Consider three force vectors F1 with magnitude 42N and direction 140°, F2 with magnitude 27N and direction -110°, and F3 with magnitude 19N and direction 120°. All direction angles 0 (0 with a line inside) are measured from the positive x axis: counter-clockwise for 0 (with a line inside)>0 and clockwise for 0 (with a line inside)< 0. What is the magnitude F of the net force vector F=F1+F2+F3? Answer in units of °
To find the magnitude of the net force vector, we need to combine the given force vectors F1, F2, and F3 using vector addition.
Step 1: Resolve each force vector into its x and y components.
F1:
Magnitude = 42 N
Direction = 140°
To find the x-component of F1: F1x = F1 * cos(angle)
F1x = 42 N * cos(140°) = -20.458 N (rounded to 3 decimal places)
To find the y-component of F1: F1y = F1 * sin(angle)
F1y = 42 N * sin(140°) = 36.27 N (rounded to 2 decimal places)
F2:
Magnitude = 27 N
Direction = -110°
To find the x-component of F2: F2x = F2 * cos(angle)
F2x = 27 N * cos(-110°) = 19.20 N (rounded to 2 decimal places)
To find the y-component of F2: F2y = F2 * sin(angle)
F2y = 27 N * sin(-110°) = -22.02 N (rounded to 2 decimal places)
F3:
Magnitude = 19 N
Direction = 120°
To find the x-component of F3: F3x = F3 * cos(angle)
F3x = 19 N * cos(120°) = -9.5 N
To find the y-component of F3: F3y = F3 * sin(angle)
F3y = 19 N * sin(120°) = 16.47 N (rounded to 2 decimal places)
Step 2: Add the x and y components of the forces to get the resultant vector.
sum of x-components = F1x + F2x + F3x
sum of x-components = -20.458 N + 19.20 N - 9.5 N = -10.758 N (rounded to 3 decimal places)
sum of y-components = F1y + F2y + F3y
sum of y-components = 36.27 N - 22.02 N + 16.47 N = 30.72 N (rounded to 2 decimal places)
Step 3: Find the magnitude and direction of the resultant vector using the Pythagorean theorem and inverse tangent.
Magnitude of the resultant vector F (net force): F = sqrt((sum of x-components)^2 + (sum of y-components)^2)
F = sqrt((-10.758 N)^2 + (30.72 N)^2) ≈ 32.250 N (rounded to 3 decimal places)
Direction (angle) of the resultant vector: F_angle = atan2(sum of y-components, sum of x-components)
F_angle = atan2(30.72 N, -10.758 N)
Now we can calculate F_angle using a calculator or an online tool. The result is approximately 107.238° (rounded to 3 decimal places).
Therefore, the magnitude F of the net force vector F = F1 + F2 + F3 is approximately 32.250 N, and the direction of F is approximately 107.238°.