Machinery's Handbook, 31st Edition
164
Force Systems Algebraic Solution of Force Systems—All Forces in the Same Plane
Finding Two Concurrent Components of a Single Force:
Case I: To find two components F 1 and F 2 at angles θ and φ , φ not being 90 ° . sin sin sin sin F F F F 1 2 z i z z i = = − ^ h Case II: Components F 1 and F 2 form 90 ° angle. sin cos F F F F 1 2 = = i i Case I: Forces F 1 and F 2 do not form 90 ° angle. sin sin sin sin cos tan cos sin R F R F R F F F F F F F 2 or or 1 2 1 2 2 2 1 2 1 i z z i z z i z z = = − = + + = + ^ h Case II: Forces F 1 and F 2 form 90 ° angle. cos sin 1 2 R F R F or or 2 1 i i = =
F 1
F 2
F 1
F 2
Finding the Resultant of Two Concurrent Forces:
F 1
F 2
F 1
i = R F F 1 2 2 2 = +
F F
tan
2 1
F 2
Finding the Resultant of Three or More Concurrent Forces: F 2 y
To determine resultant of forces F 1 , F 2 , F 3 , etc. making angles, respectively, of θ 1 , θ 2 , θ 3 , etc. with the x axis, find the x and y components F x and F y of each force and arrange in a table similar to that shown below for a system of three forces. Find the algebraic sum of the F x and F y components ( ∑ F x and ∑ F y ) and use these to determine resultant R . Force F x F y F 1 F 1 cos θ 1 F 1 sin θ 1 F 2 F 2 cos θ 2 F 2 sin θ 2 F 3 F 3 cos θ 3 F 3 sin θ 3 ∑ F x ∑ F y
F 1
F 3
2
3
4
1
– x
x
F 4
– y
y
F h ^ h y 2 +
2
R F x = ^
R R
R F y R
F F F
R
cos tan
x
=
i
R
R
– x
x
F x
R R
y
or
=
i
R
x
– y
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