figure: s.yimg.com/tr/i/0f803deaf4944fffa178bdd3c40e958c_A.png
The weight of the cylinder is 40 lb, the ropes are fixed at B, C, and D.
o x direction force equation of equilibrium? TAB + TAC + TAD + WA = 0
o y direction force equation of equilibrium? TAB + TAC + TAD + WA = 0
o z direction force equation of equilibrium? TAB + TAC + TAD + WA = 0
o Tensile force along AB, TAB ? lb
bTensile force along AC, TAC ? lb
Because tension is necessarily positive, I'd write the x-equation of equilibrium as
Tad*sin60º*sin30º - Tac*sin60º*sin30º = 0.
There is no AB term, and no W term.
(This assumes that A is on the y-axis.)
So Tad = Tac
For the y-axis
Tab*cos45º - Tad*sin60º*cos30º + Tac*sin60º*cos30º = 0
0.707*Tab - 0.75*Tad - 0.75*Tad = 0
Tab = 2.123*Tad
Vertically
Tab*sin45º + Tad*cos60º*cos30º + Tac*cos60º*cos30º - 40lb = 0
2.123*Tad + Tad*0.433 + Tad*0.433 = 40 lb
Tad = 40lb / 2.99 = 13.4 lb ◄
Tac = 13.4 lb ◄
Tab = 2.123*13.4lb = 28.4 lb ◄
To solve for the unknowns in this problem, we'll start by breaking down the forces acting on the cylinder in the x, y, and z directions.
1. X direction force equation of equilibrium:
In the x direction, we have TAB acting towards the left. There are no other forces in the x-direction, so the equation becomes:
TAB - WA = 0
2. Y direction force equation of equilibrium:
In the y direction, we have TAB, TAC, and TAD acting upwards, and WA acting downwards. The equation becomes:
TAB + TAC + TAD - WA = 0
3. Z direction force equation of equilibrium:
In the z direction, we have TAB, TAC, and TAD acting upwards, and WA acting downwards. The equation becomes:
TAB + TAC + TAD - WA = 0
Now let's solve for TAB and TAC:
4. Tensile force along AB, TAB:
From the x-direction equilibrium equation (TAB - WA = 0), we can solve for TAB:
TAB = WA
5. Tensile force along AC, TAC:
From the y-direction equilibrium equation (TAB + TAC + TAD - WA = 0), we can isolate TAC:
TAC = -TAB - TAD + WA
Note: The negative sign is included because TAC is acting in the opposite direction (downwards) compared to TAB and TAD.
Therefore, the tensile force along AB (TAB) is equal to the weight of the cylinder (WA), and the tensile force along AC (TAC) is equal to -TAB - TAD + WA.
Please let me know if I can help you with anything else.