Correcting Sextant Measurements For Dip

Posted on 21-October-2015 by mathscinotes

Quote of the Day

Patience is also a form of action.

— Auguste Rodin, Sculptor

Introduction

Figure 1: Sextant is Used by Navigators To Make Celestial Observations.

Figure 1: Sextant is Used by
Navigators To Make Celestial
Observations (Source).

I love to read stories of the sea and about the voyages made during the age of sail. I personally have never thought that I would have an opportunity for ocean sailing, but I recently began working with an engineer who is an avid sailor and teacher of sailing. He has sailed all over the world and recently trained another engineer in my group to sail. This newly trained sailor just returned from a trip to Bora Bora, which he found to be enjoyable and the sailing uneventful.

My discussions with this sailing instructor have given me some hope that I may yet get to do some sailing. This hope has reawakened my interest in some of the traditional technologies of sailing, like ropework and celestial navigation.

This post is the first in a series on correcting sextant (Figure 1) measurements for impairments. There are a number of measurements impairments: index error, refraction, and dip. This post will look at the impairment known as dip.

Background

Definitions

altitude
The angular distance of a celestial object above an observer's horizontal plane . There are published tables of the altitudes of various celestial objects. However, it is difficult to obtain a stable horizontal reference on a moving ship. The horizon provides a very stable reference and we can use a sextant to accurately measure the altitudes of celestial objects with respect to the horizon. Figure 2 shows the relationship between the horizontal and the horizon. The are related through the angle called dip.

Figure 2: Illustration of Dip With Respect to Horizontal.

Figure 2: Illustration of Dip With Respect to Horizontal (source).

dip of the horizon
Dip of the horizon is the the angular depression of the horizon below the horizontal plane. If we were taking sights on a stable land site, we could measure altitudes using  a theodolite with its horizontal established using a bubble level. At sea, however, nothing is stable except the horizon. We use a sextant at sea to measure the altitude of a celestial object with respect to the horizon and then use our dip calculation to change the altitude reference from the horizon to the horizontal.
dip short of the horizon
There are times when it may be necessary to measure the altitude of a celestial object even though we do not have a visible horizon because of an obstruction. However, if we the distance to the obstruction we can calculate a dip value that will allow us to correct our sextant measurement. We refer to this dip value as the dip short of the horizon. Typically, we will measure the altitude relative to the water level at the obstruction, which is a relatively stable reference.

Objective

I will be deriving commonly used formulas for dip to the horizon and dip short of the horizon. I should point out that I have seen many formulas for these dip values and I arbitrarily picked two formulas to investigate more closely here.

Analysis

Standard Dip Short of the Horizon Table and Formula

Common Presentation

Figure 3 shows the values for the dip short of the horizon using a common table (Norie's) and formula (Bowditch). They give similar, though not identical results. In Figure 3, I show how you can take Norie's tabular data, fit it to a Bowditch-like formula, and you find that Norie's table and Bowditch's formula produce similar results.

I should note that the results for dip short the horizon used in both Norie and Bowditch ignore refraction. This makes sense because most obstructions used for navigation purposes would be relatively near, and the effects of refraction will be minimal.

Figure M: Dip Table and Formula.

Figure 3: Dip Short of Horizon Table and Formula.

Figure 3 also shows a snippet of Bowditch's formula dip short of the horizon, which is given by Equation 1.

Eq. 1 \displaystyle d_\text{short}=0.42857\cdot d+0.56514\cdot \frac{h}{d}

where

  • d is the distance to the obstruction [nautical miles].
  • h is the height above sea level of the observer's eye [feet].
  • dshort is the dip short of the horizon [arcmin].

Derivation

Figure 4 shows the geometry of the dip short of the horizon situation.

Figure 4: Dip Short of Horizon Measurement Scenario.

Figure 4: Dip Short of Horizon Measurement Scenario.

Norie's Table Reference Voyager 1 current status

Figure 5 shows how you can derive an expression very similar to that of Bowditch's.

Figure 5: Derivation of Bowditch's Dip Short of Horizon Formula.

Figure 5: Derivation of Bowditch's Dip Short of Horizon Formula.

Amount of Earth's Curve

While the expression shown in Figure 5 is not identical to that in Bowditch, it is quite close.

Dip to the Horizon Formula

The dip to the horizon formula is shown in Figure 6 (source).

Figure 6: Commonly Used Dip of the Horizon Formula.

Figure 6: Commonly Used Dip of the Horizon Formula.

The derivation of this formula is similar to the dip short of horizon formula, but now we will model refraction. For the dip to the horizon, I have only seen formulas that are functions of the square root of the observer's height h.

Equation 2 shows a common expression for dip of the horizon, dhorizon.

