Determine the resultant of the system of forces:
Four forces act on a body as illustrated in Figure. Determine the resultant of the system of forces.
Resolving all of forces along x-axis, we attain
Rx = ∑ Fx
= F1 cos θ1 + F2 cos θ2 + F3 cos θ3 + F4 cos θ4
= 40 cos 30o + 50 cos 315o + 30 cos 180o + 20 cos 240o
Note: The angle created by 50 N forces is measured in anticlockwise direction from positive x axis after making the force work away from O by principle of transmissibility of the force.
∴ Rx = 40 cos 30o + 50 cos 45o - 30 cos 0o - 20 cos 60o
θ ≤ 90o can be chosen in appropriate quadrant with correct signs as indicated above.
Rx = 34.64 + 35.36 - 30.00 -10.00
Rx = 30 N
Likewise, resolving all of the forces along y axis, we obtain
Ry = ∑ Fy
= F1 sin θ1 + F2 sin θ2 + F3 sin θ3 + F4 sin θ4
= 40 sin 30o + 50 sin 315o + 30 sin 180o + 20 sin 240o
= 20.00 - 35.36 + 0.00 - 17.32
Ry = - 32.68 N
Therefore, the resultant in vector form can be expressed as
R¯ = (30 N) i¯ + (- 32.68 N) j¯
The magnitude of the resultant is specified by following
= 44.36 N
The direction θ may be worked out as,
θ = tan -1 (Ry /Rx)
= tan -1 (- 32.68/30) = - 47o 26′ 53′′
= 312o 33′ 7′′
The resultant contain a magnitude of 44.36 N and is working in IVth quadrant making an angle of 312o 33′ 7′′ in anticlockwise direction from positive x axis.