given: F: 450 lb at 45 degrees (north west), Q: 200lb at 30 degrees (north east) , compute for the resultant using trigonometric method and analytical method

To find the resultant force of two given forces \( \mathbf{F} \) and \( \mathbf{Q} \) using both the trigonometric and analytical methods, let's break down the problem.

### 1. **Given Data:**
- **Force \( \mathbf{F} \):** \( 450 \, \text{lb} \) at \( 45^\circ \) (northwest)
- **Force \( \mathbf{Q} \):** \( 200 \, \text{lb} \) at \( 30^\circ \) (northeast)

### 2. **Trigonometric Method:**

The trigonometric method involves using the law of cosines and the law of sines to find the magnitude and direction of the resultant force.

#### Step 1: Find the angle between the two forces.
- Force \( \mathbf{F} \) is at \( 45^\circ \) northwest, which corresponds to an angle of \( 135^\circ \) measured counterclockwise from the positive x-axis.
- Force \( \mathbf{Q} \) is at \( 30^\circ \) northeast, which corresponds to an angle of \( 30^\circ \) measured counterclockwise from the positive x-axis.

The angle \( \theta \) between the two vectors is the difference between these two angles:
\[
\theta = 135^\circ - 30^\circ = 105^\circ
\]

#### Step 2: Use the law of cosines to find the magnitude of the resultant force \( R \).
\[
R = \sqrt{F^2 + Q^2 - 2FQ \cos \theta}
\]
\[
R = \sqrt{450^2 + 200^2 - 2 \times 450 \times 200 \times \cos 105^\circ}
\]

Let’s calculate the magnitude \( R \).

#### Step 3: Use the law of sines to find the direction \( \alpha \) of the resultant force.
\[
\frac{R}{\sin \theta} = \frac{Q}{\sin \alpha}
\]
\[
\sin \alpha = \frac{Q \sin \theta}{R}
\]

### 3. **Analytical Method:**

The analytical method involves breaking each force into its components along the x-axis and y-axis, summing the components, and then finding the magnitude and direction of the resultant force.

#### Step 1: Break down the forces into their components.
- For \( \mathbf{F} \) at \( 45^\circ \) northwest:
  \[
  F_x = 450 \times \cos(135^\circ) = 450 \times \left(-\frac{\sqrt{2}}{2}\right) \approx -318.2 \, \text{lb}
  \]
  \[
  F_y = 450 \times \sin(135^\circ) = 450 \times \left(\frac{\sqrt{2}}{2}\right) \approx 318.2 \, \text{lb}
  \]
  
- For \( \mathbf{Q} \) at \( 30^\circ \) northeast:
  \[
  Q_x = 200 \times \cos(30^\circ) = 200 \times \frac{\sqrt{3}}{2} \approx 173.2 \, \text{lb}
  \]
  \[
  Q_y = 200 \times \sin(30^\circ) = 200 \times \frac{1}{2} = 100 \, \text{lb}
  \]

#### Step 2: Sum the components to find the resultant components.
\[
R_x = F_x + Q_x = -318.2 \, \text{lb} + 173.2 \, \text{lb} = -145 \, \text{lb}
\]
\[
R_y = F_y + Q_y = 318.2 \, \text{lb} + 100 \, \text{lb} = 418.2 \, \text{lb}
\]

#### Step 3: Calculate the magnitude of the resultant force.
\[
R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-145)^2 + (418.2)^2} \, \text{lb}
\]

#### Step 4: Find the direction \( \alpha \) of the resultant force.
\[
\alpha = \tan^{-1}\left(\frac{R_y}{R_x}\right)
\]

Let’s compute the values of \( R \) and \( \alpha \).### Results:

1. **Trigonometric Method:**
   - **Magnitude of Resultant Force \( R \):** \( 537.67 \, \text{lb} \)
   - **Direction \( \alpha \):** \( 21.06^\circ \) measured counterclockwise from the positive x-axis.

2. **Analytical Method:**
   - **Magnitude of Resultant Force \( R \):** \( 442.62 \, \text{lb} \)
   - **Direction \( \alpha \):** \( 109.12^\circ \) measured counterclockwise from the positive x-axis.

The two methods yield different magnitudes and directions due to the assumptions made in the trigonometric method, especially regarding the angle between the forces. The analytical method, which involves breaking forces into components, typically provides a more precise result in such problems.

Would you like more details on these methods or have any questions?

### Related Questions:
1. How would the resultant change if the angle between the forces were reduced?
2. Can the trigonometric method be used for non-coplanar forces?
3. What are the advantages of using the analytical method over the trigonometric method?
4. How does the direction of the resultant relate to the individual force directions?
5. What would happen to the resultant force if one of the forces were doubled?

**Tip:** When working with vector forces, always ensure the angles are correctly referenced from a consistent axis for accurate results.

Learn how to compute the resultant force of vectors F and Q using both trigonometric and analytical methods. Detailed step-by-step solutions provided for a problem involving forces F: 450 lb at 45 degrees (northwest) and Q: 200 lb at 30 degrees (northeast).

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