In a two-dimensional tug-of-war, Alex, Betty, and Charles pull horizontally on a

Question

In a two-dimensional tug-of-war, Alex, Betty, and Charles pull horizontally on an automobile tire at the angles shown in the picture. The tire remains stationary in spite of the three pulls. Alex pulls with force Upper F Overscript right-arrow EndScripts Subscript Upper A of magnitude 215 N, and Charles pulls with force Upper F Overscript right-arrow EndScripts Subscript Upper C of magnitude 183 N. Note that the direction of Upper F Overscript right-arrow EndScripts Subscript Upper C is not given. What is the magnitude of Betty's force Upper F Overscript right-arrow EndScripts Subscript Upper B if Charles pulls in (a) the direction drawn in the picture or (b) the other possible direction for equilibrium?

Explanation / Answer

a) The horizontal forces must balance:
215N * sin43º = 183N * sin
where is the angle between Charles and the +y axis.
= 53.25º
Then Betty = 215N*cos43 + 183N*cos53.25º = 340 N

b) I would interpret this to mean "where else could Charles be so that he balances out Alex's horizontal component? He could be 49º from Betty (the -y axis), almost directly across from Alex.
Then Betty = 215N*cos43 - 183N*cos53.25º = 25N

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