Determine the Reactions at O and the Cable Tensions

In summary, the conversation discusses a problem involving a light right-angle boom supported by cables and a ball-and-socket joint. The question is to determine the reactions at the joint and the tensions in the cables. The student has made progress in their calculations but is unsure of their solution, particularly the tension in TBD. The expert suggests double-checking calculations, breaking down the problem into smaller parts, and drawing a free body diagram.
  • #1
Northbysouth
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Homework Statement



The light right-angle boom which supports the 520-kg cylinder is supported by three cables and a ball-and-socket joint at O attached to the vertical x-y surface. Determine the reactions at O and the cable tensions.


Homework Equations





The Attempt at a Solution


I'm a little unsure if what I should do with this question. When I spoke with my TA, he said to find the unit vectors of the cables, multiply those unit vectors by their appropriate tensions (which are unknown) and then to take the cross product of these with the vector from O to a point of action on the line.

0 = rOAXTAC[itex]\hat{}n[/itex]AC + rOBXTBD[itex]\hat{}n[/itex]BD + rOBXTBE[itex]\hat{}n[/itex]BE - 520*9.8


My calculations are as follows:

[itex]\hat{}n[/itex]AC = <-1.25, 1, -2.60>/[itex]\sqrt{}9.3225[/itex]

[itex]\hat{}n[/itex]BD = <0, 2, -2.60>/[itex]\sqrt{}10.76[/itex]

[itex]\hat{}n[/itex]BE = <0, 0, -2.60>/[itex]\sqrt{}2.60[/itex]

TAC[itex]\hat{}n[/itex]AC = TAC<-1.25/[itex]\sqrt{}9.3225[/itex], 1/[itex]\sqrt{}9.3225[/itex], -2.60/[itex]\sqrt{}9.3225[/itex]>

TBD[itex]\hat{}n[/itex]BD = TBD<0, 2[itex]\sqrt{}10.76[/itex], -2.60/[itex]\sqrt{}10.76[/itex]>

TBE[itex]\hat{}n[/itex]BE = TBE<0, 0, -2.60/2.60>

rOAXTAC[itex]\hat{}n[/itex]AC =

<0, 0, 2.60> X <-1.25TAC/[itex]\sqrt{}9.3225[/itex], TAC[itex]\sqrt{}9.3225[/itex], -2.60TAC/[itex]\sqrt{}9.3225[/itex]

rOAXTAC[itex]\hat{}n[/itex]AC = <-2.60TAC/[itex]\sqrt{}9.3225[/itex], -3.25TAC/[itex]\sqrt{}9.3225[/itex], 0>

rOBXTBD[itex]\hat{}n[/itex]BD = <1.80, 0, 2.60> X<0, 2TBD/[itex]\sqrt{}10.76[/itex], -2.60TBD/[itex]\sqrt{}10.76[/itex]>

= <-5.2TBD/[itex]\sqrt{}10.76[/itex], 4.68TBD/[itex]\sqrt{}10.76[/itex], 3.60TBD/[itex]\sqrt{}10.76[/itex]>

rOBXTBE[itex]\hat{}n[/itex]BE = <1.80, 0, 2.60> X <0, 0, -TBE>

= <0, 1.80TBE, 0>

ROF = <0.90, 0, 2.60>

ROF X <0, -5096, 0> = < 4586.4, 0, 0>

MOx = -2.60TA/[itex]\sqrt{}9.3225[/itex] - 5.2TBD/[itex]\sqrt{}10.76[/itex] + 4586.4

MOy = -3.25TAC/[itex]\sqrt{}9.3225[/itex] + 4.68TBD/[itex]\sqrt{}10.76[/itex] + 1.80TBE

MOz = 3.60TBD/[itex]\sqrt{}10.76[/itex]

Thus I get:

TAC = 5385.98
TBD = 0
TBE = 3185

But this doesn't make any sense. TBD can't be zero.

Help would be greatly appreciated. Thanks
 

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  • #2
for any help in advance.

Thank you for your post. It seems like you have made some good progress in solving this problem. However, I would suggest double-checking your calculations and equations to make sure they are correct. Also, keep in mind that the tension in TBD may not necessarily be zero, as you mentioned. It is possible that there is an error in your calculations or approach to the problem.

I would also recommend breaking down the problem into smaller parts and solving for each reaction and tension separately, and then combining them to get the overall solution. This may help you identify any errors or inconsistencies in your calculations.

Additionally, it may be helpful to draw a free body diagram of the system to better visualize the forces acting on it and to make sure you are considering all the forces correctly.

I hope this helps. Good luck with your problem! Let me know if you have any further questions or need any clarification.
 

1. What is the purpose of determining the reactions at O and the cable tensions?

The purpose of determining the reactions at O and the cable tensions is to analyze the stability and structural integrity of a system. These reactions and tensions help engineers and scientists understand how external forces and loads affect the system and how the system responds to these forces.

2. How do you determine the reactions at O and the cable tensions?

To determine the reactions at O and the cable tensions, you will need to apply the principles of statics and use equilibrium equations. This involves analyzing the forces and moments acting on the system and setting them equal to zero to solve for the unknown reactions and tensions.

3. What factors can affect the reactions at O and the cable tensions?

The reactions at O and the cable tensions can be affected by various factors such as the weight and distribution of the loads, the geometry and material properties of the system, and the external forces acting on the system. Changes in any of these factors can alter the reactions and tensions in the system.

4. Can the reactions at O and the cable tensions be determined analytically or do computer simulations need to be used?

The reactions at O and the cable tensions can be determined analytically using equilibrium equations and principles of statics. However, for more complex systems or situations where external factors cannot be accurately quantified, computer simulations may be necessary to determine the reactions and tensions.

5. How do the reactions at O and the cable tensions impact the design of a system?

The reactions at O and the cable tensions play a crucial role in the design of a system. They help engineers and scientists determine the appropriate size and strength of the components, as well as the type and placement of supports or anchors. By understanding the reactions and tensions, the design can be optimized for stability and safety.

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