Eq. 2 \displaystyle d_{horizon}=1.932\text{ arcsec}\cdot \sqrt{h}

Figure 8 shows how you can derive Equation 2.

Figure 8: Derivation of Dip to Horizon formula.

Figure 8: Derivation of Dip to Horizon formula.

distance to horizon Amount of Earth Curvature

My derivation yielded a function similar in form to the that shown in Figure 6, but with a larger coefficient term. The coefficient term is a function of the atmospheric conditions, and I assume that my atmospheric assumptions are different. The results differ by ~10%, which is well within the bounds of atmospheric variation.

Conclusion

I believe that I understand where the dip tables and formulas come from. I also understand that some dip equations take refraction into account (Figure 7) and some do not (Figure 3). Dip was a good impairment to use to kickoff my learning process – next are index error and refraction.

Posted in Naval History, Navigation | 8 Comments

My Phone Stops a Road Rage Incident

Posted on 15-October-2015 by mathscinotes

Quote of the Day

Remember to look up at the stars and not down at your feet. Try to make sense of what you see and wonder about what makes the universe exist. Be curious. And however difficult life may seem, there is always something you can do and succeed at. It matters that you don't just give up.

- Stephen Hawking

Figure 1: I Do Not Understand Road Rage.

Figure 1: I Do Not Understand Road Rage (Source).

Road rage is everywhere these days. It's surprising how easily people can come to trading blows or declaring DEFCON 1 over minor car collisions (and I mean minor, at most I'd need to find a service that does dent repair near me in Denver to buff it out). I don't know if it's something in the water, or if the ensuing fight is the result of someone's really bad day coming to a head. The world may never know. I witnessed one such road rage argument a while ago and were it not for my "intervention" (I barely did anything to be honest) it could have blown into a full fight.

Every night I walk a 6.4-mile path around the Lake in Maple Grove, the epitome of an upper-middle-class suburb. While on the home-bound leg of my walk, I saw two men in two cars jousting in the parking lot of the Maple Grove Applebee's. One man, about 45 years old, soon stopped his car, rolled his windows up, and immediately made a phone call – I later confirmed he was calling the police. The other man, about 35 years old, got out of his car and began pacing outside the car of the man who was calling the police. The pacing man was having a total meltdown – yelling, finger-pointing, veins sticking out – I will refer to him as "the Brute." The man inside the car looked terrified. The Brute was outside his car looking very threatening.

I saw what was occurring and decided to place myself between a couple of hedges. I then did what every red-blooded American male would do -- I started filming the incident with my phone. The Brute did not notice me for about a minute and continued to throw a very impressive display of anger.

From inside the car, the 45ish man then pointed me out to the Brute. The Brute became even more enraged and marched over to me; stopping about 10 feet away. I braced for an attack. Instead, he screamed at me saying "If you can hide in the bushes and film me, I can film you!" He then pulled out his phone and began filming me. At this point, we had an odd form of standoff. I am still laughing at the image of two grown men filming each other -- High Noon it was not. The standoff lasted about five seconds, and the Brute decided that he should call the police himself. My film at this point is dull because the Brute is just on the phone to the police.

While the Brute was calling the police, the frightened man in the car rolled down has a window and thanked me for staying. He also said that he had called the police, and they would arrive in about a minute.

After about a minute, a Highway Patrol officer came and separated the two men. He asked me if I had any direct involvement in the incident. I said no and that I was going home -- which I immediately did. There was no visible damage to the cars, no one had been hurt, and the police officer just wanted to get everyone on their way.

I am certain that I stopped a man from getting beaten – all because I was standing in the bushes holding a phone. The presence of the phone made the Brute reconsider what he was doing.

Tonight, my phone and I will again walk the mean streets of Maple Grove.

POST SCRIPT

Here is the response to this post that I received from one of the engineers in my group.

I know what you're thinking. "Did he film six seconds or only five?" Well, to tell you the truth, in all this excitement I kind of lost track myself. But being as this is a smart phone, the most powerful phone in the world, and would film your head clean off, you've got to ask yourself one question: "Do I feel lucky?" Well, do ya, punk?

Posted in Personal | Comments Off on My Phone Stops a Road Rage Incident

Rogue Software Engineers and Responsibility, Accountability, and Authority

Posted on 9-October-2015 by mathscinotes

Quote of the Day

Plans are merely a platform for change.

— Israeli Defense Forces

Figure 1: Old-Time VW Beetle.

Figure 1: Old-Time VW Beetle (Wikipedia). I loved working on this car.

I have always liked Volkswagen (VW) cars – I actually rebuilt a Beetle engine during a shop class in high-school (Figure 1). My respect for VW took a serious downturn this morning when read the following headline,"Top U.S. VW Exec Blames 'A Couple of Software Engineers' for Scandal".

