To find the sum of the four vector forces you've described, we can break each force down into its horizontal (x) and vertical (y) components. This method, known as the component method, allows us to add vectors more easily. Let’s go through the process step by step.
Step 1: Define the Axes
First, we need to establish a coordinate system. We'll use the standard Cartesian coordinates where the right direction is positive x, and the upward direction is positive y. This will help us keep track of the components of each force.
Step 2: Break Down Each Force into Components
Now, let's analyze each force and calculate its components:
- Force 1: 10.5 N at 35.0° above the horizontal
- Fx1 = 10.5 N * cos(35°)
- Fy1 = 10.5 N * sin(35°)
- Force 2: 31.0 N to the left at 54.5° above the horizontal
- Fx2 = -31.0 N * cos(54.5°) (negative because it's to the left)
- Fy2 = 31.0 N * sin(54.5°)
- Force 3: 8.60 N to the left at 35.0° below the horizontal
- Fx3 = -8.60 N * cos(35°) (negative because it's to the left)
- Fy3 = -8.60 N * sin(35°) (negative because it's below the horizontal)
- Force 4: 24.0 N to the right at 50.0° below the horizontal
- Fx4 = 24.0 N * cos(50°)
- Fy4 = -24.0 N * sin(50°) (negative because it's below the horizontal)
Step 3: Calculate the Components
Now, let's compute the values of these components using a calculator:
- For Force 1:
- Fx1 = 10.5 * cos(35°) ≈ 8.59 N
- Fy1 = 10.5 * sin(35°) ≈ 5.99 N
- For Force 2:
- Fx2 = -31.0 * cos(54.5°) ≈ -17.36 N
- Fy2 = 31.0 * sin(54.5°) ≈ 25.40 N
- For Force 3:
- Fx3 = -8.60 * cos(35°) ≈ -7.05 N
- Fy3 = -8.60 * sin(35°) ≈ -4.93 N
- For Force 4:
- Fx4 = 24.0 * cos(50°) ≈ 15.43 N
- Fy4 = -24.0 * sin(50°) ≈ -18.29 N
Step 4: Sum the Components
Next, we add up all the x-components and y-components separately:
- Total x-component (Fx):
- Fx = Fx1 + Fx2 + Fx3 + Fx4
- Fx = 8.59 - 17.36 - 7.05 + 15.43 ≈ -0.39 N
- Total y-component (Fy):
- Fy = Fy1 + Fy2 + Fy3 + Fy4
- Fy = 5.99 + 25.40 - 4.93 - 18.29 ≈ 8.17 N
Step 5: Resultant Force Calculation
Now that we have the total components, we can find the magnitude and direction of the resultant force using the Pythagorean theorem and trigonometry:
- Magnitude:
- R = √(Fx² + Fy²)
- R = √((-0.39)² + (8.17)²) ≈ 8.17 N
- Direction:
- θ = tan⁻¹(Fy / Fx)
- θ = tan⁻¹(8.17 / -0.39) ≈ 270° + 1.76° (since Fx is negative)
- θ ≈ 271.76° from the positive x-axis.
Final Result
The resultant force is approximately 8.17 N directed at an angle of about 271.76° from the positive x-axis. This means it points slightly below the negative x-axis, indicating a downward direction with a slight leftward component. This comprehensive breakdown illustrates how to effectively sum vector forces using the component method.
