Two forces working on an object with mass m are F1 = 34 N 25° North of West and F2 = 37 N East. By using the perpendicular component method, calculate the resultant of these two forces and express the answer in unit vector notation
First, we need to resolve F1 into its perpendicular components, one pointing directly North and the other pointing directly West:
F1perp = F1 * cos(25°) = 30.98 N
F1west = F1 * sin(25°) = 14.36 N (to the left)
Now we can add up the forces in the x and y directions separately:
Fx = F1west + F2 = 14.36 N - 37 N = -22.64 N (to the left)
Fy = F1perp = 30.98 N (upwards)
The resultant force is the vector sum of these two forces:
F = √(Fx^2 + Fy^2) = √((-22.64 N)^2 + (30.98 N)^2) = 37.86 N
To express this answer in unit vector notation, we need to find the direction of the resultant force. We can use the inverse tangent function to find the angle between the resultant force vector and the positive x-axis:
θ = tan^-1(Fy/Fx) = tan^-1(-30.98 N/-22.64 N) = 53.22° (measured counterclockwise from East)
Finally, we can express the resultant force in unit vector notation:
F = 37.86 N at 53.22° North of East
Or, in component form:
F = (37.86 N) * cos(53.22°) i + (37.86 N) * sin(53.22°) j
= 23.63 i + 30.09 j