Particle in equilibrium (balancing forces on an object on an incline)

AI Thread Summary
The discussion centers on understanding the equilibrium of forces acting on an object on an incline, particularly when only magnitudes are provided. Participants emphasize the importance of defining the coordinate system to correctly apply force equations, noting that the angle θ in the equations must correspond to the correct axes. It is clarified that equilibrium occurs when the net force is zero, and the components of forces must be analyzed accordingly. The relationship between the angle of the incline and the forces is highlighted, indicating that as the incline angle increases, the component of the force parallel to the slope becomes significant for achieving balance. Overall, a clear understanding of the coordinate system and force components is essential for solving the problem accurately.
Justin_Lahey
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Homework Statement
Find the magnitude and angle
Relevant Equations
F1= cos(theta)38.4i+ sin(theta)38.4j
F2= cos(theta)52.7i + sin(theta)52.7j
Hi, I’m wondering if someone can help me understand this question. I can find a resultant force/vector when given an initial angle but I’m stuck here when the only information is the two magnitudes. I think I’m solving for the unknowns but a little lost on how or what equation I should be using. In the pic this is how I normally start by finding the x and y components but without theta I’m a bit lost. Thanks for any help.
0B71EB58-E39E-4864-8B94-D0ADFFA07733.jpeg
 
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If I am understanding the term "equilibrium" (in this context) correctly, shouldn't the net force simply be 0? So you know that F1 + F2 + R = 0.

Can you find ##\theta## from there?
 
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First of all you got to tell us your coordinate system. I think none of your two equations are correct (especially if we take the coordinate system with the i-direction parallel to the incline and the j direction perpendicular to the incline).

I assume you have been taught the following equation $$\vec{F}=|\vec{F}|\cos\theta\hat i+|\vec{F}|\sin\theta\hat j$$
which you apply it in a wrong way in this problem. You got to be careful what the angle ##\theta## is in this equation. It is the angle that the force vector ##\vec{F}## makes with the x-axis (or i-axis should i say, and that's why i asked what is your coordinate system). It is not the same ##\theta## for all forces (each force has its own ##\theta## in other words ) and it is not the angle ##\theta## that is given in the problem statement as the angle of the incline.
 
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Delta2 said:
I think none of your two equations are correct
Well, the F1 equation is right if ##\hat i## is horizontally to the right and ##\hat j## is vertically up; the F2 equation is right if ##\hat i## is normal to the slope and down to the right, and ##\hat j## is parallel to the slope and down to the left.
 
haruspex said:
Well, the F1 equation is right if ##\hat i## is horizontally to the right and ##\hat j## is vertically up; the F2 equation is right if ##\hat i## is normal to the slope and down to the right, and ##\hat j## is parallel to the slope and down to the left.
E hehe @haruspex you did some sort of reverse engineering to find coordinate systems ( you got me, i could never think of the i direction as normal to the slope, good one) that each equation is true, Still there is no single coordinate system that both simultaneously are true . Thats my main point that's why i first asked what is the coordinate system he is using.
 
Welcome, Justin!
The way I would see this problem:
There would not be equilibrium for the case of a slope with very little θ angle, since ##F_1## would accelerate the car.
The magnitude of ##F_2## remains always the same.
As the angle of the slope increases little by little, a component of ##F_2## that is parallel to the surface of the slope an in line with ##F_1## appears and also increases little by little.

Your angle is the angle at which the magnitude of that component reaches the magnitude of ##F_1## and the balance is achieved, so the car does not accelerate in any direction.

Force R and the component of F2 that is perpendicular to the surface of the slope will naturally balance each other (Third law of Newton).
 
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Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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