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 186 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.

Explanation / Answer

Net force on tire is eual to zero

So

Fa + Fb + Fc = 0

The direction of F b along - be Y axis so we will resolve the rest two forces along X and Y axis .

And since Fa + Fc = - Fb

So net resultant of Fa and Fc must be along positive Y axis . or we can say that the X component of both the force must cancel each other .

Angle made by Fa with negative X axis = (138-90)° = 48°

Let angle made by Fc with positive X axis = C

Than - Fa ( cos48° ) +Fc (cosC) = 0

Thus. Fc cosC = 141.8557N

cosC = 141.8557/186 = 0.763

Thus angle C = 40.27°

Now both the forces have Y component positive

So. Fc sinC + Fa sin48° = |Fb|

So |Fc| = 212 sin48° + 186 sin40.27° = 277.775 N

The direction of Fc is 40.27° from X axis ( so 90°+ 40.27°= 130.27° from negative Y axis or we can say 130.27° from Fb )

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