The following forces are acting at a point.
1)450N force towards north-east.
2)350N force towards east.
3)250N force at 30º west of north.
4)300N force at 60º south of west.
Find magnitude and direction of the resultant.
EF = 450 cos 45 + 350 - 250 sin 30 - 300 cos 60
NF = 450 sin 45 + 0 + 250 cos 30 - 300 sin 60
magnitude = sqrt (EF^2 + NF^2)
tan direction East of North = EF/NF
or if you do math not navigation then
tan angle North of East = NF/EF
To find the magnitude and direction of the resultant force, we need to find the vector sum of all the given forces.
Step 1: Resolve each force into its horizontal and vertical components.
Let's assume east direction as positive x-axis, and north direction as positive y-axis.
Force 1 (450N towards north-east):
To resolve this force, we can split it into its x and y components using trigonometry.
Magnitude of the force, F1 = 450N.
Angle of the force with the horizontal axis, θ1 = 45 degrees (north-east).
The x component (F1x) can be calculated as F1 * cos(θ1):
F1x = 450N * cos(45°)
F1x = 450N * 0.7071 ≈ 318.64N (east direction)
The y component (F1y) can be calculated as F1 * sin(θ1):
F1y = 450N * sin(45°)
F1y = 450N * 0.7071 ≈ 318.64N (north direction)
Force 2 (350N towards east):
Since this force is already in the east direction, its x component will be 350N, and the y component will be zero.
Force 3 (250N at 30 degrees west of north):
To resolve this force, we need to consider the opposite angle of 30 degrees.
Magnitude of the force, F3 = 250N.
Angle of the force with the horizontal axis, θ3 = 180° - 30° = 150 degrees (south-east).
The x component (F3x) can be calculated as F3 * cos(θ3):
F3x = 250N * cos(150°)
F3x = 250N * (-0.866) ≈ -216.5N (west direction)
The y component (F3y) can be calculated as F3 * sin(θ3):
F3y = 250N * sin(150°)
F3y = 250N * 0.5 ≈ 125N (north direction)
Force 4 (300N at 60 degrees south of west):
To resolve this force, we need to consider the opposite angle of 60 degrees.
Magnitude of the force, F4 = 300N.
Angle of the force with the horizontal axis, θ4 = 180° - 60° = 120 degrees (south-east).
The x component (F4x) can be calculated as F4 * cos(θ4):
F4x = 300N * cos(120°)
F4x = 300N * (-0.5) ≈ -150N (west direction)
The y component (F4y) can be calculated as F4 * sin(θ4):
F4y = 300N * sin(120°)
F4y = 300N * 0.866 ≈ 259.8N (north direction)
Step 2: Add up the x and y components separately.
Sum of the x components (Rx) = F1x + F2x + F3x + F4x
Rx = 318.64N + 350N + (-216.5N) + (-150N)
Rx ≈ 302.14N (east-west direction)
Sum of the y components (Ry) = F1y + F2y + F3y + F4y
Ry = 318.64N + 0N + 125N + 259.8N
Ry ≈ 703.44N (north-south direction)
Step 3: Find the resultant magnitude and direction.
The magnitude of the resultant force, R, can be calculated using Pythagoras' theorem:
R = sqrt(Rx^2 + Ry^2)
R = sqrt((302.14N)^2 + (703.44N)^2)
R ≈ 768.72N
The direction of the resultant force, θr, can be calculated using inverse tangent:
θr = tan^(-1)(Ry / Rx)
θr = tan^(-1)(703.44N / 302.14N)
θr ≈ 66.86°
Therefore, the magnitude of the resultant force is approximately 769N, and its direction is approximately 66.86° (north-east) from the positive x-axis.