Give me a break. There is no way that a couple of rogue software engineers did this on their own – major technology corporations have processes that provide checks and balances against this kind of behavior. Someone in management had to know about this. Don't they have code reviews?  Who did the test and evaluation of the emissions system? Were all emissions test done on a dynamometer that the software was designed to deceive? Who did the systems engineering that set the emissions system requirements – these folks usually set the test requirements as well?

I see VW management trying to claim that they should not be held accountable for something they did not know about. I would argue that it is management's  responsibility to create a culture where integrity is a key value, and VW failed in this regard. They also need to be held accountable for the processes that failed to catch this deception.

I have always tried to understand the relationship between the words

  • Accountability
  • Responsibility
  • Authority

I believe that if I am going to be held accountable for a project,  then I must be informed of my responsibility and given the authority to work the problem. However, I have frequently seen people held accountable for problems that they lacked either the responsibility or authority to deal with. Many years ago, I had a manager who was particularly bad in this regard. He frequently would punish someone for a problem that they were not primarily responsible. We used the phrase "Bring Me the Head of Willie the Mailboy" to describe when he made an example of an innocent person.

In my youth, I worked for a defense contractor. While working for this contractor, I was surprised when I heard a software manager introduce himself as "the designated scapegoat". He turned out to be correct, and he was blamed for that program's problems. In reality, that program's problems had more to do with his staff being allotted inadequate time and resources for the defined software tasks, which was an upper management decision.

I hope in the end the correct people are held accountable, but I would not count on it.

 

Posted in Management | 2 Comments

Samsung S5 Field of View

Posted on 5-October-2015 by mathscinotes

Quote of the Day

I went through the whole gamut of organization, which was a vain effort so to say, because in the end, I ended with electronics, which have no life at all. But I don't think it vain because to understand life, one must understand electrons too.

— Albert Szent-Györgyi, Nobel Prize winner in Medicine. His research focused on the role of electrons in biological processes.

Introduction

Figure 1: Samsung S5 Android Phone.

Figure 1: Samsung S5 Android Phone (source).

I use a Samsung S5 phone for my daily smart phone work. Recently, I have even started to use its camera/video system for some rough measurement work (examples here and here). This work has made me a bit curious about the how the camera subsystem was designed. In this post, I will document a very rough experiment that I performed over lunch today in which I measured the S5 camera's Field of View (FOV). I will also compute the field of view using some information that Samsung provides. The agreement between the results seem reasonable considering my crude approach.

I am always surprised at the quality of the images that I can get from this phone, however, there are serious optical compromises required to put a camera into this phone's tiny form factor. While I prefer to use my DSLR, I rarely have it with me when I need to take a quick photo – most of my photos are of whiteboard discussions. In these situations, any photo is better than trying to copy the board quickly.

Background

Definitions

As usual, I will use the Wikipedia as my source for the definitions of terms.

Field of View (symbol FOV)
In photography, FOV describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the term Angle of View (AOV).
Crop Factor (symbol CF)
In digital photography, a crop factor is related to the ratio of the dimensions of a camera's imaging area compared to a reference format; most often, this term is applied to digital cameras, relative to 35 mm film format as a reference.
In the case of digital cameras, the imaging device would be a digital sensor. The crop factor is also commonly referred to as the Focal Length Multiplier ("FLM") since multiplying a len's focal length by the crop factor or FLM gives the focal length of a lens that would yield the same field of view if used on the reference format.
Focal Length
The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus. I will not be using the phone camera's focal length because Samsung has not published that number. Instead, I will use the equivalent 35 mm focal length, which they do publish.
Equivalent Focal Length (Symbol f)
The focal length of a phone camera is usually stated in terms of an equivalent 35 mm focal length. The equivalent focal length is the focal length in a 35 mm SLR that a would provide the same FOV as the camera's lens/sensor system.

S5 Technical Characteristics

In Table 1, I gathered the following S5 technical information from this web page. I will use this information to compute the S5's FOV (horizontal and vertical) based on the lens and image sensor dimensions.

Table 1: Samsung S5 Critical Optical Subsystem Parameter.
Parameter Value
Effective Focal Length 31 mm
Image Sensor Horizontal Dimension 5.08 mm
Image Sensor Vertical Dimension 3.81 mm

Note that the image sensor dimensions have a ratio (horizontal/vertical) of 16/9 ≈ 1.78. This is the aspect ratio seen with HDTV and is different than the 3/2 ratio used in standard 35 mm film format.

35 mm Film Reference

Figure 2: 35 mm Film Reference.

Figure 2: 35 mm Film Reference (Source).

Focal length is a characteristic of the camera's lens. The S5's focal length of 31 mm is always described as "effective", which means that the lens has a focal length provides an image similar to that of a 31 mm lens on a 35 mm analog camera. As a point of reference, 35 mm analog cameras often had a 50 mm focal length lens  because this gave the camera an FOV similar to the human eye.

