Statics/Equilibrium Physics question-- Two weights and a single pulley

[zubb999]

Homework Statement

The 60 lb collar "A" is on a friction-less vertical rod and is connect to a 65 lb counterweight "C". Determine the value "h" (which is the height between "A" and "B") for which the system is in equilibrium. Also, find the horizontal force "N" acting on collar "A".
Here is a rough typed up picture of what the picture looks like, the zeros (0) are empty space

<--15in-->
00000000B0 ^
0000000/0|0 |
000000/00|0 |
00000/000|0 |
0000/000 C0 h
000/0000000 |
00/00000000 |
0/000000000 |
A0000000000v

The horizontal distance between A and B is 15 in. There is a pulley at B and a rope/string going from a A to B, and B to C. From what I can tell, the rope is continuous from points A to C. I have no idea where to get started on this or how to ultimately find h or the horizontal force acting on point A.
A is a 60 lb weight.
C is a 65 lb weight.

Homework Equations

All I have so far are these few things:
∑F_x = 0
∑F_y = 0
T_BC = 65 lb
T_BA = 65 lb

The Attempt at a Solution

I really don't know where to begin with this problem. I'm thoroughly confused as to how I should go about finding "h" or the horizontal force at "A", and I haven't been able to find similar problems to this one. If anyone can help me out, it'd be much appreciated.

 

[BvU]

Hello Zubb, and welcome to PF.

Nice picture..
As a first step it might be nice to consider only A and the section AB of the rope. What vertical force is needed to keep A from moving up (or down) ?
The rope can only exercise a force in a direction along the rope. With one component known and the tangent of the angle expressed in h and the 15 in, you have an expression for the total tension in the rope. Then it's time to bring B and C back in the considerations.

 

[zubb999]

Ok...

So according to what you just said, I should have this at my disposal:
T_BAy = 60 lb.
tanθ = h/15 in. --> θ=tan^-1(h/15 in.)
If T_BA = T_BAy x sinθ
then T_BA = T_BAy x sin(tan^-1(h/15 in.))
If T_BA = T_BAx x cosθ
then T_BA = T_BAx x cos(tan^-1(h/15 in.))

Right?

And then when I bring T_BC to put all this into the ∑F_x and ∑F_y equations, and I end up with this:
∑F_x = T_BA/(cos(tan^-1(h/15 in.))) = 0? [substituted T_BAx out]
∑F_y = A[-60 lb.] + C[-65 lb.] + T_BA/(sin(tan^-1(h/15 in.))) + T_BC[65 lb.] = 0 [substituted T_BAy out]

I'm terribly sorry, but I'm not seeing how this helps me...
I really do appreciate this help though.

 

[BvU]

Well, you have T_BA,y, you have T_BA and you have a relationship with one unknown, h.
What can be easier ?

 

[HalfLostAndSearching]

I would approach this problem by first drawing a free body diagram to visualize all the forces acting on the system. From the given information, we know that there are two weights, 60 lb and 65 lb, connected by a rope passing over a single pulley at point B. The rope is continuous from points A to C, so we can assume that the tension in the rope is the same at all points.

Next, I would apply the principles of statics and equilibrium to the system. This means that the sum of all the forces in the x-direction and the y-direction must equal zero. We can write this mathematically as ∑Fx = 0 and ∑Fy = 0.

For the x-direction, we have the horizontal force acting on A, which we can label as N, and the tension in the rope, which we can label as TBA. Since there are no other forces acting in the x-direction, we can write ∑Fx = N - TBA = 0.

For the y-direction, we have the weight of collar A, which we can label as W, and the tension in the rope, which we can label as TBA. We also have the weight of counterweight C, which we can label as Wc. Again, since there are no other forces acting in the y-direction, we can write ∑Fy = W + Wc - TBA = 0.

Since we have two equations and two unknowns (N and TBA), we can solve for both of them. Once we have the values for N and TBA, we can use trigonometry to find the height h, which is the distance between points A and B.

To find the horizontal force at A, we can simply substitute the value of TBA into the equation for N. This will give us the value for N in terms of known quantities.

In summary, to solve this problem, I would first draw a free body diagram and then apply the principles of statics and equilibrium to write equations for the x-direction and y-direction. From there, I would solve for the unknowns, N and TBA, and then use trigonometry to find the height h. Finally, I would use the value of TBA to find the horizontal force at A.