# Find cable tension in three dimensions

## Homework Statement

A piece of machinery of weight W is temporarily supported by cables AB, AC, and ADE. Cable ADE is attached to the ring at A, passes over the pulley at D and back through the ring, and is attached to the support at E. Knowing that the tension in cable AB is 300 N, determine (a) the tension in AC, (b) the tension in ADE, (c) the weight W. (Hint: The tension is the same portions of cable ADE.)

Here is a link to the picture of the drawing just with a different problem (On page 63 Fig. P2.123):

## Homework Equations

$$\Sigma$$F = 0
F = FU (u is a unit vector pointing in the direction of F)
U = AB/AB (AB is a distance vector for example describing the length and oreintation of the cables in this problem)

## The Attempt at a Solution

I first am finding U for cables AC, AE, AD, and AB and then I multiply that vector by the unknown magnitude of the tension. I then sum all the x, y, and z components of each cable tension and the weight of the machine giving me 3 equations with three unknowns. Solving these equations does not give me the correct answers to this problem. I think I am solving it in the correct way I just can't figure out what I am doing wrong. Is my way of solving this problem correct?

KEØM

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Here is a scanned picture of my work.

KEØM

#### Attachments

• Statics Problem#1.pdf
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djeitnstine
Gold Member
Dont forget that cable AD is not just one cable but has to be treated as 2 because it is looped through the pulley. your expression for summation of forces will have a factor of 2 to account for forces in AE and AD i.e. $$\vec{T_{AC}} + \vec{T_{AB}} + 2 \vec{T_{AD}} + \vec{T_{AE}} = 0$$

(The pdf is pending approval so we cannot view it. Upload it on a filesharing site to show us much quicker)

Since $$T_{AE} = T_{AD}$$ you will have some cancellation of components I believe

Edit: This was just the quick and dirty, don't forget to account for the directions of the components etc...

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Thank you for the reply. I did not account for the second cable but I don't understand why you would write it as $$2\vec{T_{AD}} 2\vec{T_{AE}}$$.

I have a link to my solution on a file sharing website. I don't have my answers to the problem on this sheet though and I also used $$\lambda$$ instead of U for my unit vector. One other thing is that the dimensions on my problem are different than that of the picture on the google books page.

http://jumalafiles.info/showfile2-91015519004423306881628597981475444/statics_problem_1.pdf" [Broken]

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Coordinates
A ( 0, -2.4, 0)
B ( -2.7, 0, -3.6)
C ( 0, 0, 1.8)
D ( 1.2, 0, -0.3)
E ( -2.7, 0, 1.2)

$$\Sigma\vec{F} = 0$$
$$\Sigma\vec{F} = \vec{T_{AC}} + \vec{T_{AE}} + \vec{T_{AB}} + \vec{T_{AD}} + \vec{W} = 0$$

$$\vec{T_{AC}} = T_{AC}\vec{U_{AC}}$$

$$\vec{U_{AC}} = \vec{AC}/AC = \frac{(0-0)\vec{i} + (0 - -2.4)\vec{j} + (1.8 - 0)\vec{k}}{\sqrt{(0)^2 + (2.4)^2 + (1.8)^2}}$$

$$\vec{T_{AC}} = 0T_{AC}\vec{i} + 0.8T_{AC}\vec{j} + 0.6T_{AC}\vec{k}$$

I continued this process for each cable in the sum. I then summed all of their components giving me three equations.

$$\Sigma\vec{F_{x}} = 0T_{AC} - 0.5294T_{AB} - 0.7903T_{AE} + 0.4444T_{AD} = 0$$

$$\Sigma\vec{F_{y}} = 0.8T_{AC} + 0.4706T_{AB} + 0.6305T_{AE} - 0.8889T_{AD} - W = 0$$

$$\Sigma\vec{F_{z}} = 0.6T_{AC} - 0.7056T_{AB} + 0.3152T_{AE} - 0.1111T_{AD} = 0$$

Then knowing that $$T_{AE} = T_{AD}$$ and that $$T_{AB} = 300N$$ I simplified the equation and solved for the unknowns.

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djeitnstine
Gold Member
Thank you for the reply. I did not account for the second cable but I don't understand why you would write it as $$2\vec{T_{AD}} 2\vec{T_{AE}}$$.

Srry late night typo...will fix

djeitnstine
Gold Member
Ok so when you sum each component, you want to write the Forces in the x direction as $$T_{AC}C_1 + T_{AB}C_2 + T_{AE} C_3 + 2T_{AD} C_4$$ where $$C_n$$ is the coefficient of i'th component of that force. So lets consider the equality as mentioned before... Let $$T_{AE} = T_{AD}=T_{ADE}$$

We then have: $$T_{AC}C_1 + T_{AB}C_2 + T_{ADE}( C_3 + 2C_4)$$

I understand, so really all I am missing is the 2 in front of the $$\vec{T_{AD}}$$ component right? I just didn't account for the the second cable coming around the pulley.

nvn
Homework Helper
KEØM: First, you are dropping far too many significant digits in your first calculation, lamba_ac. Always maintain four significant digits throughout all your intermediate calculations. Secondly, your equations on p. 1 are correct except it appears you did not double-check your calculations. You calculate completely wrong values from correct formulas. Double-check your calculations.

Thanks nvn. I will make sure to put in those digits and double check my calculations. Other than those mistakes and not accounting for the second cable am I correct?

djeitnstine
Gold Member
everything else looks fine...your equations are correct.

Thanks again for all of your help djeitnstine and nvn.

KEØM