I find the use of effective focal length confusing – a focal length of 31 mm ≈ 1.25 inches is much larger than the S5 is thick. I was unable to find a drawing of the optical path for an S5, but you can get a feel for the complexity present in a cell phone camera by looking at a typical reference design I found while searching on the web (Figure 3). This lens assembly contains one glass and two plastic lenses.

Figure 3: Typical Cell Phone Camera Optical Path (Source).

Figure 3: Typical Cell Phone Camera Optical Path (Source).

Figure M: 35 mm Format Film.

Figure 4: 35 mm Format Film.

35 mm film (Figure 4) has a much larger image gathering area than a phone camera's image sensor. For example, the horizontal dimension of 35 mm film is 36 mm, but the corresponding dimension of the S5's image sensor is 5.08 mm. This means that the S5 will only capture the central portion of the image (Figure 5). The overall aspect ratio of 35 mm film is 3/2 (i.e. 36 mm/ 24 mm). Modern aspect ratios (e.g. HDTV, DSLRs) are moving to 4/3 or even larger – my Sony α55 DSLR is configurable for either 4/3 and 16/9.

Figure 4: Demonstration of the Effect of Crop Factor.

Figure 5: Demonstration of the Effect of Crop Factor (Source).

Unfortunately, Figure 5 does not show an image sensor the same size as the S5, which is 1/5" wide (horizontal). This means that the S5 sensor area would comfortably fit within the 1/3" box. This also means that the image captured by the S5 is only a tiny portion of the image that a 35 mm analog camera would capture.

Many people choose to view the cropping that occurs as a magnification because the image does look magnified. You can view what is  happening any way you want. But you need to remember that all of your sensor's pixels are in that small box in the center of the image – the rest of the image area is unused.

This becomes important in DSLR cameras when you use a lens designed for an 35 mm analog system with an image sensor with a smaller capture area than 35 mm film.

FOV Formula

The Wikipedia give Equation 1 as the formula for the field of view. The discussion there is excellent, and I will elaborate no further here on this formula.

Eq. 1 FOV =2\cdot \arctan \left( {\frac{d}{{2 \cdot f}}} \right)

where

  • FOV is the field of view along a given direction (e.g. horizontal or vertical)
  • d is the sensor length along a specific direction (e.g. horizontal or vertical)
  • f is the focal length of the lens.

Analysis

I will process the same data using two different methods and see how the results vary.

Theoretical FOV Calculation

Figure 6 shows my application of Equation 1 to compute the horizontal and vertical FOVs of the Samsung F5.

Figure 5: Using Standard FOV Equation to Compute S5 Horizontal and Vertical FOVs.

Figure 6: Using Standard FOV Equation to Compute S5 Horizontal and Vertical FOVs.

My Measurement of the S5's FOV

Angle Measurements

It is lunch time and I took a couple of minutes (literally) to take photographs from known distance to a letter-sized sheet of paper at known distances that I taped to the wall of my cube. I then used a generic graphic program (PicPick) to measure the dimensions (in pixels) of the letter-sized sheet of paper as seen in the photographs. Figure 7 shows two photograph examples.

24inches 20151002_112022
Figure 7(a): Sheet of Letter-Sized Paper at 24 inches. Figure 7(b): Sheet of Letter Sized Paper at 48 inches.

Note that my measurements might not be very accurate because my camera hold at the various distances was only approximate and I was not holding the camera's line-of-sight perfectly perpendicular to the paper.

My raw measurements are documented in the Mathcad screenshots included below.

Average of FOV Calculations

This approach is related to that used in this blog post. Equation 2 shows a modified version of Equation 1 that is a bit more useful for this type of testing.

Eq. 2 \displaystyle FOV=2\cdot \arctan \left( {\frac{{{{d}_{{ImagePixel}}}\cdot {{d}_{{Object}}}}}{{2\cdot {{r}_{{Object}}}\cdot {{d}_{{ObjectPixel}}}}}} \right)

where

    • dImagePixel is the image size measured in pixels.
    • dObjectPixel is the object size (e.g. sheet of paper) in pixels.
    • dObject is the object size in standard linear units (e.g. inches, mm).

Figure 8 shows how to derive Equation 2.

Figure 7: Derivation of FOV Formula.

Figure 8: Derivation of FOV Formula.

Figure 9 shows my application of Equation 2 to multiple data inputs and averaging the results.

Figure 7: FOV Estimates By Averaging Multiple Measurements.

Figure 9: FOV Estimates By Averaging Multiple Measurements.

Least Squares to FOV Curve

A related approach is to assume that the object range is long enough that we can ignore triangles and use spherical measurements (i.e. FOV = dObject/rObject). This approach is very similar that used in this blog post.

