Statics: System in equilibrium - determine height (h)

[gate13]

Homework Statement

Good day to all,
I have been given a problem in statics, ore specifically a system that is in equilibrium, but a system I am having some trouble with. I have attached (in pdf) the problem's context and the free body diagram I have drawn.

The data we are give are as follows:
r = 0.575 m, a=1.05 m, W=4.905 N, Wc = 4.905 N

Homework Equations

We are asked to find h.

The Attempt at a Solution

What I have so far is as follows:

Using the fact the the hoop is in equilibrium, I obtained the two equations:

T_d cos(θ) - T_ccos(α) = 0 (along x-axis: $\rightarrow$)
T_d sin(θ) + T_csin(α) - W= 0 (along y-axis: $\uparrow$)

We know W and Tc and so we have two equations with three unknowns so far. Then I thought of using the problem's geometry to obtain a third equation.

Ans so using the sine law, I obtained:
$\frac{sin180-α-θ}{a}$ = $\frac{sin(α)}{r}$

But when I try solving the system with the values, I get:
α=90, θ=88.17 and T_d = -1* 10^-81

which I am quite certain are wrong. I can't seem to pinoint the third equation to help solve the problem. Any help would be greatly appreciated.

Thank you.

 

[tiny-tim]

hi gate13!

… But when I try solving the system with the values, I get:
α=90, θ=88.17 and T_d = -1* 10^-81

your method looks ok …

show us how you got those figures ​

 

[gate13]

Hello tiny-tim,

First of all, I wish to thank you for taking time to answer my question. The values I listed above were obtained using a TI Voyage 200 calculator. What I entered (in the calclulator) was the following:
solve(Td*cos(theta)-Tc*cos(alpha)=0 and Td*sin(theta)+Tc*sin(alpha)-W = 0 and r*(180-alpha-theta) = a*sin(apha), {alpha, theta, Td})
I just realized I should be entering:
solve(Td*cos(theta)-Tc*cos(alpha)=0 and Td*sin(theta)+Tc*sin(alpha)-W = 0 and r*sin(180-alpha-theta) = a*sin(apha), {alpha, theta, Td}).
Upon correcting this, I obtained:
Td = 5.06 N, alpha = 27.99 degrees, theta = 31.00 degrees.

Once more I wish to thank you for your help ( and I feel a bit embarassed with respect to my mistake).