Consider four vectors F1, F2, F3, and F4, with magnitudes are F1 = 41 N, F2 = 24 N, F3 =27 N, and F4 = 51 N, and angle 1 = 150 deg, angle 2 = −160 deg, angle 3 = 39 deg, and angle 4 = −57 deg, measured from the positive x axis with the counter clockwise angular direction as positive.
What is the magnitude of the resultant vector F, where F = F1 + F2 + F3 + F4?
What is the meaning of interior
To find the magnitude of the resultant vector F, we need to add the individual vectors F1, F2, F3, and F4.
First, we'll break each vector into its x and y components using the given magnitudes and angles.
For F1:
Magnitude (F1) = 41 N
Angle (1) = 150 degrees
To find the x-component (F1x) of F1, we can use the formula:
F1x = F1 * cos(angle 1)
F1x = 41 N * cos(150 degrees)
Using the trigonometric identity cos(150 degrees) = -cos(180 degrees - 150 degrees),
F1x = 41 N * -cos(30 degrees)
Simplifying further,
F1x = -41 N * √3/2
F1x ≈ -35.498 N
To find the y-component (F1y) of F1, we use the formula:
F1y = F1 * sin(angle 1)
F1y = 41 N * sin(150 degrees)
F1y ≈ 20.996 N
Repeat this process for the remaining vectors:
F2:
Magnitude (F2) = 24 N
Angle (2) = -160 degrees
F2x = F2 * cos(angle 2)
F2x = 24 N * cos(-160 degrees)
Using the trigonometric identity cos(-160 degrees) = cos(180 degrees + 160 degrees),
F2x = 24 N * cos(20 degrees)
F2x ≈ 22.076 N
F2y = F2 * sin(angle 2)
F2y = 24 N * sin(-160 degrees)
F2y ≈ -6.487 N
F3:
Magnitude (F3) = 27 N
Angle (3) = 39 degrees
F3x = F3 * cos(angle 3)
F3x = 27 N * cos(39 degrees)
F3x ≈ 20.671 N
F3y = F3 * sin(angle 3)
F3y = 27 N * sin(39 degrees)
F3y ≈ 16.927 N
F4:
Magnitude (F4) = 51 N
Angle (4) = -57 degrees
F4x = F4 * cos(angle 4)
F4x = 51 N * cos(-57 degrees)
F4x ≈ 25.743 N
F4y = F4 * sin(angle 4)
F4y = 51 N * sin(-57 degrees)
F4y ≈ -42.602 N
Now, we can add the x and y components of each vector to find the resultant vector components:
F_total_x = F1x + F2x + F3x + F4x
F_total_y = F1y + F2y + F3y + F4y
F_total_x ≈ -35.498 N + 22.076 N + 20.671 N + 25.743 N
F_total_x ≈ 32.992 N
F_total_y ≈ 20.996 N + (-6.487 N) + 16.927 N + (-42.602 N)
F_total_y ≈ -11.166 N
Now, we can find the magnitude of the resultant vector F using the Pythagorean theorem:
|F| = sqrt(F_total_x^2 + F_total_y^2)
|F| = sqrt((32.992 N)^2 + (-11.166 N)^2)
|F| ≈ sqrt(1089.404 N^2 + 124.563 N^2)
|F| ≈ sqrt(1213.967 N^2)
|F| ≈ 34.834 N
Therefore, the magnitude of the resultant vector F is approximately 34.834 N.