To achieve equilibrium and thus, not lose the game, what is the force Maria must exert and at what angle?

Question

Sally is playing tug of war against two friends. She is pulling with a force of 75 N at 150 degrees. Allison exerts a force of 55 N at 55 degrees. Maria exerts a force as well.

Answer 1

Sally is applying force against Allison & Maria.

We should draw the diagram of all forces acting.

 

Horizontal components of forces----

 

75 cos(30)=55cos(55)+F cos(theta)

 

Vertical components of forces-------

 

75 sin (30)=F sin(theta)-55sin(55)

 

F cos(theta)=33.4

 

Fsin(theta)=82.55

 

Tan (theta)=2.47

 

Theta= 67.9

 

F=88.8 N

 

For guidance of figure---

 

Force of Sally is in second quadrant

Force of Allison is in first quadrant.

Force of Maria is in fourth quadrant

Answer 2

the horizontal and vertical forces must add to zero.

 

Sally's horizontal force is 75cos30 and vertical force is 75sin30

 

Alison's horizontal force is 55cos55 and vertical force is 55sin55

 

Maria's horizontal force is Fcosθ and vertical force is Fsinθ

 

So we get two equations.

 

Horizontal: 75cos30 = 55cos55 + Fcosθ

Vertical:     75sin30 + 55sin55 = Fsinθ

 

64.95 = 31.55 + Fcosθ

37.5 + 45.05 = Fsinθ

 

Fcosθ = 33.4

Fsinθ = 82.55

 

Fsinθ/Fcosθ = 2.47

 

tanθ = 2.47

 

θ = 68 degrees

 

F ≈ 89 N