Find the magnitude of the torque about P if an F = 28-lb force is applied as shown. (Round your answer to the nearest whole number.)

A rigid body is given. Starting at the point P, the rigid body goes horizontally right 4 feet, makes a 90° turn, and goes down vertically 4 feet. A vector labeled F begins at the bottom of the rigid body and goes down and left at an angle of 30° from the horizontal.

To calculate the magnitude of the torque (\( \tau \)) about point \( P \), we can use the formula:

\[
\tau = r \cdot F \cdot \sin(\theta)
\]

Where:
- \( r \) is the distance from the point \( P \) to the point where the force \( F \) is applied (the length of the rigid body),
- \( F \) is the magnitude of the force,
- \( \theta \) is the angle between the position vector \( r \) and the force vector \( F \).

### Step-by-Step Process:

1. **Determine the distance \( r \):**  
   The rigid body is composed of two perpendicular segments, each 4 feet in length. To find the distance from \( P \) to the point where the force is applied, we can use the Pythagorean theorem:

   \[
   r = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66 \, \text{feet}
   \]

2. **Identify the force \( F \):**  
   The force applied is given as \( F = 28 \, \text{lb} \).

3. **Determine the angle \( \theta \):**  
   The force vector is directed 30° from the horizontal, but the angle \( \theta \) for the torque calculation is the angle between the position vector and the force vector. Since the position vector is diagonal, making a 45° angle with the horizontal, the angle between the position vector and the force vector is:

   \[
   \theta = 45^\circ + 30^\circ = 75^\circ
   \]

4. **Calculate the torque:**  
   Now, substitute the values into the torque formula:

   \[
   \tau = r \cdot F \cdot \sin(\theta) = 5.66 \cdot 28 \cdot \sin(75^\circ)
   \]

   Using \( \sin(75^\circ) \approx 0.9659 \):

   \[
   \tau = 5.66 \cdot 28 \cdot 0.9659 \approx 153.1 \, \text{lb-ft}
   \]

### Final Answer:
The magnitude of the torque about point \( P \) is approximately **153 lb-ft** (rounded to the nearest whole number).

Would you like further details or clarifications?

Here are 5 related questions for further exploration:
1. How does the angle between the force and the position vector affect the torque?
2. What happens to the torque if the force is applied at a different angle?
3. How would the torque change if the force were doubled?
4. How does the length of the rigid body influence the torque?
5. What is the difference between torque and moment?

**Tip:** When calculating torque, always ensure the angle used is the one between the position vector and the force vector, not just the angle from the horizontal.

Find the magnitude of the torque about P if an F = 28-lb force is applied as shown. (Round your answer to the nearest whole number.)

Learn how to calculate the magnitude of the torque when a 28-lb force is applied to a rigid body at a 30° angle from the horizontal. This detailed solution covers key physics concepts such as torque, trigonometry, and the Pythagorean theorem. Suitable for high school students in Grades 10-12.

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