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Let's solve the problem step by step. The force applied is \( F = 50 \, \text{lb} \), and we are asked to:

1. **Graph the relationship \( M_O \) vs. \( \theta \) (2.5 points)**
2. **Determine the value of \( \theta \) that maximizes \( M_O \) and the maximum moment value (1 point)**

### A. Graph the Relationship \( M_O \) vs. \( \theta \)

The moment \( M_O \) is generated by the force acting at a perpendicular distance from the point \( O \). The magnitude of the moment is given by:

\[
M_O = F \times d_{\perp}
\]

Where \( d_{\perp} \) is the perpendicular distance from point \( O \) to the line of action of the force. This can be computed as:

\[
d_{\perp} = 10 \, \text{in} \times \sin(\theta)
\]

Thus, the moment becomes:

\[
M_O = 50 \, \text{lb} \times (10 \, \text{in}) \times \sin(\theta)
\]

So the expression for \( M_O \) is:

\[
M_O = 500 \, \text{lb-in} \times \sin(\theta)
\]

### B. Determine \( \theta \) for Maximum Moment and the Maximum Value

The moment \( M_O \) is maximized when \( \sin(\theta) \) is maximized. Since \( \sin(\theta) \) reaches its maximum value of 1 when \( \theta = 90^\circ \), the maximum moment will occur at this angle. Therefore:

\[
M_O = 500 \, \text{lb-in} \times \sin(90^\circ) = 500 \, \text{lb-in}
\]

### Summary:
- The moment equation is \( M_O = 500 \sin(\theta) \).
- The maximum moment occurs at \( \theta = 90^\circ \), and its value is \( M_O = 500 \, \text{lb-in} \).

Would you like me to generate the graph for \( M_O \) vs. \( \theta \)? Let me know if you have any questions or need further clarification!

Here are some relative questions to expand the topic:
1. How does the distance from point \( O \) affect the moment generated by a force?
2. What role does the angle \( \theta \) play in determining the effectiveness of the force?
3. How would the moment change if the force applied were not perpendicular to the lever arm?
4. What happens to the moment if the length of the lever arm changes?
5. How can we apply the concept of moments to real-life mechanical systems?

**Tip:** The sine function governs the relationship between force direction and moment generation; for maximum efficiency, the force should be perpendicular to the lever arm.

La fuerza de 50 lb genera el momento Mo con respecto al punto O. Se pide: A. Grafique la relación Mo vs θ. B. Determine cual es el valor de θ que maximice Mo, y cuál es el valor máximo del momento.

Solve a mechanics problem involving a 50 lb force and the calculation of the moment about a point. Learn how to graph the relationship between Mo and θ, and determine the angle that maximizes the moment. Suitable for college-level engineering students.

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