Solve F2 Components & Find Magnitudes U & V Axes

[Kaura]

Homework Statement

Resolve F2 into components along the u and v axes and determine the magnitudes of these components. Note that the magnitudes are positive numbers.

Homework Equations

Basic Trigonometrics

The Attempt at a Solution

I correctly calculated 150cos(150)N or 130N as the positive magnitude for F2 along the U axis however I cannot figure out what it is along the V axis.
I tried 150sin(150)N or 75N at first and got it wrong. Then I realized that the axis are not orthogonal so I corrected for that and got 150cos(75)N or 38.8N but that is also counted as wrong.
I am mildly annoyed since this is basic stuff so I am assuming I am just missing something obvious. Thanks for any help.

 

[cnh1995]

You know the angle between u and v axes. You also know the angle between F2 and u.
So what is the angle between F2 and v?

 

[Kaura]

I thought it was 75 degrees (105-30) is that not right?

 

[Doc Al]

I thought it was 75 degrees (105-30) is that not right?

That's the angle between u and v. But what's the angle between F2 and v?

 

[Kaura]

I thought the angle between u and v was 105 and then you subtract 30 to get 75 for the angle between F2 and v

 

[Doc Al]

I thought the angle between u and v was 105 and then you subtract 30 to get 75 for the angle between F2 and v

Careful. The angle between +u and v is 75. But what's the angle between -u and v?

 

[Kaura]

105 right? I am confused

 

[Chestermiller]

I would represent each force by $$\mathbf{F}=F_u\mathbf{i_u}+F_v\mathbf{i_v}$$ where the i's are unit vectors in the u and v directions. Then I would dot the equation with each of the unit vectors in turn to obtain two equations in two unknowns for ##F_u## and ##F_v##.

 

[Doc Al]

105 right? I am confused

That's correct. And you were also correct before (I wasn't paying attention!): The angle between F2 and v is 75. (Sorry about that! )

But you used the wrong trig function in finding the component along v.

 

[Kaura]

I thought I tried both cos and sin

 

[Doc Al]

Then I realized that the axis are not orthogonal so I corrected for that and got 150cos(75)N or 38.8N but that is also counted as wrong.

I would say that this answer is correct. (And I should have read your first response more carefully.)

Is this from a textbook?

 

[Kaura]

It is an online assignment
These are the answers I submitted and it seems to say that term 2 is incorrect but I have no idea why

Submitted Answers
ANSWER 1: Deduction: -5%
(F2)u, (F2)v =
130,75
N
Term 2: Not quite. Check through your calculations; you may have made a rounding error or used the wrong number of significant figures.
ANSWER 2: Deduction: -5%
(F2)u, (F2)v =
130.,75
N
Term 2: Not quite. Check through your calculations; you may have made a rounding error or used the wrong number of significant figures.
ANSWER 3: Deduction: -5%
(F2)u, (F2)v =
130.,75.0
N
Term 2: Not quite. Check through your calculations; you may have made a rounding error or used the wrong number of significant figures.
ANSWER 4: Deduction: -5%
(F2)u, (F2)v =
130.,38.8
N

 

[Doc Al]

ANSWER 4: Deduction: -5%
(F2)u, (F2)v =
130.,38.8
N

If they claim that is wrong they are busting your chops over significant figures. F2 is only given to 2 sig figs, so round your answers accordingly.

 

[Kaura]

I should also mention that the problem states to use 3 significant figures
"Express your answers using three significant figures separated by a comma."

 

[Doc Al]

I should also mention that the problem states to use 3 significant figures
"Express your answers using three significant figures separated by a comma."

Interesting. Did you get feedback on that answer like the others?

 

[Kaura]

No, which is weird

 

[Doc Al]

Although not quite correct, I would have put 130,38.8 (leaving out the decimal in the first number). (But this system clearly has problems.)

 

[Doc Al]

(By the time you "guess" what they want you'll be down to no points anyway.)

 

[Kaura]

It was 77.6 I have no clue why but sadly I missed this one. Would be nice if homeworks had more than 1 problem per.