Equation 4 shows the how the ratio between the paper and image size varies with 1/rObject.

Eq. 3 \displaystyle k=\frac{{{{d}_{{ObjectPixel}}}}}{{{{d}_{{ImagePixel}}}}}=\frac{{\frac{{{{d}_{{Object}}}}}{{{{r}_{{Object}}}}}}}{{FOV}}=\frac{{{{d}_{{Object}}}}}{{FOV}}\cdot \frac{1}{{{{r}_{{Object}}}}}

where

  • k is the ratio of the object size to the image size.

The data of Figure 10 will plot linearly versus 1/r, which allows us to generate a least-square linear fit – I am specifically interested in the slope m. The slope of the line, m = \frac{d_{Object}}{FOV}. Since we know the size of the sheet of paper, we can compute the FOV. The results obtained are similar to the previous two methods, at least considering how crudely I took the measurements.

Figure 10: Least Squares Analysis of the FOV Data.

Figure 10: Least Squares Analysis of the FOV Data.

Conclusion

All three methods produced good agreement for the vertical FOV (37°). There was a slight discrepancy between the horizontal FOV values (60° versus 67°). I do not consider this discrepancy significant considering how crudely I took the measurement, especially in the horizontal dimension (i.e. my camera position was not exactly perpendicular to the sheet of paper).

I am amazed at how well these cell phone camera's perform, particularly when you consider how inexpensive they must be. I plan on buying an Arduino shield camera (example) with cameras to do some experimenting on my own.

Posted in optics | 5 Comments

My Sister in Math Class

Posted on 30-September-2015 by mathscinotes

Quote of the Day

While an open mind is priceless, it is priceless only when its owner has the courage to make a final decision which closes the mind for action after the process of viewing all sides of the question has been completed. Failure to make a decision after due consideration of all the facts will quickly brand a man as unfit for a position of responsibility. Not all of your decisions will be correct. None of us is perfect. But if you get into the habit of making decisions, experience will develop your judgment to a point where more and more of your decisions will be right. After all, it is better to be right 51% of the time and get something done, than it is to get nothing done because you fear to reach a decision.

— H. W. Andrews

My sister posted this video on her Facebook page while stating it described her in math class. There is some truth to this statement. It could also have been used to describe any of her four brothers in nearly every classroom or church situation.

Posted in Humor | 5 Comments

Mass-Energy Conversion Example

Posted on 20-September-2015 by mathscinotes

Quote of the Day

Fathers, like mothers, are not born. Men grow into fathers and fathering is a very important stage in their development.

— David Gottesman. His statement was true for me.

Introduction

Figure 1: Photograph of Little Boy Bomb Casing.

Figure 1: Photograph of Little Boy Bomb Casing.

Since 2015 is the 70th anniversary of the end of World War 2 (WW2), C-SPAN has been running a number of oral history interviews with people who worked on the Manhattan Project. I have found these interviews very interesting. You can find them on YouTube and watch them for yourself.

After watching one interview with a worker from the Y12 enrichment plant, I decide to look for some additional background material on the work done on the Manhattan Project. It was during this research that I watched a video in which engineer made the statement that only 900 grams of {}_{{92}}^{{235}}U underwent fission in the Little Boy bomb. I thought this was an interesting number that I could show is consistent with the reported amount of energy released.

Background

The following video states that 900 grams out of 60 kg of {}_{{92}}^{{235}}U underwent fission in the Little Boy bomb. It was this statement that got me thinking about where the energy of these weapons come from – the binding energy that holds the uranium nucleus together.

Analysis

Figure 2 shows my calculation for the energy released by splitting 900 grams of {}_{{92}}^{{235}}U. Most sources put the yield of the Little Boy device in the 16 kiloton range.

Figure 2: Energy Released From 900 Grams of {}_{{92}}^{{235}}U.

Elementary Charge Avogadro's Number Electron-Volt TNT Equivalent Fission Energy

The video states that 900 grams of the 60 kilograms of {}_{{92}}^{{235}}U underwent fission, which means about 1.5% of the fissile material was consumed. That is a very tiny amount of mass for a huge amount of energy.

It is useful to compare the energy released from the fission of single {}_{{92}}^{{235}}U nucleus to that of the explosion of a single molecule of TNT (Figure 3). Observe that a single TNT molecule exploding is ~20 million times less energetic than a single {}_{{92}}^{{235}}U fission.

Figure 3: Relative Energy Release of a {}_{{92}}^{{235}}U Fission and TNT Reaction.

I should note that you will find a number of values listed for TNT's energy release – the value varies by how you define the release characteristics. For example, while the heat of combustion is listed as 14.5E6 Joules per kilogram, the energy released in the act of exploding is listed as 4.184E6 Joules per kilogram.

