Finding Equilibrium Force in 2D Vector Addition

[Jrlinton]

Homework Statement

Homework Equations

[/B]
xcomponent of vector= magnitude*cosine of theta
ycomponent of vector= magnitude*sine of theta
Basic trig operations after this. Honestly I think that I am just having trouble with the algebra

The Attempt at a Solution

Because the magnitude of the added vectors a and b must equal the magnitude of c, I added the vectors, with the exponent x as the magnitude for force b, and used Pythagorean theorem

to come up with the magnitude of the sum. I acted as if force a was acting along the positive x axis, putting the angle of force b at 144. Set this equal to the magnitude of force c. See below

186=((210cos(0)+xcos(144)^2+(210sin0+xsin(144))^2)^0.5
34596=(210-.81x)^2+.588x^2
34596=1.006x^2-340.2x+44100
0=1.006x^2-340.2x+4504
x=13.8,324.7
I took the 324.7 being the magnitude of force b, obviously if this was right I wouldn't be here
Many thanks for the help

 

[Chestermiller]

What are the x and y components of Alex's force? Let $\theta$ be the angle that Charles' force makes with the x axis. In terms of $\theta$, what are the x and y components of Charles' force? What is the force balance in the x direction?

 

[Jrlinton]

So the x component of Alex's force is 210 and the y component is 0. x and y components of Charles' force would be 186cosθ and 186sinθ respectively. With their added forces being 180 degrees from 144 degrees or 324°, we can say the arctan of (186sinθ/210+186cosθ)=324. So 186sinθ/(210+186cosθ)=-.727.

 

[Chestermiller]

So the x component of Alex's force is 210 and the y component is 0.

No. Does it look that way to you from the diagram?

x and y components of Charles' force would be 186cosθ and 186sinθ respectively.

Correct.

With their added forces being 180 degrees from 144 degrees or 324°, we can say the arctan of (186sinθ/210+186cosθ)=324. So 186sinθ/(210+186cosθ)=-.727.

I asked for the force balance in the x direction. Once you know the x components of Alex's force and Charles' force, writing this equation should be easy.

 

[Jrlinton]

Equilibrium Force in x= -(-123.43+186cosθ)

 

[Jrlinton]

Forgive me, I had rearranged the problem so that alex's force was acting upon the positive x axis, that has been corrected.

 

[Chestermiller]

Equilibrium Force in x= -(-123.43+186cosθ)

I get $$186\cos{\theta}-123.43 = 0$$Do you see this? From this equation, what is the value of $\theta$?

 

[Jrlinton]

Placing the image on a grid, with Betty's force being at 270°, this would make the balancing force be at 90° making the added x components of Alex's and Charles' force be 0. Making the x component of Force C be 123.43, taking the arccos of the magnitude (186) divided by 123.43 gives an angle of 47.12°. Thanks for the help

 

[Chestermiller]

I get 48.4 degrees.

Now you need to do the force balance in the y direction, which includes Betsy's force.

 

[Jrlinton]

or 48.425. my mistake

 

[Jrlinton]

Adding those two forces gives a force in the positive y direction of 309 N, with the magnitude of Betty's force being equal in the opposite direction.

 

[Jrlinton]

Although I am unsure about "the other possible direction for equilibrium"

 

[Chestermiller]

Adding those two forces gives a force in the positive y direction of 309 N, with the magnitude of Betty's force being equal in the opposite direction.

I get 301.

 

[Jrlinton]

210sin126=169.9 186sin48.4=139.1 169.9+139.1=309

 

[Chestermiller]

Suppose Charles' force was pointing 48.4 degrees below the x-axis instead of above the x axis. Wouldn't the x component of his force be the same?

 

[Chestermiller]

210sin126=169.9 186sin48.4=139.1 169.9+139.1=309

Oh, I used 200 for some reason.

 

[Jrlinton]

Adding the y components using 210N at 126° for Alex and 186N at -48.4° for Charles I get 30.8 N with the equilibrium force having a magnitude equal but in the -y direction

 

[Chestermiller]

Adding the y components using 210N at 126° for Alex and 186N at -48.4° for Charles I get 30.8 N with the equilibrium force having a magnitude equal but in the -y direction

Nicely done.