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

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In summary, Mitch found that the rod is at a distance of 6/7 a below the ceiling and found the inclination of BD. He also found that the modulus of elasticity of each string is in terms of W.
  • #1
gnits
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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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  • #2
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?
 
  • #3
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.
 

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

1. How do you measure the distance of a horizontally suspended rod from a ceiling?

To measure the distance of a horizontally suspended rod from a ceiling, you can use a measuring tape or ruler and measure the vertical distance between the rod and the ceiling. Alternatively, you can use a laser distance measurer for more precise measurements.

2. Why is it important to know the distance of a horizontally suspended rod from a ceiling?

Knowing the distance of a horizontally suspended rod from a ceiling is important for safety reasons. If the distance is too short, the rod may hit or damage the ceiling. If the distance is too long, the rod may not be stable and could potentially fall.

3. What factors can affect the distance of a horizontally suspended rod from a ceiling?

The distance of a horizontally suspended rod from a ceiling can be affected by several factors, such as the weight of the rod, the material and strength of the ceiling, and the method of suspension (e.g. hooks, brackets, etc.). It is important to consider these factors when determining the appropriate distance.

4. Can the distance of a horizontally suspended rod from a ceiling be adjusted?

Yes, the distance of a horizontally suspended rod from a ceiling can be adjusted by changing the method of suspension or using different types of hardware. However, it is important to ensure that the new distance is safe and stable for the rod.

5. Are there any safety precautions to keep in mind when measuring the distance of a horizontally suspended rod from a ceiling?

Yes, it is important to take proper safety precautions when measuring the distance of a horizontally suspended rod from a ceiling. This may include using a ladder or step stool to reach the rod, wearing protective gear, and having someone assist you to ensure stability and accuracy.

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