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 using both the trigonometric and analytical methods, we need to follow a step-by-step approach. We'll first break down the forces into their components and then find the resultant.

### Given:
- **Force \( F \) = 450 lb** at **45 degrees** (North-West)
- **Force \( Q \) = 200 lb** at **30 degrees** (North-East)

### 1. Trigonometric Method (Graphical Addition):

This method involves adding the vectors by using the law of cosines and the law of sines.

#### Step 1: Determine the angle between the forces
- The angle between the vectors \( F \) and \( Q \) is \( 45^\circ + 30^\circ = 75^\circ \).

#### Step 2: Use the Law of Cosines to find the magnitude of the resultant \( R \):

\[
R = \sqrt{F^2 + Q^2 - 2 \cdot F \cdot Q \cdot \cos(\theta)}
\]

Substituting the given values:

\[
R = \sqrt{(450)^2 + (200)^2 - 2 \cdot 450 \cdot 200 \cdot \cos(75^\circ)}
\]

Let's compute this:

\[
R = \sqrt{202500 + 40000 - 180000 \cdot 0.2588}
\]
\[
R = \sqrt{202500 + 40000 - 46602}
\]
\[
R = \sqrt{195898} \approx 442.53 \, \text{lb}
\]

#### Step 3: Use the Law of Sines to find the direction of the resultant:

\[
\frac{\sin(\alpha)}{Q} = \frac{\sin(\theta)}{R}
\]

\[
\sin(\alpha) = \frac{Q \cdot \sin(\theta)}{R} = \frac{200 \cdot \sin(75^\circ)}{442.53}
\]

\[
\alpha = \arcsin\left(\frac{200 \cdot 0.9659}{442.53}\right) \approx \arcsin(0.4361) \approx 25.8^\circ
\]

So the angle of the resultant \( R \) with respect to the force \( F \) is \( 25.8^\circ \) north of west.

### 2. Analytical Method (Component Method):

This method involves resolving each force into its horizontal and vertical components and then summing these components to find the resultant vector.

#### Step 1: Resolve the forces into components.

For force \( F = 450 \, \text{lb} \) at \( 45^\circ \) (North-West):
- \( F_x = 450 \cos(135^\circ) = 450(-0.7071) = -318.2 \, \text{lb} \) (West)
- \( F_y = 450 \sin(135^\circ) = 450(0.7071) = 318.2 \, \text{lb} \) (North)

For force \( Q = 200 \, \text{lb} \) at \( 30^\circ \) (North-East):
- \( Q_x = 200 \cos(30^\circ) = 200(0.866) = 173.2 \, \text{lb} \) (East)
- \( Q_y = 200 \sin(30^\circ) = 200(0.5) = 100 \, \text{lb} \) (North)

#### Step 2: Sum the components:

- **Resultant in \( x \)-direction \( R_x \)**:
\[
R_x = F_x + Q_x = -318.2 + 173.2 = -145 \, \text{lb} \quad (\text{West})
\]

- **Resultant in \( y \)-direction \( R_y \)**:
\[
R_y = F_y + Q_y = 318.2 + 100 = 418.2 \, \text{lb} \quad (\text{North})
\]

#### Step 3: Compute the magnitude of the resultant \( R \):

\[
R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-145)^2 + (418.2)^2}
\]
\[
R = \sqrt{21025 + 174903.24} = \sqrt{195928.24} \approx 442.52 \, \text{lb}
\]

#### Step 4: Find the direction of the resultant \( R \):

\[
\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{418.2}{-145}\right) \approx \tan^{-1}(-2.883)
\]

\[
\theta_R \approx -70^\circ \quad (\text{measured clockwise from East})
\]

Since \( R_x \) is negative and \( R_y \) is positive, the resultant lies in the North-West quadrant.

So the direction is \( 70^\circ \) North of West.

### Summary:
- **Magnitude of the resultant force**: \( \approx 442.5 \, \text{lb} \)
- **Direction**: \( 70^\circ \) North of West

Would you like to go into more detail or have any questions? Here are some related problems:

1. What would happen if the forces had the same magnitude but opposite directions?
2. How does the angle between the forces affect the resultant?
3. What if the second force was acting directly east? How would the solution change?
4. How can we verify the analytical method result using a different approach?
5. How would these calculations change if the forces were in different quadrants?

**Tip:** Always check which quadrant your resultant force lies in when determining the angle to ensure the direction is correct.

Learn how to compute the resultant force of vectors using both trigonometric and analytical methods. This problem involves applying the Law of Cosines and Law of Sines to find the magnitude and direction of the resultant force. Suitable for advanced high school students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation