Write the equation for each Force in Cartesian (unit vector) form. DO NO CALCULA

Question

Write the equation for each Force in Cartesian (unit vector) form. DO NO CALCULATOR MATH: leave results in terms of the trigonometry shown. Include i, j, and k in your solution. F1 = F2 = F3 = Take F1 = 200 lb, F2 = 350 lb, and F3 = 150 lb. a) Represent the resultant force (F_R = F1 + F2 + F3) as a Cartesian vector (i, j, and k, components) b) Calculate the magnitude of the resultant force c) Determine the projection angles of the resultant vector: alpha, beta gamma measured from the positive x, y, and z axes, respectfully.

Explanation / Answer

Part 1 :

F1 = F1(-sin45o i + cos 45o j + sin 75o k) = F1(-0.707 i + 0.707 j + 0.966 k)

F2 = F2 (coss 55o i - cos60o j + cos 50o k) = F2(0.573 i - 0.5 j + 0.643 k)

Coordinate of points through which F3   are (0,0,0) and (3,6,-2)

vector along F3 = 3i+6j-2k

unit vector along this = (3i+6j-2k)/sqrt(32+62+22) = 0.429i+0.857j-0.286k

F3 = F3 (0.429i+0.857j-0.286k)

Part 2:

a)F1 = 200 lb ; F2 = 350 lb; F3 = 150 lb

vectorially,

F1 = -141.4 i + 141.4 j +193.2 k lb

F2 = 200.55 i -175 j + 225.05 k lb

F3 = 64.35 i + 128.55 j - 42.9 k lb

Resultant force = Fr = 123.5 i + 94.95 j + 375.35 k lb

b) Magnitude of resultant force = sqrt(123.52+94.952+375.352) = 406.39 lb

c) alpha = cos-1(123.5/406.39) = 72.3o

beta = cos-1(94.95/406.39) = 76.5o

gamma = cos-1(375.35/406.39) = 22.5o

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