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Let's break down and solve this problem in parts.

### Question 4
A wrench is used to tighten bolts onto a plate, with the following details:

- The force applied at point \( B \) is \( 50 \, \text{lbs} \).
- The angle between the force and the wrench is \( 60^\circ \).
- The length from \( A \) to \( B \) is \( 10 \, \text{in} \), and the additional distance is \( 4 \, \text{in} \).

---

### Part a) Finding the Moment at point \( A \)
The moment \( M \) is given by the equation:

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

Where:
- \( F = 50 \, \text{lbs} \) is the force applied,
- \( r = 10 \, \text{in} \) is the distance from \( A \) to \( B \),
- \( \theta = 60^\circ \) is the angle between the force and the wrench.

Let’s calculate the moment.

### Part b) Maximum Moment Orientation
The moment is maximized when the angle between the force and the lever arm is \( 90^\circ \), as \( \sin(90^\circ) = 1 \). We’ll confirm this mathematically after solving part a.

### Part c) Other Changes to Increase Moment
There are two primary ways to increase the moment:
- Increase the length of the wrench (i.e., the distance \( r \)),
- Apply the force at a perpendicular angle to the wrench (i.e., at \( 90^\circ \)).

Let me calculate part (a) first.### Part a) Moment Calculation

The moment applied to the bolt at point \( A \) due to the force at point \( B \) is approximately \( 433.01 \, \text{in-lbs} \).

---

### Part b) Orientation for Maximum Moment
To maximize the moment, the force must be applied perpendicular to the wrench (i.e., at \( \theta = 90^\circ \)). In this case, the moment would be:

\[
M_{\text{max}} = F \cdot r \cdot \sin(90^\circ) = F \cdot r
\]

Let’s calculate that value for confirmation.### Part b) Maximum Moment

The maximum moment occurs when the force is applied at a \( 90^\circ \) angle, resulting in a moment of \( 500 \, \text{in-lbs} \).

---

### Part c) Other Ways to Increase the Moment
To further increase the moment:
1. **Increase the length of the wrench**: Increasing \( r \) will proportionally increase the moment.
2. **Apply the force more perpendicular**: Ensuring the force is applied at a \( 90^\circ \) angle maximizes the sine factor, achieving the highest possible moment for a given force and distance.

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Would you like further clarifications on any part? Here are some related questions:

1. What is the physical significance of the angle in moment calculations?
2. How does increasing the wrench length affect the torque applied?
3. Why does applying the force perpendicular maximize the moment?
4. Can you provide an example of a real-life scenario where torque optimization is critical?
5. How does moment relate to rotational equilibrium?

**Tip:** Always ensure the applied force is perpendicular to maximize the efficiency of torque in practical applications.

A wrench is used to tighten bolts onto a plate. Find the moment applied to the bolt at A due to the force at point B (50 lbs at an angle of 60°). What orientation would maximize the moment? What other change could increase the moment?

Learn how to calculate the moment applied to a bolt using a wrench with a 50 lbs force applied at a 60° angle. Understand the trigonometry involved and how to maximize the moment by adjusting the angle or distance. Suitable for high school and college-level physics.

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