Conclusion

It is hard to believe the amount of energy that is released by splitting atoms. An even more unimaginable number is the amount of energy released by the complete annihilation of matter, such as with matter/antimatter interactions.

Save

Posted in General Science, History of Science and Technology, Military History | Comments Off on Mass-Energy Conversion Example

Coal Train Daydreams

Posted on 19-September-2015 by mathscinotes

Quote of the Day

Always do sober what you said you'd do drunk. That will teach you to keep your mouth shut.

— Ernest Hemingway

Introduction

Figure 1: Sherco Coal-Fired Power Plant in Minnesota.

Figure 1: Sherco Coal-Fired Power Plant in Minnesota (Wikipedia).

Because of where I work and live, I regularly wait at train-road crossings while coal trains pass in front of me. I get bored while sitting there and I start to think about the amount of coal that is being consumed by one of our local coal-fired power plant known as Sherburne County Generating Station or Sherco for short. I will present my estimates here and show that they agree with the numbers our local power producer is stating for this plant.

Background

Sherco Power Plant Trivia

We need to know two things about the Sherco plant to obtain some useful results:

  • Nameplate capacity of 2,129 MW (source).

    The nameplate capacity is the intended, sustained full-load capacity. No plants run at full capacity all the time.

  • The plant uses over 9 million tons of coal per year (source).

    I will assume an average daily consumption based 9 million tons divided by 365 days.

Coal Energy Density

The Sherco plant uses coal from the Powder River Basin, which straddles Montana and Wyoming. The coal from this region is referred to as sub-bituminous coal. On average, this coal generates 8500 BTU/lb of coal. This form of coal does not have have the energy density of other forms of coal (e.g. anthracite has an energy density of ~14000 BTU/lb). Sub-bituminous coal has the advantage of being plentiful, cheap, and relatively near.

Train Characteristics

Figure 2: Steel Coal Car (source).

Figure 2: Steel Coal Car (source).

Figure 2 shows the typical coal car that I seepass by me nearly every day. According to BNSF, each of these cars carries 102 tons of coal – I will assume that all of them are full.

A coal train typically has 115 of these coal cars – I have been so bored as to count these cars. If you want more details, see this document.

Analysis

Figure 3 shows my analysis of the number of train cars and trains needed per day to fuel the Sherco power plant. It needs between two and three trains of 115 cars each to supply enough coal to meet their stated specifications. My calculated range of daily train arrivals agrees with their written documentation.

Figure 3: My Estimate of the Number of Trains Per Day Into the Sherco Power Plant.

Figure 3: My Estimate of the Number of Trains Per Day Into the Sherco Power Plant.

Plant Capacity Utilization Thermal Output More Thermal Output Coal Per Coal Car Energy Per Pound of Coal

Conclusion

I can easily believe that between 2 and 3 coal trains service this power plant every day – I seem to wait for a train every five or six trips over the tracks.

 

Posted in General Science | Comments Off on Coal Train Daydreams

Circular Saw Depth-Of-Cut Formula

Posted on 15-September-2015 by mathscinotes

Do unto others 20% better than you would expect them to do unto you, to correct for subjective error.

— Linus Pauling, Nobel Prize winner in both Chemistry and Peace.

Introduction

Figure 1: Milwaukee 6.5 inch Battery-Powered Circular Saw (source).

Figure 1: Milwaukee M18, 6.5 inch Diameter,
Battery-Powered, Circular Saw (source).

I recently bought a battery powered, 6.5-inch diameter, circular saw from Milwaukee. I REALLY like this saw. I have been using it at my cabin in Northern Minnesota, a place where dragging around electrical cords is painful. This saw has quickly become one of my workhouse tools.

One initial concern I had with this saw had to do with the reduced depth of cut that I would get with a 6.5-inch diameter blade versus a 7.25-inch diameter blade. I decided to calculate a table of depth of cut values versus the angle of the saw blade. I will keep this on my phone so I always know my depth of cut.

It has turned out that the 6.5-inch blades more limited depth of cut has not been an issue at all. Overall, this is one of the best tool purchases that I have made.

Background

Figure 2 shows the manufacturer's specifications for the saw's depth of cut at 90° and 45°.

Figure 2: Manufacturer's Specification for the 6.5-in Depth of Cut.

Figure 2: Manufacturer's Specification for the 6.5-in Depth of Cut.

For comparison, I have included the depth of cut specifications for the 7.25-inch version of this saw. I prefer the 6.5-inch saw because it is significantly smaller and lighter. However, sometimes you need a bit more depth of cut and 7.25-inch is needed.

Figure 3: Depth of Cut for a 7.5-inch Version of This Saw.

