This problem involves the equilibrium of forces acting on a boom supported by a tension cable and a pivot at the bottom. To solve for the tension $T$ in the cable and the force $F$ at the pivot, we can apply principles from statics, particularly considering both force equilibrium and moment equilibrium.

Step 1: Force Equilibrium

We begin by breaking the forces into their horizontal and vertical components:

• The tension $T$ makes a 30° angle with the horizontal. Its components are:

  • $T_x = T \cos(30^\circ)$ (horizontal)
  • $T_y = T \sin(30^\circ)$ (vertical)

• The force $F$ at the pivot has both horizontal and vertical components:

  • $F_x$ (horizontal)
  • $F_y$ (vertical)

• The weight of the load is acting vertically downward with a force of 40 lb.

Step 2: Moment Equilibrium

To satisfy rotational equilibrium (assuming the pivot at the bottom does not create any moment), we sum the moments about the pivot point. The moment created by the weight and the tension in the cable must balance.

The moment arm for the weight (40 lb) is the entire length of the boom, which acts at a distance of the horizontal component of the boom, creating a moment:

$\text{Moment of weight} = 40 \, \text{lb} \times L$

The moment arm for the tension $T$ is the vertical distance from the pivot to the point where the cable connects, and this tension creates a counteracting moment.

Step 3: Equations of Equilibrium

• Horizontal force equilibrium: $F_x = T \cos(30^\circ)$

• Vertical force equilibrium: $F_y + T \sin(30^\circ) = 40 \, \text{lb}$

• Moment equilibrium: The total moment around the pivot should sum to zero. Using the geometry of the situation, we can relate the forces to the lengths and angles provided.

Solution

Let me compute the values for $T$ and $F$ using these equations.The calculations give us the following results:

• The tension $T$ in the cable is approximately 80.00 lb.
• The horizontal component of the force at the pivot $F_x$ is approximately 69.28 lb.
• The vertical component of the force at the pivot $F_y$ is 0 lb, meaning the entire vertical force is balanced by the tension in the cable.

Would you like further details, or do you have any other questions?

Here are five related questions to extend your understanding:

1. How would the solution change if the angle of the cable was different?
2. What happens if the weight is increased from 40 lb to 50 lb?
3. How do you calculate the total force at the pivot point, considering both components $F_x$ and $F_y$?
4. Can you explain how moments are used to ensure rotational equilibrium in this problem?
5. What assumptions are made when considering the boom to have negligible weight?

Tip: Always remember to break forces into their horizontal and vertical components when solving equilibrium problems.