4.5 Determination of a Force from its Perpendicular Components
The magnitude and direction of a force can be found if its perpendicular components are known.
Applying the Pythagorean theorem to the right-angled triangle OAC (Fig. 4.11-b):
(OC)² = (OA)² + (AC)²
F² = F_x² + F_y²
F = √(F_x² + F_y²) ............ (4.4)
Hence, using Eq. (4.4), the magnitude F of the required vector F can be determined. The direction of F is given by:
tan θ = F_y / F_x ............ (4.5)
θ = tan⁻¹ (F_y / F_x)
By using a table of trigonometric ratios or a calculator, the value of θ can be determined.
Example 4.3
A force of 160 N is acting on a wooden box at an angle of 60° with the horizontal direction. Determine the values of its x and y components.
Solution:
Magnitude of force F = 160 N Angle θ = 60°
Using calculator:
- sin 60° = 0.866
- cos 60° = 0.5
x-component is given by Eq. (4.2):
F_x = F cos θ
F_x = 160 N × 0.5 = 80 N
y-component is given by Eq. (4.3):
F_y = F sin θ
F_y = 160 N × 0.866 = 138.6 N