Physics in the playground II: The tree

There's a tree standing in the Walnut Hills Park playground ( one of the playgrounds, that is). I think it's some kind of fruit tree but there are no leaves on it and I'm not a tree identification expert. Here's the tree.

The real reason I did this was the new app I downloaded. Dioptra is a software version of my surveyor's compass. It gives me all the angles utilizing the accelerometers and the magnetometers in my phone.

We looked at the accelerometers last time.

Phones that have  built-in magnetometers use them for an onboard compass. Apps like Science Journal can use them to directly measure magnetic and electric fields.

Magnetometers in phones usually sense changes in conductivity in tiny electronic devices when they are exposed to a magnetic field. 

The phone has three magnetometers in one and they're aligned, just like the accelerometers along the x, y, and z planes, so the phone can sense the direction of a magnetic field.

Here's what the Dioptra screen looks like.

I aim with the crosshairs and the scales tell me directions. The curved scale on the left is a level. It tells me how much the phone is tilting. The data above it gives me my position in space and time: date and time, latitude and longitude, altitude, and error (CEP stands for circular error probable. For those in statistics, it's a 50% confidence interval for measurements within a circle.) I can set the app for different units.

The scale at the bottom is the compass reading for line-of-sight and the numbers above it makes it explicit. Those are the azimuth and bearings. Azimuth is the degrees of the angle between line-of-sight and due north. The bearing is the same in compass directions.

The scale to the right is the incline of the sight-line to horizontal. So, here's my sight-line to the tree top.

The phone is 4 degrees off level...ah, who cares. The location data is not important to what I'm doing, so I'll ignore it.  The GPS data could be as much as 30 feet off. That won't affect the accelerometers. My line-of-sight, just in case you're interested, is nearly southwest, 232 degrees from North and 52 degrees southwest,

but that won't play in our calculations either.

What I need is the inclination, which is 43.6 degrees from horizontal. You might have noticed that I have a tape measure in the first photo above. I was 20 feet from the base of the tree when I shot this line. That means that I have a right triangle with a base (adjacent side to my position) of 20 feet and the angle where I stood was 43.6 degrees. What I want is the length of the opposite side (height of tree). I can get that in several ways but I think I'll go with tan(angle)=opposite side/adjacent side. That shuffles around to opposite side=tan(angle) times adjacent side, or the length I want is the tangent of 43.6 degrees times 20 feet.

The height of the tree above my eye level is 19.05 feet. My eye level is 5.5 feet off the ground so that makes the tree 24.5 feet high.

I did the calculations on my MC50 calculator app.

The Dioptra app is put out by Workshop512 and the MC50 calculator by Walter Stubbs.

It's pretty impressive what phone apps will do. If you need your phone to do something that it doesn't already do, check your app store. Check out the rating, reviews, and app description before downloading. Advertisements are not usually a big problem but some of these apps are clogged up with ads. Some of the apps have a price and some are free but have a better version that you can buy. The prices are usually very reasonable.

Next time, I send a phone down a spiral ramp on a crash course.

Although the Dioptra app does everything my surveyor's compass does, I'm not ready to get rid of the compass. In the first place, it's very small. Also, it will never need a battery change or charge.

--- How big is Creation Rock 1 ---

                                                    [Creation Rock from Morrison]

Surveying Creation Rock at Red Rocks....this adventure requires some planning.

I'll be using a surveyor's compass, a relatively primitive piece of equipment, a 100 foot length of cord (which I have to measure out before I go), and some surveyor flags and stakes.... and trigonometry.

I've done this kind of surveying before. In Alabama, I conceived the notion of determining the heights of waterfalls in Alabama, and did survey a couple before I realized that the number of waterfalls in Alabama would require several lifetimes. The northernmost is on the Tennessee border and the most southern is just about 20 miles north of Mobile in Mount Vernon.

One that I did survey was Falling Rock Falls near Montevallo. I hiked there with a friend, Dr. Gregory Reece, author of several books including Elvis Religion, UFO Religion, Weird Science, and Creatures of the Night. His family and a friend went with us.

                                                                [Falling Rock Falls]

This waterfall is notorious due to the number of people who have fallen from it. It's close to 90 feet tall (I calculated it to be 87 feet). The name says it all. The lip is unstable and rocks come lose from it frequently. If a person is standing on the rock....well....

So you shouldn't try to just drop a line over the edge to measure the height (although it is a popular rappelling area). I played it safe and used surveying techniques to measure the height from the bottom. In fact, I used the same surveyor's compass I plan to use at Red Rocks.

At Falling Rock, the trick was not difficult. Water falls straight down, so I had a vertical distance to measure that made a right angle with the ground. Right triangles are easy with trigonometry.

The surveyor's compass includes an inclinometer that let me measure an angle of incline from where I stood near the base of the falls up to the top. Other than the right triangle between the water and the ground, that gave me one of the other angles I needed (labeled alpha in the following diagram).

