Derivation of Height Function given an angle

In summary, The conversation discusses the research being conducted at a university involving an experiment on the optical properties of graphene. To measure a tiny change in angle, a step motor attached to a threaded rod is used, with a winged nut holding wires that connect to a laser and analyzer. The goal is to find an equation that accurately measures the height of the winged nut as a function of the desired angle of incidence, which can only vary between 90 and 30 degrees. The diagram provided shows the relevant measurements and highlights the Law of Sines as a key component in the derivation process. The thread is also mentioned to be posted in the appropriate location and apologies are given for the thread title.
  • #1
Legaldose
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6
Hey everyone, I'm currently doing research at a University, I've been working on a problem for a few hours and wrote up a quick paper that shows my derivation of a certain height based on an angle. Basically the experiment is looking at optical properties of graphene, but for this to happen we need to shine a laser onto the species, where it reflects back into an analyzer. To get a measurement from a tiny change in angle, we will use a step motor attached to a threaded rod, and at the top of the rod somewhere is a winged nut that holds a wire attached on both sides, those wires run down to the laser and analyzer on each side. I wanted to find an equation that accurately measures the height of the winged nut above the machine as a function of the desired angle of incidence. The angle can only vary between 90 and 30 degrees. I will attach a couple of pictures to the machine, as well as my derivation, which includes a crudely drawn picture of the proposed step motor system. If anyone could go over my derivation it would be much appreciated. I already sent it to the PI to be looked over, but I just want to make sure this is correct, and what better way to do that than give it to you guys :p.

Also, I don't know exactly where this thread should go, so feel free to move it where it needs to go.

Double also, sorry for the thread title, it's 2:30am.
 

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  • #2
From the diagram, the pertinent measurements are L and R2 (which are fixed) and y which varies as the step motor is turned. The angle of interest (call it θ) is the angle opposite side L of the triangle formed by lengths L, R2, and y.
By the Law of Sines, the ratio
L/sinθ = D (the diameter of the circumcircle about triangle yLR2)
D = yLR2/2A where A = area of triangle yLR2 = √(s(s-y)(s-L)(s-R2))
and semiperimeter s = (y+L+R2)/2
from all of which we can get
θ = arcsin(2/yR2)A
 

1. What is the purpose of deriving the height function given an angle?

The purpose of deriving the height function given an angle is to determine the height of an object or location by using the angle at which it is viewed. This can be useful in various applications such as navigation, surveying, and construction.

2. How is the height function derived from an angle?

The height function is derived using basic trigonometric principles. By knowing the angle of elevation or depression and the distance from the object or location, the height can be calculated using the tangent function: height = distance * tan(angle).

3. What is the difference between the angle of elevation and the angle of depression?

The angle of elevation is the angle formed between the horizontal line and the line of sight when looking up at an object or location. The angle of depression, on the other hand, is the angle formed between the horizontal line and the line of sight when looking down at an object or location.

4. Can the height function be used for any angle of elevation or depression?

Yes, the height function can be used for any angle of elevation or depression as long as the angle and distance are known. However, for angles close to 90 degrees, the margin of error may be higher due to the tangent function approaching infinity.

5. Are there any limitations to using the height function?

One limitation of using the height function is that it assumes a flat surface and does not take into account any changes in elevation. This may result in slight inaccuracies in the calculated height. Additionally, the height function may not be applicable in situations where the object or location is obstructed or not visible from the viewing point.

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