If the balloon is subjected to a net uplift force of F = 800 N, determine the tension developed in ropes AB, AC, AD.

#### Solution:

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We will first write each tension developed in Cartesian vector form. To do so, we need to write the locations of points A, B, C, and D in Cartesian vector form.

From the diagram, the locations of the points are:

B:(-1.5i-2j+0k) m

C:(2i-3j+0k) m

D:(0i+2.5j+0k) m

The position vectors for each rope are:

r_{AC}\,=\,\left\{(2-0)i+(-3-0)j+(0-6)k\right\}\,=\,\left\{2i-3j-6j\right\}

r_{AD}\,=\,\left\{(0-0)i+(2.5-0)j+(0-6)k\right\}\,=\,\left\{0i+2.5j-6j\right\}

The magnitude of each position vector is:

magnitude of r_{AC}\,=\,\sqrt{(2)^2+(-3)^2+(-6)^2}\,=\,7

magnitude of r_{AD}\,=\,\sqrt{(0)^2+(2.5)^2+(-6)^2}\,=\,6.5

The unit vectors are:

u_{AC}\,=\,\left(\dfrac{2}{7}i+\dfrac{-3}{7}j-\dfrac{6}{7}k\right)

u_{AD}\,=\,\left(0i+\dfrac{2.5}{6.5}j-\dfrac{6}{6.5}k\right)

We can now express each force in Cartesian vector form:

F_{AC}\,=\,F_{AC}\left(\dfrac{2}{7}i-\dfrac{3}{7}j-\dfrac{6}{7}k\right)

F_{AD}\,=\,F_{AD}\left(0i+\dfrac{2.5}{6.5}j-\dfrac{6}{6.5}k\right)

(further simplify this by expanding the brackets using FOIL and simplifying the fractions into decimal values)

F_{AB}\,=\,\left\{-0.231F_{AB}i-0.308F_{AB}j-0.923F_{AB}k\right\}

F_{AC}\,=\,\left\{0.286F_{AC}i-0.429F_{AC}j-0.857F_{AC}k\right\}

F_{AD}\,=\,\left\{0i+0.385F_{AD}j-0.923F_{AD}k\right\}

F\,=\,\left\{0i+0j+800k\right\}

(Force F is the net uplift force, which is applied directly upwards, thus it only has a z-component)

We can now write our equations of equilibrium. All forces added together must equal zero.

F_{AB}+F_{AC}+F_{AD}+F\,=\,0

Since all forces added together must equal zero, then all individual components (x, y, z-components) added together must also equal zero.

-0.231F_{AB}+0.286F_{AC}\,=\,0

y-components:

-0.308F_{AB}-0.429F_{AC}+0.385F_{AD}\,=\,0

z-components:

-0.923F_{AB}-0.857F_{AC}-0.923F_{AD}+800\,=\,0

Solving the three equations simultaneously gives us:

F_{AC}\,=\,202.9 N

F_{AD}\,=\,427.1 N

hey,

your solution is wrong. the points you have set up have a wrong sign.

thx

Where is it wrong? Just saying solution is wrong doesn’t help anyone, please point out what value is wrong, it’s easy to miss when checking so it would be very helpful if you can point out the error. I checked through and none of the signs seem to be incorrect.

I think that the Y component of C should be -3 instead of +3

If not, please respond why.

Thx,

Kind regards

Joris

Ahh, I see, thank you! I will fix it, however, the solution is correct, I think it was a typo because the correct values were used for the position coordinates 🙂 Many thanks!