get the resultant of the following displacements 6 ft.southwest; 10 ft. northwest; 4 ft. southeast; 12 ft. northwest; and 3 ft. southwest. what is the direction of the resultant displacement?
All angles are measured CCW from +x-axis.
Fr = 6ft[225o] + 10[135] + 4[315] + 3[225].
Fr = (-4.24-4.24i)+(-7.07+7.07i)+(2.83-2.83i)+(-2.12-2.12i),
Add all hor. components. Then add all Ver. components:
Fr = -10.6 - 2.12i = 10.8ft[11.31o} S. of W. = 10.8ft[191.31o].
To find the resultant displacement, we can add up all the displacements using vector addition. Let's break down the given displacements into their components.
The displacement 6 ft. southwest can be divided into a horizontal component and a vertical component. Since it is southwest, which is halfway between south and west, the horizontal component will be 6 ft. multiplied by cos(45°) and the vertical component will be 6 ft. multiplied by sin(45°).
Horizontal component = 6 ft. × cos(45°) = 6 ft. × √(2)/2 = 3√2 ft.
Vertical component = 6 ft. × sin(45°) = 6 ft. × √(2)/2 = 3√2 ft.
Similarly, we can calculate the horizontal and vertical components for the other displacements:
10 ft. northwest:
Horizontal component = 10 ft. × cos(45°) = 10 ft. × √(2)/2 = 5√2 ft.
Vertical component = 10 ft. × sin(45°) = 10 ft. × √(2)/2 = 5√2 ft.
4 ft. southeast:
Horizontal component = 4 ft. × cos(45°) = 4 ft. × √(2)/2 = 2√2 ft.
Vertical component = 4 ft. × sin(45°) = 4 ft. × √(2)/2 = 2√2 ft.
12 ft. northwest:
Horizontal component = 12 ft. × cos(45°) = 12 ft. × √(2)/2 = 6√2 ft.
Vertical component = 12 ft. × sin(45°) = 12 ft. × √(2)/2 = 6√2 ft.
3 ft. southwest:
Horizontal component = 3 ft. × cos(45°) = 3 ft. × √(2)/2 = 3/√2 ft.
Vertical component = 3 ft. × sin(45°) = 3 ft. × √(2)/2 = 3/√2 ft.
Now, we can add up the horizontal components and the vertical components separately:
Horizontal displacement = (3√2 + 5√2 + 2√2 + 6√2 + 3/√2) ft.
Vertical displacement = (3√2 + 5√2 + 2√2 + 6√2 + 3/√2) ft.
We can combine like terms to simplify:
Horizontal displacement = (16√2 + 3/√2) ft.
Vertical displacement = (16√2 + 3/√2) ft.
Finally, we can find the magnitude and direction of the resultant displacement using the Pythagorean theorem and trigonometry:
Magnitude of the resultant displacement = √[(horizontal displacement)^2 + (vertical displacement)^2]
Magnitude of the resultant displacement = √[(16√2 + 3/√2)^2 + (16√2 + 3/√2)^2]
To find the direction, we can use the inverse tangent function:
Direction of the resultant displacement = tan^(-1)((vertical displacement)/(horizontal displacement))
Please note that these calculations are quite complex and might give a long and approximate numerical answer.
To find the resultant of the displacements, we need to add all of the displacements together.
First, let's assign a coordinate system where north is positive Y and east is positive X.
The given displacements are:
- 6 ft. southwest
- 10 ft. northwest
- 4 ft. southeast
- 12 ft. northwest
- 3 ft. southwest
To add these vectors, we need to break them down into their X (east-west) and Y (north-south) components.
For the southwest direction, we will consider it as a combination of -45 degrees (south) and -90 degrees (west). Similarly, for the northwest direction, we will consider it as a combination of 45 degrees (north) and -90 degrees (west). Finally, for the southeast direction, we will consider it as a combination of -45 degrees (south) and 90 degrees (east).
Now we can calculate the X and Y components for each displacement:
Southwest (6 ft):
X component = 6 * cos(-45°) = 6 * √2 / 2 = 3√2 ft
Y component = 6 * sin(-45°) = -6 * √2 / 2 = -3√2 ft
Northwest (10 ft):
X component = 10 * cos(45°) = 10 * √2 / 2 = 5√2 ft
Y component = 10 * sin(45°) = 10 * √2 / 2 = 5√2 ft
Southeast (4 ft):
X component = 4 * cos(-45°) = 4 * √2 / 2 = 2√2 ft
Y component = 4 * sin(45°) = 4 * √2 / 2 = 2√2 ft
Northwest (12 ft):
X component = 12 * cos(45°) = 12 * √2 / 2 = 6√2 ft
Y component = 12 * sin(45°) = 12 * √2 / 2 = 6√2 ft
Southwest (3 ft):
X component = 3 * cos(-45°) = 3 * √2 / 2 = 3√2 / 2 ft
Y component = 3 * sin(-45°) = -3 * √2 / 2 = -3√2 / 2 ft
Now we can add the X and Y components together:
X Component Total: 3√2 + 5√2 + 2√2 + 6√2 + 3√2 / 2 = 19.5√2 ft
Y Component Total: -3√2 + 5√2 + 2√2 + 6√2 - 3√2 / 2 = 7√2 / 2 ft
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem:
Resultant magnitude = √(X Component Total^2 + Y Component Total^2)
= √((19.5√2)^2 + (7√2 / 2)^2)
= √(380.25 + 24.5)
= √(404.75)
≈ 20.12 ft
Now, let's calculate the direction of the resultant displacement using trigonometry:
Direction = atan(Y Component Total / X Component Total)
= atan((7√2 / 2) / 19.5√2)
= atan(0.179)
≈ 10.27°
The direction of the resultant displacement is approximately 10.27°.