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In this problem, you're asked to determine the moment of a force with respect to a point \( P \). The force applied is \( 80 \, \text{lb} \), and the figure shows the geometry, which is important for finding the moment.

### Given Information:
- Force \( F = 80 \, \text{lb} \) applied at a 20° angle from the horizontal.
- A 3-ft distance from point \( P \) to the line of action of the force along a 40° inclined beam.
  
The **moment** \( M \) of a force about a point is calculated as:
\[
M = F \cdot d \cdot \sin(\theta)
\]
Where:
- \( F \) is the magnitude of the force,
- \( d \) is the perpendicular distance from the point \( P \) to the line of action of the force,
- \( \theta \) is the angle between the force direction and the position vector from point \( P \).

Let me break it down and calculate the moment step by step.

### Step 1: Finding the effective angle
- The force is applied at a 20° angle to the beam. The beam itself is inclined 40° to the horizontal. Therefore, the angle between the force and the position vector is:
\[
\theta = 40^\circ - 20^\circ = 20^\circ
\]

### Step 2: Calculate the perpendicular distance
We need the perpendicular distance from the point \( P \) to the line of action of the force. Since the beam is inclined, this distance \( d \) is affected by the geometry. The horizontal distance is given as 3 ft. The perpendicular distance can be calculated as:
\[
d = 3 \, \text{ft} \cdot \sin(40^\circ)
\]

### Step 3: Calculate the moment
Now we can use the moment equation to calculate:
\[
M = 80 \, \text{lb} \cdot d \cdot \sin(20^\circ)
\]

Let me compute this.The moment of the force with respect to point \( P \) is approximately \( 52.76 \, \text{lb-ft} \).

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

Here are some related questions to deepen your understanding:
1. How would the moment change if the angle of the beam were steeper, say 60°?
2. What would happen to the moment if the force were applied vertically instead of at a 20° angle?
3. Can you explain why we use \( \sin(\theta) \) in the moment calculation?
4. How does the perpendicular distance affect the overall moment produced by a force?
5. What if the force were applied in the opposite direction? How would that affect the moment?

**Tip:** When calculating moments, always ensure that you're using the perpendicular distance to the force's line of action. This ensures your calculations reflect the true rotational effect.

Con base en la figura P4.9, determine el momento de la fuerza de 80 lb respecto al punto P.

Learn how to calculate the moment of an 80 lb force applied at an angle with respect to a point. This problem involves key concepts from statics and trigonometry, including the use of the moment equation M = F * d * sin(θ) and finding the perpendicular distance.

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