Please show all work and equations used Thanks! In a two-dimensional tug-of-war,

Question

Please show all work and equations used

Thanks!

In a two-dimensional tug-of-war, Alex, Betty, and Charles pull on ropes that are tied to an automobile tire. The ropes make angle with each other as shown in Figure 3.10, Part (a), which is a bird's eye view of the situation. Alex pulls with a force FA (220 N), while Charles exerts a force FC (170 N). What force, FB, does Betty apply to the system? The tire is stationary and the orientation of Charles' rope is not specified, i.e the angular direction, phi, must also be determined. Figure 3.10 The three forces acting on the tire, as shown in Part (a), are described by the free-body diagram of the system in Part (b).

Explanation / Answer

Fa = 220 N
Fb = ?
Fc = 170 N

As the system is in equilibrium,

Horizontal forces will counter each other.
Fa * cos(47) = Fc * cos()
= cos^-1(220 * cos(47) / 170)
=  28o

Vertical Forces must also counter each other

Fb = Fa * sin(47) + Fc * Sin()
Fb = 220 * sin(47) + 170 * sin(28)
Fb = 240.7 N

Force Betty apply on the system , Fb = 240.7 N

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