Question

Find the magnitude and direction of 4 concurrent coplanar forces:
i) 10 N 32 degrees east of north
ii) 35 N 50 degrees east of north
iii) 25 N 72 degrees east of north
iv) 85 N 40 degrees east of south:

Answer 1

Take east as +x and north as +y.

Components:
- F1 = 10 N, 32° east of north: Fx1 = 10 sin32° = 5.299 N, Fy1 = 10 cos32° = 8.480 N
- F2 = 35 N, 50° east of north: Fx2 = 35 sin50° = 26.812 N, Fy2 = 35 cos50° = 22.498 N
- F3 = 25 N, 72° east of north: Fx3 = 25 sin72° = 23.776 N, Fy3 = 25 cos72° = 7.725 N
- F4 = 85 N, 40° east of south: Fx4 = 85 sin40° = 54.636 N, Fy4 = −85 cos40° = −65.114 N

Sum components:
Fx = 5.299 + 26.812 + 23.776 + 54.636 = 110.524 N
Fy = 8.480 + 22.498 + 7.725 − 65.114 = −26.410 N

Resultant magnitude:
R = sqrt(Fx^2 + Fy^2) ≈ sqrt(110.524^2 + (−26.410)^2) ≈ 113.6 N

Direction:
angle = arctan(|Fy|/Fx) = arctan(26.410/110.524) ≈ 13.5°
So the resultant is 13.5° south of east (equivalently 76.5° east of south).

Answer: R ≈ 113.6 N, directed 13.5° south of east.