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The cable and boom shown in the figure support a load of 600 lb. Determine the tensile force T in the cable and the compressive for C in the boom

This is a typical statics problem involving the equilibrium of forces. Here’s the step-by-step solution:

### Given:
- Load (\( W \)) = 600 lb
- The angles between the forces are:
  - \( 30^\circ \) between the cable and the horizontal (force \( T \)).
  - \( 45^\circ \) between the boom and the horizontal (force \( C \)).

### To solve:
We need to determine the tensile force (\( T \)) in the cable and the compressive force (\( C \)) in the boom. This can be done using the equilibrium conditions for forces:

1. **Sum of forces in the horizontal direction** (\( \Sigma F_x = 0 \)):
   \[
   T \cos(30^\circ) = C \cos(45^\circ)
   \]
   (For equilibrium in the horizontal direction, the horizontal components of the forces must cancel out.)

2. **Sum of forces in the vertical direction** (\( \Sigma F_y = 0 \)):
   \[
   T \sin(30^\circ) + C \sin(45^\circ) = W
   \]
   (The sum of the vertical components must balance the 600 lb weight.)

### Step 1: Solve for \( T \) and \( C \)

From the first equation (horizontal equilibrium):
\[
T \cos(30^\circ) = C \cos(45^\circ)
\]
\[
T \cdot \frac{\sqrt{3}}{2} = C \cdot \frac{\sqrt{2}}{2}
\]
\[
T \cdot \sqrt{3} = C \cdot \sqrt{2}
\]
Thus, we can express \( C \) in terms of \( T \):
\[
C = \frac{\sqrt{3}}{\sqrt{2}} \cdot T
\]

### Step 2: Substitute into the vertical equilibrium equation

Now substitute \( C \) into the vertical force equilibrium equation:
\[
T \sin(30^\circ) + \left(\frac{\sqrt{3}}{\sqrt{2}} T \right) \sin(45^\circ) = 600
\]
\[
T \cdot \frac{1}{2} + \frac{\sqrt{3}}{\sqrt{2}} T \cdot \frac{\sqrt{2}}{2} = 600
\]
Simplifying:
\[
T \cdot \frac{1}{2} + T \cdot \frac{\sqrt{3}}{2} = 600
\]
Factor out \( T \):
\[
T \left( \frac{1}{2} + \frac{\sqrt{3}}{2} \right) = 600
\]
Calculate the term inside the parentheses:
\[
T \left( \frac{1 + \sqrt{3}}{2} \right) = 600
\]
Now solve for \( T \):
\[
T = \frac{600 \times 2}{1 + \sqrt{3}} \approx 366.03 \, \text{lb}
\]

### Step 3: Find \( C \)

Using the expression for \( C \) from Step 1:
\[
C = \frac{\sqrt{3}}{\sqrt{2}} \cdot 366.03 \approx 448.43 \, \text{lb}
\]

### Final Answer:
- Tensile force in the cable \( T \approx 366.03 \, \text{lb} \)
- Compressive force in the boom \( C \approx 448.43 \, \text{lb} \)

Would you like further details on any of these steps or calculations?

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**Relative Questions:**
1. What are the conditions for equilibrium of forces in any system?
2. How do angles influence the tension and compression in structures?
3. Why is it important to resolve forces into components in statics problems?
4. How can you derive the tension in a cable if the load and angles are known?
5. What assumptions are made when solving equilibrium problems in two dimensions?

**Tip:** Always draw a free-body diagram to visualize the forces acting on a structure.

The cable and boom shown in the figure support a load of 600 lb. Determine the tensile force T in the cable and the compressive force C in the boom.

Learn how to determine the tensile force in a cable and the compressive force in a boom supporting a 600 lb load using equilibrium of forces and trigonometry. This solution involves solving simultaneous equations based on horizontal and vertical force components.

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