Angular Size Lab

Enough was enough. I couldn’t handle going through the chapters in the introductory astronomy course. It was too much material and it was too fast. But I’ve already complained about this in a previous post.

I didn’t have much time to prepare, but I decided to do a lab activity in class. My idea was to have the students build something to measure angular sizes. They could then use this device to measure the size (or distance) of various objects outside. I figured it would be fun to have them actually build things.

So, here is the plan. Step one is to go over the math of angular size. That includes the relationship between the circumference and radius for a circle.

$C = 2\pi R$

Another important idea is the relationship between the angle in degrees and radians. We need to measure angles in radians, so I also explained this relationship with 360 degrees equal to 2*Pi radians.

The next part was to build an angle measuring device. I’ve done the before in a physics lab, but I wanted something a little simpler. Here’s what I suggested to the students—make some small “flag” that you can hold at arm’s length (so that the distance from eye to flag is constant) and use this to measure angles.

I gave them popsicle sticks and sticky notes and a bunch of other stuff (with the hope that they would come up with their own design). In the end, they had something like this.

This is just a sticky note on a pencil.

The next step is to “calibrate” this instrument. Put your eye a known distance (say 5 meters) from a known length. I had the students use bricks in the wall or put tape on the wall a set distance apart. From this, they can calculate the angular size of the object and make markings on their device. Oh, this diagram might help.

The distance from the eye to the object is R and the length of the object is L. If the object is small compared to the distance, then this length is approximately the same as the arc length of that part of a circle. The value for theta can be determined by dividing L by R.

Now repeat this for another object so that you can turn the angle measuring device into something useful with multiple markings on it. Now we are ready to collect some data.

Here is a large light for the football stadium, you can see it right outside the classroom.

I used Google maps to get the distance to this object (it’s 240 meters) and then had them measure the angular size and calculate the width.

Here are some other questions:

• What is the angular size of your thumb at arms length?
• What is the size of a sign on a building across the street?
• There is a doorway down the hallway. The width of the frame is 0.91 meters. How far is it?
• What is the angular field of view of your phone’s camera?

Overall, the lab went fairly well. Students have a bunch of trouble with that first step—where they build something. You can tell they don’t feel comfortable without explicit instructions.

MacGyver Season 1 Episode 14 Science Notes: Fish Scaler

Isn’t it nice that I have written enough of these MacGyver science note posts that I no longer have to give some witty introductory comment? Oh, I guess that was an intro comment. OK—next time it’s just going to jump into the science.

Picture triangulation

MacGyver is trying to track down some dude. He finds a skyline picture that he drew and assumes the guy drew it from his apartment window (somewhere in Atlanta).

Oh wait! I think I can find out where this guy lives based on the drawing. True? Yes, this is true. If the guy drew a scale drawing, then yes—it’s entirely possible to find out where he drew it from. Oh, if he does an abstract drawing then all bets are off. Right?

There is a lot here, so let me go over two important ideas needed to backwards engineer this drawing.

First—angular size. You already know about angular size. The farther away something gets, the smaller it looks. If you like, you can make it so that someones head appears to be as big as your thumb. Yes, the human would have to be much farther away than your thumb (from your eye).

If the thumb covers up someone’s head, then the two objects would have the same angular size. How about a diagram to explain angular size? Suppose some object has a length of L and is a distance r away from an observer. It might look like this.

The blue circle is the observer and the red thing is the object. Yes, I drew it as an arc of a circle. If the object is far enough away, this is very good approximation. That means I can use the arc length equation. Remember that if you go all the way around a circle, then the total length is $2\pi r$. That means I get the following:

$L = r\theta$

Assuming the angle θ is in radians and not degrees. Oh, here is a more detailed explanation of the difference between radians and degrees. But in the end, if you know two of the things (angle, distance, size) you can find the third thing.

If MacGyver sees a building that he is familiar with, he knows the size of that building (or at least he could look it up). But he doesn’t know the distance or the angular size—bummer. If this was an actual photograph, it’s possible he could determine the angular size of the building based on the angular field of view for the camera. However, this is drawing, so the entire width of the picture could be just about anything.

Now for the next idea—triangulation. Suppose you know the angular position of two objects. From those angles, you can draw two lines at those angles. Where those two lines meet—that’s your location.

But you can see the problem, right? The triangulation depends on the angular size of the drawing and so does the distance to the objects. It looks like a dead end. But it’s not. Actually, you have enough information to math-it-out if you try (and boy did I try).

I’ll be honest. I worked on this problem for quite some time. Here is one of my earlier sketches for this calculation.

But yes, it does involve some trig.

Hot wire a car

Everyone wants to steal a car. Honestly, modern cars are fairly difficult to just take. There are four or three (depending on how you count) different classes of cars. Let me list them.

• Super old cars. These have a key that starts the car. That’s it. You can steal these—BUT YOU SHOULD NOT STEAL CARS.
• Just plain old cars. These are like super old cars, but they have a steering wheel lock. Sure, you can hot wire these—but you can’t turn the steering wheel.
• Modern cars. I think it’s cars after 1997. These cars have a chip in the key. No chip, no start. Well, you might be able to start it but the car’s computer won’t pump fuel or something like this.
• Even more modern. What about those cars with the key fob and you don’t even put the key in the car? You can’t really hot wire those either.

But check it out. This guy has a great video that goes over the different types of cars and how thieves would steal them (but don’t steal cars).

So, in this case MacGyver hot wires a car. It looks like an older model—so it’s at least plausible. What about the steering wheel lock? Maybe he just yanked on the steering wheel really hard and broke the steering wheel lock.

Cleaning bottle bolo

This is pretty straight forward. MacGyver uses a string to tie two bottles of cleaning solution together. He then swings these around and throws them at a baddies legs. The thing is a bolo. It wraps up his legs and he falls like an AT-AT on Hoth (but a lot faster).

Trip wire fan

MacGyver runs a fishing wire in a hallway and then back to the room. The wire then connects to the switch in an electrical fan. When someone steps on the fishing line, it connects the switch inside the fan and turns it on.

This should work.

Bump key

The bump key is a tool used to pick locks. The main goal in lock picking is to move lock pins up out of the lock cylinder so that you can turn the key. Here is a better explanation (I’m not really an expert here).

Light explosion

How do you make a distraction in a parking garage? One way might be to jam a charger for an electric car into a power box for the overhead lights. That’s what MacGyver did.

Would this work? It’s possible. Most car chargers run at 220-240 volts, but most overhead lights are fluorescent lights that expect 120 volts. If you double the voltage, then bad things can happen.

Basically, there is an electrical ballast inside the fluorescent light. This is a transformer that takes the 120 volts and ramps it up much higher (depending on the length of the tube) so that you can make light. If the voltage is too high, the ballast could go boom.