to use
Ty = Ry

(This must be true since otherwise the bar would rotate about its center of mass.)
 

1. What is the minimum length of cable I need for my project?

The minimum length of cable needed for a project depends on several factors, such as the distance between the two connected points, the type of cable being used, and the voltage and current requirements. It is important to consult a professional or do thorough research to determine the appropriate minimum length of cable for your specific project.

2. How do I calculate the minimum length of cable needed?

To calculate the minimum length of cable needed, you will need to consider the distance between the two connected points, the cable's resistance and current carrying capacity, and the voltage drop allowed. You can use online calculators or seek assistance from a professional to accurately calculate the minimum length of cable for your project.

3. Can I use a cable that is shorter than the minimum length required?

It is not recommended to use a cable that is shorter than the minimum length required for your project. Doing so can result in a high voltage drop, which can cause damage to your equipment or affect its performance. It is always best to use the recommended minimum length of cable to ensure safe and efficient operation.

4. Is there a maximum length for cables?

Yes, there is a maximum length for cables. The maximum length is determined by several factors, such as the type of cable, voltage and current requirements, and environmental conditions. It is crucial to consult a professional or do thorough research to determine the appropriate maximum length for your project.

5. What are the consequences of using a cable that is too short?

Using a cable that is too short can result in a high voltage drop, which can cause damage to your equipment or affect its performance. It can also lead to increased resistance and heat buildup, which can result in electrical fires. It is essential to use the recommended minimum length of cable to ensure safe and efficient operation of your project.

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