# Two Dimensional Force

## Homework Statement

In a two-dimensional tug-of war, Alex, Betty, and Charles pull horizontally on an automobile
tire. The angle between Alex and Betty is 137degrees. The tire remains
stationary in spite of the three pulls. Alex pulls with a force of magnitude 220 N, and Charles
pulls with a force of magnitude 170 N. Note that the direction of the force applied by Charles
is not given. What is the magnitude of force with which Betty pulls?

## Homework Equations

F1 + F2 + F3 = 0 ?

## The Attempt at a Solution

I do not know how to even approach this problem. Charles could be pulling with Betty, or with Alex, or in some other random direction. What formula could I use that would compensate for Charles?

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Doc Al
Mentor
Hint: What must be the magnitude of the sum of Alex's force + Betty's force?

You are incredible.
<3

HallsofIvy
Set up a coordinate system. let's set it up so that Alex is pulling along the positive x-axis: his force vector is <220, 0>. Then Betty is pulling at an angle of 137 degrees has force vector <Bcos(137), Bsin(137)> where "B" is the magnitude of the force (which is what you are asked). Finally, we know that Charles is pulling with magnitude 170 but don't know the angle: let's call the angle $\theta$ degrees. His force vector is $<170 cos(\theta),170 sin(\theta)>$. The total force vector is $<220+ Bcos(137)+ 170cos(\theta), Bsin(137)+ 170sin(\theta>= <0, 0>$. That gives you two equations to solve for B and $\theta$.