Four forces are acting on an object. Force 1 has a magnitude of 5N at an angle of 70° above the +x-axis; force 2 has a magnitude of 8N at an angle of 35° above the +x-axis; force 3 has a magnitude of 4N at an angle of 20° above the -x-axis; force 4 has a magnitude of 7N at an angle of 25° to the left of the -y-axis. Find the magnitude and direction of the resultant of these four forces. Find the resultant force, direction angle (in degrees), and direction.
Fr = 5N[70o] + 8[35] + 4[20] + 7[245]
X=5*Cos70 + 8*Cos35 + 4*Cos20 + 7*Cos245
= 9.1 N.
Y=5*sin70 + 8*sin35 + 4*sin20 +7*sin245
= 4.31 N.
Tan A = Y/X = 4.31/9.1 = 0.47583
A = 25.4o CCW = Direction.
Fr = X/Cos A = 9.1/Cos25.4 = 10.0 N. @
25.4o = Resultant force.
£fy=4•31100N
£fx=9.063N
Direction =24 degrees
To find the magnitude and direction of the resultant force, you need to combine all four forces using vector addition. Here's how you can do it step by step:
Step 1: Resolve each force into its x and y components.
- Force 1: This force has a magnitude of 5N at an angle of 70° above the +x-axis.
- x-component: 5N * cos(70°)
- y-component: 5N * sin(70°)
- Force 2: This force has a magnitude of 8N at an angle of 35° above the +x-axis.
- x-component: 8N * cos(35°)
- y-component: 8N * sin(35°)
- Force 3: This force has a magnitude of 4N at an angle of 20° above the -x-axis.
- x-component: 4N * cos(180° - 20°)
- y-component: 4N * sin(180° - 20°)
- Force 4: This force has a magnitude of 7N at an angle of 25° to the left of the -y-axis.
- x-component: 7N * cos(90° - 25°)
- y-component: -7N * sin(90° - 25°) [negative sign because it is downward]
Step 2: Add up the x and y components separately.
- x-component total: x-component of Force 1 + x-component of Force 2 + x-component of Force 3 + x-component of Force 4
- y-component total: y-component of Force 1 + y-component of Force 2 + y-component of Force 3 + y-component of Force 4
Step 3: Find the magnitude and direction angle of the resultant force.
- Magnitude: √(x-component total)^2 + (y-component total)^2
- Direction angle: arctan((y-component total)/(x-component total))
Step 4: Determine the direction of the resultant force based on the direction angle.
The direction angle calculated above will give you the angle of the resultant force with respect to the positive x-axis. To determine the direction in terms of compass headings (e.g., north, south, east, west), you can use the standard compass angles:
- If the direction angle is between 0° and 90°, the resultant force is in the northeast quadrant.
- If the direction angle is between 90° and 180°, the resultant force is in the northwest quadrant.
- If the direction angle is between -90° and 0°, the resultant force is in the southeast quadrant.
- If the direction angle is between -180° and -90°, the resultant force is in the southwest quadrant.
Keep in mind that the angles given in the problem statement might need to be converted to the correct range based on the quadrant reference.