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 214 N, and Charles pulls with force Upper F Overscript right-arrow EndScripts Subscript Upper C of magnitude 182 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

net force Fa + Fb + Fc = 0

Fb = -Fa - Fc

Fby *sin 90 = -Fa cos 133 - Fc cos phi

Fbx = -Fa -Fcx

Fb cos (-90) = Fa cos 133 -Fc cos phi

0 = -214 cos 133 - 182 cos phi

phi = 36.77 deg

now

Fb sin(-90) = -214 sin 133 - 180 sin 36.77

Fb = 264.25 N

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