Question

To find the resultant of multiple forces using Cartesian components and to determine the direction of this resultant from its components. As shown, three forces act on the tip of a pole. F1 = ( -60 i + 145 j + 45 k ) N. F2 = 140 N and forms the following angles with the x, y, and z axes, respectively: α = 55.2 ∘ , β = 65.6 ∘ , and γ = 135.2 ∘ . F3 = 120 N , forms the angle ϕ = 26 ∘ with the z axis, and forms the angle θ = 27 ∘ between the x axis and the projection of F3 in the xy plane.

Answer 1

F1 = ( -60 i + 145 j + 45 k ) N

Components of F2 :

F2x = 140*cos(55.2) = 79.89 N ;

F2y = 140*cos(65.6) = 57.83 N;

F2z = 140*cos(135.2) = (-99.33) N;

So, F2 = ( 79.89 i + 57.83 j + (-99.33) k ) N;

Components of F3 :

F3x = (Projection of F3 on xy plane)*(x-component of F3's projection on xy plane) = (120*sin(26))*cos(27) = 46.87 N ;

F3y = (Projection of F3 on xy plane)*(y-component of F3's projection on xy plane) = (120*sin(26))*sin(27) = 23.88 N

F3z = 120*cos(26) = 107.85 N;

So, F3 = ( 46.87 i + 23.88 j + 107.85 k ) N;

Therefore,

Resultant R = F1 + F2 + F3 = ( 66.76 i + 226.71 j + 53.52 k ) N  

So, Magnitude of R = || R || = 242.32 N ;

Direction of R : If Angle of R with z-axis is Q then : cosQ = 53.52 / 242.32 ; So Q = 77.24 degree ;

If Angle of R with x-axis is P then : cosP = 66.76 / 242.32 ; So P = 74 degree ;

If Angle of R with y-axis is T then : cosT = 226.71 / 242.32 ; So T = 20.68 degree ;

So, Angle with XY plane = 90 - 77.24 = 12.76 degree .;

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