To find the distance of a horizontally suspended rod from a ceiling

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The discussion centers on proving that a horizontally suspended rod is positioned at a distance of (6/7)a below the ceiling while maintaining equilibrium with a particle attached. The participant has successfully calculated the tensions in the strings and the angle of inclination but struggles to demonstrate the distance from the ceiling. Suggestions include using geometry to analyze the vertical lines through points C and D, leading to the lengths AE and BF in terms of distance d. The participant acknowledges the guidance received and expresses gratitude for the assistance. The conversation highlights the importance of geometric reasoning in solving the problem.
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Homework Statement
To find the distance of a horizontally suspended rod from a ceiling
Relevant Equations
Moments and equating of forces
Please could I ask for help with the following?

Two points, A and B on a horizontal ceiling are at a distance of 2a apart. A uniform rod CD of length a and weight W is suspended from A and B by two light strings AC, BD. A particle of weight (2/5)W is attached to the rod at D, and the system hangs in equilibrium with the rod horizontal. Prove that the rod is at a distance (6/7)a below the ceiling and find the inclination of BD. If both strings are elastic and of natural length (1/2)a, find the modulus of elasticity of each string in terms of W.

I have correctly found the inclination of BD (agree with answer in back of book) and if I assume the answer of (6/7)a then I can find the elastic moduli.

I cannot see how to prove that the distance from the ceiling is (6/7)a. Here's a diagram:

rod.png
I have found that T1 = 5W/8 and T2 = 39W/40 and that Ɵ = atan(12/5). This last answer agrees with the book answer. (This I did by equating forces horizontally, vertically and by taking moments about D and solving these equations for T1, T2 and Ɵ)

I have tried to involve d by taking moments about points on the line AB (for instance about the point on AB vertically above D) but this always leads to 1 = 1.

Thanks for any help in showing me how I can prove that d = (6/7)a.

Mitch.
 
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gnits said:
Homework Statement:: To find the distance of a horizontally suspended rod from a ceiling
Relevant Equations:: Moments and equating of forces

tried to involve d by taking moments
Having found those angles, there is no need for any further mechanical analysis; it's just geometry.
Take vertical lines through C, D to meet AB at E, F. In terms of d, how long are AE and BF?
 
haruspex said:
Having found those angles, there is no need for any further mechanical analysis; it's just geometry.
Take vertical lines through C, D to meet AB at E, F. In terms of d, how long are AE and BF?
Thanks very much, Just wasn't seeing it. Much obliged.
 

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