Learning Goal:

To replace force$-$couple systems with an equivalent force by specifying the magnitude and direction of the equivalent force and the location where the force needs to be applied.

Certain force$-$couple systems can be replaced by a single force. The requirement is that the lines of action of the resultant force, FR

$,$ and the resultant couple moment, $($MR$)$O$,$ are perpendicular to each other. This is equivalent to saying the following types of force systems can be simplified to a single force:

Concurrent force system: This is a system where all the lines of action of the applied forces intersect at a common point. Because this means all the forces point either directly toward or away from the common point, there is no net moment about that point.

Coplanar force system: This is a system where all the lines of action of the applied forces lie in the same plane. That the resultant force and resultant couple moment are perpendicular to each other is obvious from the definition that the individual forces and moments are perpendicular.

Parallel force system: This is a system where all the forces act along lines parallel to a common line, usually chosen to be the positive z axis. Again, because the individual forces and moments are mutually perpendicular, the resultant force and resultant couple moment will be perpendicular.

Part A

The object in the diagram below is called a Warren truss. $($Figure $1)$For the system of parallel forces F$1=430$N$,$ F$2=924$N$,$ and F$3=1600$N

$,$ determine the equivalent resultant force of the system by finding both the magnitude of the force and the distance from point A where the force needs to be applied.

Express your answers, separated by a comma, to three significant figures.

Part B

For the bracket shown below, F$1=174$lb$,$ F$2=58$lb$,$ and M$=400$lb$$ft

$.($Figure $2)$Replace this force and couple system with an equivalent force by specifying the magnitude of the force, the direction in which the force acts, and where the force's line of action intersects member AB as measured from A$.$

Express your answers, separated by commas, to three significant digits. The angle should be measured counterclockwise from the positive x axis $($parallel to segment CB$).$

Part D

The concrete slab shown in the picture is subject to four forces, F$1=225$lb$,$ F$2=280$lb$,$ F$3=555$lb$,$ and F$4=350$lb$.$

$($Figure $3)$

The dimensions are d$1=7\mathrm{.}0$ft$,$ d$2=20$ft$,$ d$3=5\mathrm{.}0$ft$,$ d$4=16$ft$,$ and d$5=9\mathrm{.}0$ft$.$

Determine the equivalent resultant force by specifying its magnitude and its location $($x$,$y$)$

on the slab.

Express your answers, separated by commas, to three significant figures.