Determine the maximum weight of the crate that can be suspended from cables AB, AC, and AD so that the tension developed in any one of the cables does not exceed 270 lb
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To determine the maximum weight of the crate that can be suspended from cables AB, AC, and AD so that the tension developed in any one of the cables does not exceed 270 lb, we need to analyze the forces acting on the crate.
Looking at the figure, we can see that the crate is in equilibrium, meaning that the sum of the forces acting on it is equal to zero in both the horizontal and vertical directions.
Let's break down the forces acting on the crate:
1. Tension in cable AB (TAB): This force is acting upwards and to the left.
2. Tension in cable AC (TAC): This force is acting upwards and to the right.
3. Tension in cable AD (TAD): This force is acting directly upwards.
4. Weight of the crate (W): This force is acting downward.
Since the crate is in equilibrium, the sum of the forces in the vertical direction must be zero. This means that the upward forces (TAB, TAC, and TAD) must balance out the downward force (W).
So we can write the following equation:
TAB + TAC + TAD = W
We know that we want to find the maximum weight of the crate, which means we want to maximize W. To do this, we need to determine the maximum tension that any one of the cables can handle without exceeding 270 lb.
Let's assume that TAB is the maximum tension and set it equal to 270 lb. We can then solve for W:
270 + TAC + TAD = W
Now we need to find the maximum tension for the other two cables, TAC and TAD.
Looking at the figure, we can see that TAC and TAD form a right triangle with the vertical and horizontal components of TAB. Using trigonometry, we can find the values of TAC and TAD in terms of TAB.
TAC = TAB * cos(30°)
TAD = TAB * sin(30°)
Substituting these values into the equation, we get:
270 + TAB * cos(30°) + TAB * sin(30°) = W
Now we can solve for W. We have the maximum tension in one of the cables, TAB = 270 lb, and we can calculate the values of cos(30°) and sin(30°). Once we solve for W, we will have the maximum weight of the crate that can be suspended from cables AB, AC, and AD without exceeding the tension limit of 270 lb.