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 +150 j +45 k lb. F2=120 lband forms the following angles with the x, y, and zaxes, respectively: a=52.7°, ß=68.1°, and Φ=134.6°. F3=90 lb, forms the angle θ=30° with the z axis, and forms the angle θ=27° between the x axis and the projection of F3 in the xy plane.
Part A - Resultant of adding F1 and F3 Find the resultant of adding F1 and F3. Express your answer in component form. R13=?
Part B - Resultant of adding F1 and F2 and F3 Find the resultant of adding F1, F2, and F3. Express your answer in component form. R123=?
Part C - Direction of a resultant For the given forces, F1, F2, and F3, find the three direction angles aR, ßR, and ?R between the resultant force R123 found in Part B and the x, y, and z axes, respectively. aR, ßR, ?R=?
Part D - Find the direction angle a when given direction angles ß and ? As shown, a force of 280 lb acts at the tip of a tower. Two of the force's direction angles are known, ßR1=39° and ?R1=63°, that define the angles between the force and the y and z axes, respectively. Find aR1, the angle between the x axis and the force.
Given
Fl = -60i + 150j + 45k
F2 = 120(cos(α)i + cos(β).i + cos(γ)k)
F2 = 72.72i + 44.76j - 83.36k
F3 = F3[(COS(Φ)k sin(Φ)cos(θ)i - sin(Φ)sin(θ)j]
The projection of F3 on the z-axis is F3 cos(phi) and on the xy plane is F3 sin(phi) Now resolve the F3 sin(phi) into two components on the x and y axis respectively
F3 = 40.1i - 20.43j + 77.94k
A) Resultant of 1 and 3 are
R13 = F1 + F3 = -601+150 j + 45 k + 40.11-20.43 j + 77.94 k
R13 = -19.91+ 129.57 j + 122.94 k
B) The resultant is
R=F1+F2 +F3=R13 +F2= -19.91+ 129.57j+ 122.94 k+ 7272 i+ 4476j-8336 k R = 52.82 1+ 174.33 j + 39.58 k