                                             [Trigonometry of Falling Rock Falls]

I needed at least one length of a leg of the triangle to figure out the rest of the triangle. The distance from me to the water would have been nice, but I would have gotten very wet trying to measure that. I could use another triangle. I struck off a line at 90 degrees from the line to the waterfall and measured 20 feet. From the end of that line, I measured the angle back to the waterfall (call that angle beta).

Now, I had all I needed.

A trigonometric function is constant for a particular angle. No matter the size of a triangle, the sine of a given angle will always be the same. The sine of a 30 degree angle, regardless of what the two sides are, is 0.5.

The longest of the sides of a right triangle is called it's hypotenuse. The other side connected to the angle is called the adjacent angle. The side opposite the angle is called the opposite side (duh!).

So, the sine of an angle is the ratio of the opposite side length to the length of the hypotenuse. I would be using another trigonometric function called the tangent, which is the ratio of the length of the opposite side to the length of the adjacent side.

I've lost my original data, but I think you can see that, having the 20 feet between my two survey points (the adjacent side) and the angle beta, I could figure out my distance from the waterfall (the opposite side). Then, knowing the distance from the waterfall and the angle of inclination, I could use the tangent again to figure out the height of the waterfall.

The point is that I can't do that with Creation Rock. I can't sight any vertical to the summit. If water was falling straight down from there, I could sight to the bottom of the waterfall but Creation Rock is dry as....well, rock.

So, what am I to do?

There is a technique used by weather watchers to determine the height of clouds (again, no vertical line to the cloud is available) that uses some of the more advanced formulas of trigonometry. A point on the cloud has to be sighted by two surveyors at the same time (clouds move). You don't have to have a right triangle to use the law of cosines and the law of sines. They apply to any triangle.

So, if I could make a sighting from one point and a point 100 feet away (the base of the triangle I'm forming is 100 feet long), then I would have two angles and the side between - enough to figure out the other angles and sides of the triangle. Here's a diagram.

                                                          [Surveying Creation Rock]

Angle G is the angle of EGH. Angle H is the angle EHG. I know the length of the base of the triangle, side e. I should be able to determine the angle E because the sum of all the angles in a triangle is 180 degrees. Knowing all the angles and one side, I can easily determine the length h or g. The law of sines states that:

                                                                  [Law of sines]

I know G, e, and E, so I can figure out the distance from my first sight to the summit, but that's not the vertical distance from the summit to the ground, so I have more work to do. If I measure the angle of inclination from the first sight up to the summit, I will have a right triangle - in the diagram, GEF (F is the point directly under the summit). I know h, the length of the hypotenuse of the right triangle,  the length of the adjacent side to angle G, and the angle G. The sine of G is the length of the opposite side (the distance I want) over the hypotenuse (h). So, I can find the height of Creation Rock.

That's all pretty jumbled but I'll go through the calculations in How Big is Creation Rock 2 and it should make more sense.

So, I will need to measure the angle of inclination at one point (G), the angle from the summit to that point to a point 100 feet away, and the angle from the summit to the second point back to the first point.

The surveyors that determine property lines, areas of lots, paths for roadways and power lines use an array of complicated equipment but, essentially, they measure angles and distances and they use trigonometry to figure out everything else. The tools they use are just elaborations on the compass, which measures angles in respect to the earth's magnetic field, and a cord stretched between two stakes. The elaborations improve the convenience of the tools in the field and their precision.

Here is my compass, a Konustar surveyor's compass that I got for less than thirty dollars.

                                                                    [My compass]

The compass would be recognized by any Boy Scout as the navigation tool that aligns itself with the earth's magnetic field to point toward the magnetic north and south poles. The dials around the compass mark off the 360 degrees of the circle. The reticle and sights help to line up land features and surveying markers with the compass.

This compass has an added bubble level to indicate when it's....well, level, and an inclinometer which can measure angles of sight upward or downward. The inclinometer is just a little pendulum that drops straight down to show the direction to the center of the earth. The scale, then measures the tilt of the pendulum to the vertical.

The eyepiece is a cool addition. It focuses on the edge of the compass disk, which is graduated from 0 to 360 degrees, 0 degrees indicating a north bearing. That means that, once I get a sighting, I can drop my eye a little to get it's bearing in relation to north. Here's what it looks like.

                                                   [What I see in the compass eyepiece]

For a right triangle, all I need is one angle (other than the 90 degree angle) and one side length to figure out the lengths of all the other sides and the measure of the other angle. For all other triangles, I just need two angles and a side. That's the core of surveying. How do I measure distance at Red Rocks?

I have a 100 foot length of cord.

I placed two stakes in my yard 20 yards across (as measured by a tape measure) and wrapped the cord five times around it.

I suspect that my calculated height of Creation Rock will be somewhat off the known height because real surveyors figured that out using much more precise equipment than what I will use. There are several places where imprecision enters my measurements and calculations. Can you see some of them?

Surveying is the way of measuring inaccessible measurements such as widths of rivers or the distance between the rims of a canyon, heights, or outlines of a plot of land that is longer than a tape measure. Even though the surveyor's compass and tape measure are not nearly as precise as the tools that a professional surveyor uses, they are precise enough for many uses. You might want to buy a set and play around with them. Neither the compass or tape are expensive.