Tug of war in two dimensions.

In summary, the problem asks for the minimum magnitude of Betty's force Fb in a two-dimensional tug-of-war scenario. With Alex pulling with force Fa of magnitude 208 N at an angle of 39 degrees, Charles pulling with force Fc of magnitude 178 N at an unknown angle, and the tire remaining stationary, the minimum magnitude of Fb can be found using the cosine formula with the given angles and forces. Solving for Fb, we get a minimum value of approximately 377.08 N. Other methods, such as using the fact that acceleration is 0, could also be used to solve this problem. However, the given method is the most straightforward and leads to the same equation.
  • #1
AnkhUNC
91
0

Homework Statement


In a two-dimensional tug-of-war, Alex, Betty, and Charles pull horizontally on an automobile tire at an angle of 141d (Alex and Betty's angle). The tire remains stationary in spite of the three pulls. Alex pulls with force Fa of magnitude 208 N, and Charles pulls with force Fc of magnitude 178 N. Note that the direction of Fc is not given. What is the minimum magnitude of Betty's force Fb ?

Sorry don't have a picture :(

Homework Equations





The Attempt at a Solution



OK so for this I get
x = -Fa cos(39)[141-180]+0+Fc cos (Theta)
y = Fa sin(39)-Fb+Fc sin (theta)

Solving for Theta I get 24.75240251.
So when I try to plug this into the equation for y to find Fb I find the magnitude to be 56.37042988 (There is no magnitude in the x direction). But this is incorrect. Any idea where I went wrong?
 
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  • #2
Apply the cosine formula directly:

Fb^2 = Fa^2 + Fc^2 + 2 FaFcCos B. Now find the min value of Fb.
 
  • #3
So I get Fb^2 = 142192.8858 -> sqrt = 377.0847196 is that correct?
 
  • #4
I am terribly sorry, but apply the formula for the resultant of the force opp the angle which is given. The angle which is given is between Fa and Fb, i.e., angle C=141 deg . (Draw a diagram of three concurrent forces, to be on the safe side.)

Fc^2 = Fb^2 + Fa^2 + 2FbFaCos C.

Now you have a quadratic expression in Fb, and can minimize it. All the other terms are known.

(How did you get the value you have given? Cos B was unknown.)
 
  • #5
Can't I just use the fact that acceleration is 0 to make this an easier problem? Was the way I was trying to solve completely incorrect?
 
  • #6
AnkhUNC said:
Can't I just use the fact that acceleration is 0 to make this an easier problem? Was the way I was trying to solve completely incorrect?

The method I've given should be the easiest.

AnkhUNC said:

Homework Statement



The Attempt at a Solution



OK so for this I get
x = -Fa cos(39)[141-180]+0+Fc cos (Theta)
y = Fa sin(39)-Fb+Fc sin (theta)

Solving for Theta I get 24.75240251.

Write properly in which direction you are choosing the x-axis etc and the components of the forces along the axes. Ultimately any method should lead you to the same eqn.
 

1. What is "Tug of war in two dimensions"?

"Tug of war in two dimensions" is a physical phenomenon that involves two teams pulling on opposite ends of a rope in two-dimensional space. It is a popular game and also has applications in physics and engineering.

2. How does the tension in the rope affect the outcome of the game?

The tension in the rope is the force that each team applies to the rope in order to pull the other team towards their side. The team with the greater tension will have an advantage and will be able to pull the other team towards their side, winning the game.

3. What factors can affect the outcome of "Tug of war in two dimensions"?

The factors that can affect the outcome of "Tug of war in two dimensions" include the strength of each team, the friction between the rope and the ground, the weight of the rope, and the length of the rope. These factors can influence the tension and ultimately determine which team will win.

4. How is "Tug of war in two dimensions" used in physics and engineering?

In physics, "Tug of war in two dimensions" can be used to demonstrate the concept of force and the relationship between force, mass, and acceleration. In engineering, it can be used to simulate real-life scenarios where forces act in different directions, such as in bridge or building construction.

5. Are there any strategies or techniques that can help a team win in "Tug of war in two dimensions"?

Yes, there are various strategies and techniques that can help a team win in "Tug of war in two dimensions". Some common techniques include having a strong and coordinated team, using proper body positioning and weight distribution, and pulling in short, quick bursts rather than sustained pulling. Additionally, teams can also use tactics such as distracting the other team or pulling at an angle to create an advantage.

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