Finding an unknown force at an unknown angle

In summary, the problem involves determining the magnitude of a force and its angle in a triangle based on given values for the force's components and angles. The student makes several attempts at solving the problem, using various trigonometric identities and equations, but struggles to find the correct answer. They eventually realize that a simpler approach using trigonometric substitution can lead to the correct solution.
  • #1
Xovvo
5
0

Homework Statement



So we have a force of unknown magnitude acting on these struts at an angle θ measured from strut AB.
The component of the force acting along AB is 600lb, and the magnitude of the force acting along BC is 500lb.
If Φ = 60°, what is the magnitude of F and the angle θ?

Hibbler.ch2.p13.jpg


Homework Equations



Fcos(θ) = 600lb

The Attempt at a Solution



Ok. So, it'll probably help if I knew the third angle of the triangle formed.
180° = ɣ + (60° + 45°)
180° - 105° = ɣ
75° = ɣ

Great. So, I know that Fcos(θ) = 600lb, and Fcos(75° - θ) = 500lb
hm. Fcos(θ)/600 = 1 = Fcos(75 - θ)/500
500Fcos(θ) = 600Fcos(75° - θ)
5Fcos(θ) = 6Fcos(75° - θ)
5cos(θ) = 6cos(75° - θ)
0 = 5cos(θ) - 6cos(75° - θ)

Originally I tried finding where z = 5cos(θ) - 6cos(75° - θ) intersected with z = θ + η where η = 75° + θ, but I couldn't get Wolfram Alpha to understand what I was talking about. Here, I see I should have just left η as
75° - θ, but even still, I have to *ask* Wolfram Alpha what θ works for 0 = 5cos(θ) - 6cos(75° - θ) when
0<=θ<=75° (it gives me an angle of ~30.7°).

Worse, since I couldn't figure it out, I figures if I gave in on the magnitude of F, I could still find the angle. Mastering Engineering told me F = 870lb. So, if Fcos(θ) = 600, θ=arccos(600/F) and arccos(600/870) ≈ 46.4°.
Which was wrong. The θ it wanted was ~34°

Which means everything I did was wrong.
So what triangle magic do I do to get from the initial problem, to the final F=870 θ=34°, without doing something so complicated I need Wolfram Alpha to crunch it out?
 
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  • #2
A hint: cos(x-y) = cos(x)cos(y) + sin(x)sin(y). Apply that to cos(75° - θ).
 
  • #3
Xovvo said:
Great. So, I know that Fcos(θ) = 600lb, and Fcos(75° - θ) = 500lb
Good. I suggest using a trig identity to simplify the expression Fcos(75° - θ).
 
  • #4
While that does get everything in simpler terms of θ, that doesn't address that 870cos(34) = 721.3 and not 600.
My error is a lot farther up.
 
  • #5
Xovvo said:
While that does get everything in simpler terms of θ, that doesn't address that 870cos(34) = 721.3 and not 600.
My error is a lot farther up.
Start over with the two simpler equations. You'll get a different value for θ and F.
 
  • #6
what do you mean by that? Because F=870 and θ≈34° are set in stone Those are the answers for this problem.
 
  • #7
Xovvo said:
what do you mean by that? Because F=870 and θ≈34° are set in stone
And yet you know that:
Xovvo said:
870cos(34) = 721.3 and not 600.

Xovvo said:
Those are the answers for this problem.
Says who?

Why not just solve it using that trig substitution. You'll solve it easily without needing Wolfram.
 
  • #8
Doc Al said:
Says who?
Says The homework. Because I've already gotten this question wrong, and the answers were displayed. I'm asking about this question here so I know what to do when this sort of problem comes up again.
 
  • #9
Xovvo said:
Says The homework. Because I've already gotten this question wrong, and the answers were displayed.
Do you not agree that the given answers do not work? That they contradict the problem statement?

Xovvo said:
I'm asking about this question here so I know what to do when this sort of problem comes up again.
I would solve it in the manner described above, so you can get a straightforward answer.
 
  • #10
The given answers don't work for Fcos(θ)=600, no. But that could mean either Mastering Engineering flubbed or, more likely, I flubbed the starting equations.
 

FAQ: Finding an unknown force at an unknown angle

1. What is an unknown force at an unknown angle?

An unknown force at an unknown angle is a force that is acting on an object at an angle that is not known. This means that the magnitude and direction of the force are both unknown.

2. How do you find an unknown force at an unknown angle?

To find an unknown force at an unknown angle, you can use the laws of physics, such as Newton's laws of motion, to analyze the motion of the object and determine the force acting on it.

3. Why is it important to find an unknown force at an unknown angle?

It is important to find an unknown force at an unknown angle because it can help us understand the forces that are acting on an object and how they are affecting its motion. This information is crucial in many fields, such as engineering and physics.

4. What are some common techniques for finding an unknown force at an unknown angle?

Some common techniques for finding an unknown force at an unknown angle include using vector diagrams, resolving forces into components, and using equations such as F = ma and ΣF = 0.

5. Can an unknown force at an unknown angle be negative?

Yes, an unknown force at an unknown angle can be negative. This indicates that the force is acting in the opposite direction of the chosen coordinate system. It is important to pay attention to the signs and directions of forces when solving for an unknown force at an unknown angle.

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