Shoia all on your work mahe clear your a ssumptions vea Soni

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Solution

consider DA as x-axis and DB as y-axis and the mast DO as z-axis. let O be our origin.

we will take g= 32ft/s²

A = (8,0, 5), B= (0,8,5)

Force Fa = 495g k^

Force Fb = 375g k^

Fa will produce a moment about y-axis, it is zero about x and z

moment of Fa

    = 495g*8 j^ = 126720 j^

similarly Fb will produce a moment about x-axis and 0 for y and z

moment of Fb

    = 375g*6 i^ = 72000 i^

weight of the mast will make 0 moment about any axis as we considered O as the origin and OD as z-axis

resultant moment of the weigthts

= 72000i^ + 126720j^

Let the cable be fixed at apoint P on the ground, it shall be fixed to ground somewhere along (–x, -y ,0) position so that the tension in the cable will produce a moment to counter the resultant moment

Let the tension

T = pi^ + qj^ + rk^

suppose the cable makes an angle with the horizontal then component TSin will act along z-axis and TCos , will act in the x-y plane

hence r = TSin and

p² +q² = TCos

TCos is the projection of T on xy-plane

let it make an angle with the x-axis

the p = TCos Cos

and q = TCos Sin

suppose L is the length of the cable used then

(LCos Cos , LCos Sin ) will be the point P where the cable will be pegged to the ground.

x-component p and y-component q of the tension will produce zero moment as they pass through the point D and the arm length=0

only z-component r=TSin, will produce a moment and can counter the moment of the weights.

moment of r about x-axis

r LCos Sin i^= 7200i^

and about y-axis

r LCos Cos j^=126720 j

Tan = 72000/126720

= 29.6⁰

wehave to choose the point P such Op makes an angle 29.6⁰ with the x-axis, we have no other choice.

substituting r= TSin

TLSin Cos = 72000/Sin(29.6⁰)

TLSin Cos = 145746

we have an option to choose L such that T-tension in the cable is minimum, maximum of L = 60

height of the mast OD = 5

Sin() = (5/L)

TL(5/L)Cos = 145746

T Cos = 145746/5 = 29149

T is minimum when Cos is maximum ideally 1 but we shall have infinet length of the cable we can have maximum of L= 60

(Cos )² = 1-25/60² = 0.993

= 4.78⁰

T = 29149/Cos(4.78) = 29259 N

z-component   r = TSin(4.78) = 2438N

x-component p = 29259Cos (4.78)Cos(29.6) = 25352 N

y-component q = 29259Cos (4.78)Sin(29.6) = 14402 N