Figure 3: Depth of Cut for a 7.5-inch Version of This Saw.

Analysis

Graphical View

Figure 4 shows the saw blade at three common angles: 90°, 60°, and 45°. The drawings also show the depth of cut. The depth of cut for the 90° and 45° cases agrees with the manufacturer specifications shown in Figure 1 – the 60° case was not specified by the manufacturer.

90deg 60deg 45deg
Figure 4(a): 90° Cut Angle. Figure 4(b): 60° Cut Angle. Figure 4(c): 45° Cut Angle.

Analysis

I used a bit of trigonometry to derive a formula for the depth of cut. I show this formula and my depth of cut table in Figure 5.

Figure 5: Table of Depth of Cut Values.

Figure 5: Table of Depth of Cut Values.

Conclusion

I now now my saw's depth of cut for a large number of possible cutting angles. I will keep this table on my phone so that I always have it near.

Posted in Construction, Geometry | 16 Comments

Tree Height Measuring Example

Posted on 10-September-2015 by mathscinotes

Quote of the Day

Your grades, whatever is your GPA, rapidly becomes irrelevant in your life. I cannot begin to impress upon you how irrelevant it becomes. Because in life, they aren’t going to ask you your GPA ....If a GPA means anything, it’s what you were in that moment — and it so does not define you for the rest of your life.

— Neil deGrasse Tyson, astrophysicist, during a commencement address at the University of Massachusetts Amherst. I often cite quotes like this to young people who have a grade problem. A low GPA may limit some options early on, but it should not be viewed as an insurmountable obstacle.

Introduction

Figure 1: Silver Maple, a Common Tree in My Area.

Figure 1: Silver Maple, a Common
Tree in My Area (source).

I have been testing a number of Android applications that are intended to measure the size of objects knowing their range or vice versa. One application that I have found particularly useful is called Baumhöhenmesser – Tree Height Meter (my translation) – which is an application written by a German developer. I have found this application particularly useful, and I thought I would review its operation here. It is part of a suite of Android applications intended for forestry management. This app makes excellent use of the Android's ability to measure angles.

My plan in this post is to present one of the formulas used in the application and illustrate this formula's use with an example. While the app is focused on measuring the height of trees, I have been using this application to measure the height of many objects. I do occasionally need to determine the height of a tree – usually just before I cut it down. I want to know the tree's height so that I can clear a spot on the ground of appropriate size.

Background

There are two formulas used in this application. My focus here will be on the formula used to measure the height of a tree using a fixed height reference that is placed against the tree and three angle measurements. Equation 1 shows the formula used in this measurement scenario.

Eq. 1 \displaystyle h=L\cdot \frac{{\tan \left( {{{\alpha }_{1}}} \right)-\tan \left( {{{\alpha }_{3}}} \right)}}{{\tan \left( {{{\alpha }_{2}}} \right)-\tan \left( {{{\alpha }_{3}}} \right)}}

where

  • h is the tree height
  • α1 angle from horizontal to the top of the height reference (counterclockwise positive)
  • α2 angle from horizontal to the top of the height reference (counterclockwise positive)
  • α3 angle from horizontal to the bottom of the tree (counterclockwise positive)
  • L is length of the height reference.

I show how to derive this formula in Figure 2.

Example

Figure 2 shows a simple example of how Equation 1 can be used to determine the height of a tree. In Figure 2, I define a range term R, but it will be used only as a temporary variable. The tree height is computed in terms of the reference length L.

Figure 2: Tree Height Measurement Example.

Figure 2: Tree Height Measurement Example.

Conclusion

Just a quick example illustrating how a useful height-measurement application works.

Posted in General Mathematics, Geometry | 3 Comments

Glacier Melting Math

Posted on 9-September-2015 by mathscinotes

I'm not a Star Trek writer, I'm a science fiction writer. I like building my own worlds more than share-cropping in someone else's.

— David Gerrold, well known science fiction author who has contributed to Star Trek (e.g. wrote "The Trouble with Tribbles" episode).

Introduction

Figure 1: Mendenhall Glacier (Photograph by Me).

Figure 1: Mendenhall Glacier (Photograph by Me).

My wife and I are currently on an Alaskan cruise with friends that used to be our neighbors when our children were young. Our cruise ship is the Millennium, which is part of the Celebrity fleet. We are currently moored in Juneau, where we visited the Mendenhall glacier. While at the glacier, I talked with a local Forest Service guide about the rate of glacier melting. I also made a few measurements using my phone and a bit of math then ensued, which I will discuss here.

Figure 2: Nugget Falls (Photograph by Me).

Figure 2: Nugget Falls (Photograph by Me).

