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 FA of magnitude 212 N, and Charles pulls with force Fc of magnitude 176 N. Note that the direction of Fc is not given. What is the magnitude of Betty's force FB if Charles pulls in (a) the direction drawn in the picture or (b) the other possible direction for equilibrium? Alex Charles Betty 140°

Explanation / Answer

FB = - F2 j

FA = 212 (cos(270 - 140)i + sin(270 - 140)j)

FA = - 136.3i + 162.4 j

FC = 176(cos(theta)i + sin(theta)j)

for equilibrium,

FA + FB + FC = 0

(-136.3 + 176 cos(theta))i + (- F2 + 162.4 + 176 sin(theta))j = 0

cos(theta) = 136.3 / 176

theta = 40 deg

- F2 + 162.4 + (176 sin40)= 0

F2 = 275.5 N .......Ans (Magnitude of betty'force)

(A) angle between alex and charles = 270 - 140 - 40 = 90 deg

(B) other possible direction:

taking theta = - 40 deg

angle between alex and charles = (270 - 140) + 40

= 170 deg

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