I get lowest four dropped though so no worries I guess. I just wish the problem had made more sense.

 

[Kaura]

I actually have yet to understand why it is 77.6 N and if anyone can offer insight why I would be quite pleased, cheers!

 

[Doc Al]

I actually have yet to understand why it is 77.6 N and if anyone can offer insight why I would be quite pleased, cheers!

I'd say it's just wrong.

 

[Kaura]

If that is the case then I will have a good laugh because this question seems so absurd.

 

[I like Serena]

I actually have yet to understand why it is 77.6 N and if anyone can offer insight why I would be quite pleased, cheers!

Let's see. We have:

From the sine rule it follows that:
$$\frac{F_u}{\sin 75^\circ} = \frac{F_2}{\sin 75^\circ}$$
$$\frac{F_v}{\sin 30^\circ} = \frac{F_2}{\sin 75^\circ}$$
With ##F_2=150\text{ N}## we get that ##F_u = 150\text{ N}## and ##F_v = 77.6 \text{ N}##.

 

[Doc Al]

Let's see. We have:
View attachment 209325

Interesting, but I'm not sure what F_u and F_v correspond to in your diagram.

 

[I like Serena]

Interesting, but I'm not sure what F_u and F_v correspond to in your diagram.

The problem statement says: 'Resolve F2 into components along the u and v axes and determine the magnitudes of these components.'
F_u and F_v are the magnitudes of the components of F₂ along the u and v axes.

 

[Doc Al]

The problem statement says: 'Resolve F2 into components along the u and v axes and determine the magnitudes of these components.'
F_u and F_v are the magnitudes of the components of F₂ along the u and v axes.

How can the component of F₂ along u equal F₂ itself?

 

[Doc Al]

And the angle that F₂ makes with the vertical is 75°, so F_v = F₂ cos(75°). (What you have labeled F_v in your diagram is not the vertical component of F₂.)

 

[I like Serena]

How can the component of F₂ along u equal F₂ itself?

We can also write:
$$\mathbf F_2 = \mathbf F_{2,u} + \mathbf F_{2,v} = F_{2,u} \mathbf{\hat u} + F_{2,v} \mathbf{\hat v}$$
where ##\mathbf{\hat u}## and ##\mathbf{\hat v}## are the unit vectors along the u and v axes.
After which we have ##F_u = |F_{2,u}|## and ##F_v = |F_{2,v}|##.

We can have a component of F₂ along u equal F₂ since u and v do not form an orthogonal basis.
The vector decomposition is effectively shown in the diagram.

And the angle that F₂ makes with the vertical is 75°, so F_v = F₂ cos(75°). (What you have labeled F_v in your diagram is not the vertical component of F₂.)

That would only be the case if we had an orthogonal basis (which is admittedly usually the case but not here).

 

[Doc Al]

We can have a component of F₂ along u equal F₂ since u and v do not form an orthogonal basis.

The component of F₂ along u can equal F₂ only if F₂ is parallel to u. That's not the case here.

 

[I like Serena]

The component of F₂ along u can equal F₂ only if F₂ is parallel to u. That's not the case here.

Untrue.
See for instance https://math.stackexchange.com/ques...mpose-a-vector-into-non-orthogonal-components how to decompose a vector into a non-orthogonal basis.
(I was looking for a nice reference on wiki or wolfram but couldn't find it quickly enough.)

 

[Doc Al]

Interesting. I was not viewing u and v as a non-orthogonal basis, but as simply two unrelated axes/directions. You are probably correct in your interpretation!

 

[I like Serena]

Interesting. I was not viewing u and v as a non-orthogonal basis, but as simply two unrelated axes/directions. You are probably correct in your interpretation!

When seen in the standard orthonormal basis, consider that we have to solve:
$$\binom{150\cos(165^\circ)}{150\sin(165^\circ)} = \mathbf F_2 = F_{2,u} \mathbf{\hat u} + F_{2,v} \mathbf{\hat v}
= F_{2,u} \binom{\cos(15^\circ)}{\sin(15^\circ)} + F_{2,v} \binom{0}{1}$$