I will be estimating the amount of glacial runoff that was occurring today and will estimate the total contribution of the glacial runoff to the flow of the Mendenhall River. Much of the glacial meltwater enters the Mendenhall River by way of Nugget Falls (Figure 2). I will estimate the contribution of glacial meltwater to the flow of the Mendenhall River by estimating the flow from Nugget Falls – I assume that all other glacial sources into the Mendenhall River are ignorable. I obtained the daily flow of the Mendenhall River from a Forest Service guide.

As usual, my calculations are rough, but do give a feel for the amount of melting that is occurring.  After I performed my calculations, I found a peer-reviewed journal article that presents similar data. I have included this article with this post.

Background

Basic Information

Here is what I learned from the Forest Service guide:

  • The Mendenhall River carries about 10 billion (x109) gallons of water every day from the Mendenhall glacier region (source: Forest Service guide).
  • Approximately 200 feet of glacier is melting away each year (source: Forest Service guide). I will not use this value in my calculations, but I record it here as being an interesting number.
  • Approximately 50% of the Mendenhall River flow is made up of melt water from the glacier during the peak of summer heat (source: peer-reviewed paper). I have included the abstract from this paper here.
  • Mendenhall Lake has not been there all that long. The Wikipedia states that it first formed in 1958 (source: Wikipedia). Note that some sources say it started forming in the 1930s – still not that long ago relative to the age of the glacier.
Figure 2: Mendenhall Glacier's Retreat versus Time.

Figure 3: Mendenhall Glacier's Retreat versus Time (source).

Figure 3 provides the best illustration that I could find of the extent and rate of the Mendenhall glacier's retreat. You can see that the rate of retreat is accelerating. For example, the amount of retreat from 1760 to 1832 is about equal to that from 1982 to 1996. In 2004, it retreated 200 m (656 feet).

A number of Alaska's glaciers are retreating – there are about 100,000 glaciers in Alaska (source). This could have long-term implications for a number of Alaskan cities that depend on the glacial meltwater for their freshwater supply.

Measurements

I have a couple of applications on my phone that helped me make a few measurements:

  • Mil Rangefinder: This application allows you to use your phone estimate the size of something given its distance from you or vice versa. With respect to Nugget Falls, I will use this app to measure the width of the falls.
  • Clinometer: This application allows you to measure the angle of an object. Given the inclination of an object and its distance from you, just a bit of trigonometry will give you its height. I will use this app to measure the height on the falls that I can use to measure the time of fall of water from an elevated point to Mendenhall Lake.
  • Stopwatch and Timer: I used this to estimate the time it took the water to fall a given distance.

Analysis

Figure 4 shows how I generated my estimate for the Nugget Falls flow rate and its contribution to the flow of the Mendenhall River. My results are consistent with one published estimate for the Nugget Falls flow rate – I could not find any other flow rate estimates.

Figure 3: Estimate of Nugget Falls Flow Rate.

Figure 4: Estimate of Nugget Falls Flow Rate.

Conclusion

My estimate for the flow of water from Nugget Falls is in the same range as an estimate given in a peer-reviewed journal article. Generating my flow rate estimate proved to be a good test of my phone's distance and size measuring applications.

These apps are not complicated, and I will blog about how some of them work in posts to follow.

Appendix A: Quote on Glacial Melt Percentage in Mendenhall River

The abstract from this peer-reviewed paper gave me a rough number for the glacial melt percentage of the Mendenhall River.

Mendenhall Glacier is a dynamic maritime glacier in southeast Alaska that is undergoing substantial recession and thinning. The terminus has retreated 3 km during the 20th century and the lower part of the glacier has thinned 200 m or more since 1909. Glacier-wide volume loss between 1948 and 2000 is estimated at 5.5 km3. Wastage has been the strongest in the glacier’s lower reaches, but the glacier has also thinned at higher elevations. The shrinkage of Mendenhall Glacier appears to be due primarily to surface melting and secondarily to lake calving. The change in the average rate of thinning on the lower glacier, >1 m a-1 [annum] between 1948 and 1982 and >2 m a-1 since 1982, agrees qualitatively with observed warming trends in the region. Mean annual temperatures in Juneau decreased slightly from 1947 to 1976; they then began to increase, leading to an overall warming of ~1.6 °C since 1943. Lake calving losses have periodically been a small but significant fraction of glacier ablation. The portion of the terminus that ends in the lake is becoming increasingly vulnerable to calving because of a deep pro-glacial lake basin. If current climatic trends persist, the glacier will continue to shrink and the terminus will recede onto land at a position about 500 m inland within one to two decades. The glacier and the meltwaters that flow from it are integral components of the Mendenhall Valley hydrologic system. Approximately 13% of the recent average annual discharge of the Mendenhall River is attributable to glacier shrinkage. Glacier melt contributes 50% of the total river discharge in summer.

Posted in Geology, Personal | 2 Comments