Three forces act on an object at the same time. F1 = 100. N at 30.0degrees north of east, F2 = 200. N at 45.0degrees north of west, and F3 = 100. N at 30.0degrees east of south. What are the magnitudes and direction of both the resultant force and equilibrant force?
To find the magnitudes and directions of the resultant force and equilibrant force, we need to add up the given forces using vector addition.
Step 1: Convert all the given angles to a common reference direction. Let's use the east direction as the common reference direction.
Given:
- F1 = 100 N at 30.0 degrees north of east
- F2 = 200 N at 45.0 degrees north of west
- F3 = 100 N at 30.0 degrees east of south
Converting angles:
- F1 is 30.0 degrees north of east, which means it is 60.0 degrees south of north. (180 - 30 = 150 degrees)
- F2 is 45.0 degrees north of west, which means it is 45.0 degrees east of north. (180 + 45 = 225 degrees)
- F3 is 30.0 degrees east of south, which means it is 60.0 degrees south of east. (180 - 30 = 150 degrees)
So now we have the following angles:
- F1 = 100 N at 150.0 degrees
- F2 = 200 N at 225.0 degrees
- F3 = 100 N at 150.0 degrees
Step 2: Resolve the forces into their horizontal (x-component) and vertical (y-component) vectors.
For each force:
- Fx = magnitude * cos(angle)
- Fy = magnitude * sin(angle)
Calculating the x and y components:
- F1x = 100 N * cos(150.0 degrees)
- F1y = 100 N * sin(150.0 degrees)
- F2x = 200 N * cos(225.0 degrees)
- F2y = 200 N * sin(225.0 degrees)
- F3x = 100 N * cos(150.0 degrees)
- F3y = 100 N * sin(150.0 degrees)
Step 3: Add up the x-components and y-components of the forces separately to find the resultant force.
Resultant force:
- Rx = F1x + F2x + F3x
- Ry = F1y + F2y + F3y
Step 4: Calculate the magnitude of the resultant force using Pythagorean theorem.
Resultant force magnitude:
- R = √(Rx^2 + Ry^2)
Step 5: Calculate the direction of the resultant force using inverse tangent (tan^-1) function.
Resultant force direction:
- θ = tan^-1(Ry / Rx)
Now that we have the resultant force, we can find the equilibrant force by reversing the direction of the resultant force and keeping the magnitude the same.
Equilibrant force:
- Magnitude of equilibrant force = magnitude of resultant force
- Direction of equilibrant force = resultant force direction + 180 degrees
Finally, we can substitute the values and calculate the magnitudes and directions of both the resultant force and equilibrant force based on the given forces.