Whatever floats your boat…

Well now. That was a long lasting spontaneous blog hiatus…

Sorry. It’s not that I’ve run out of ideas, it’s mainly lack of time… well, since time is relative, maybe it’s just that I perceive myself as having less time than I actually do. But enough excuses… let’s get back to the physics.

I saw a really neat (3 page!) article on arXiv today called “Google Earth Physics“. I love google, and I love physics, so naturally I had to check it out. The abstract reads,

Google Earth photographs often show ships and their wakes in great detail. We discuss how the images can be used to calculate the velocity of these ships.

Did someone say, do-it-yourself-physics? I knew I had to post this. Naturally, ZapperZ beat me to it… but I’ll go one step further and actually try it out.

The actual article is only three pages long and is very easy to read, so if you want more details I’ll direct you straight to the source. But the main suggestion of the article is that using a very simple formula and making two measurements, you can get a rough estimate of the velocity of a boat. The formula is simply,

boat velocity =  1.25 * Sqrt(wl)/Sin(ang)

; wl is wavelength of wake in meters,

ang is angle between wake and boat direction of motion

The 1.25 comes from some dimensionful constants in front involving Pi and g, but if you use standard units (meters) for the wavelength of the wake, those factors just become 1.25. (The fine print also says this assumes the boat is in deep water. In shallow water this equation isn’t accurate.) You can check out the article for the real formula. To make the measurements you just need to measure an angle and the wavelength shown in the picture up at the top. You can use a protractor for the angle and a ruler for the wavelength and then use the scale given in google maps to convert between your “image length” and the actual length.

I found a boat in my area (toronto) on google maps here. Then I used gimp to measure the wavelength and angle of the wake. I then converted my image length to a real length by measuring the image length of the google scale and a simple ratio:

real distance = wavelength * real length of gscale/img length of gscale

= 76px *  10m / 93px

= 8.17m

And  I measured an angle of around 35 degrees. Plugging that into the equation I get:

boat velocity =  6.23 m/s = 22.4 km/h

I’d say that’s a fair velocity for a boat… but what do I know about boats. Try it for yourself and see if you get the same answer!

Physics explained through a Drinking (Dippy) Bird

As illustrated by the image to the right, physics is a serious subject.

It is a schematic diagram of a Drinking Bird toy (with hat). I didn’t make it. I just decided one day that I was curious about the physics of this timeless toy. Looking around on the Internet, I found many qualitative explanations for the Drinking Bird. I wanted something with more equations. Fortunately, I stumbled onto the article, “Experiments with the drinking bird”, by J. Güémez et al. Unfortunately, now I need a new Ph.D. thesis topic… Seriously though, it’s a fun paper to read.

If you aren’t familiar with the Drinking Bird, for shame! It is a toy that is remarkable in its simplicity. Given a glass of water in which to dip its beak, it will bob up and down with no batteries required! Surely a few dollars is a worthy price to pay for this perpetual entertainment machine.

…hey, wait a minute. Perpetual motion machines don’t exist!, you say?

You’re right. I’m just being provocative to keep you on your toes. It’s just that many people hint (or worse, even blatantly assert) that this is a perpetual motion machine. They are, of course, wrong. If you ever observe some contraption that seems to exhibit perpetual motion (that is, without an energy source), you are probably certainly overlooking the source of energy.

If you haven’t thought about it before, take a second now and try to make a rough deduction as to where the bird gets its energy (no googling!). To help you out, here’s a video of a drinking bird.

Ok, so you must have realized that the glass of water has something to do with it. But what’s up with the colored liquid inside the bird? And why do they all wear the same kind of hat? Well, I’ll just mention right now that, despite its inclusion in the “serious diagram”, the hat has more to do with the bird’s dignity and self-image than its functionality.

First, let me give you a rough sketch of what’s going on, then I’ll elaborate on a few specific phenomena. The colored liquid inside the bird is not water, it is Methylene Chloride (CH₂Cl₂). The gas inside the bird is not air, it’s (surprise!) Methylene Chloride vapor! The Methylene Chloride is called a volatile liquid. This means that it has a boiling point very close to room temperature. As a result, the Methylene Chloride inside the bird is in, what we call, thermal equilibrium resulting in a coexistence of its gas phase and its liquid phase. Next, you need to know that the bird’s head is a glass bulb (like the bottom) but the head is covered in fabric that absorbs a bit of water.

So, to start the drinking process the bird’s head must be covered in water. Once this happens, the water on the head begins to evaporate and cools the head a little bit. This decrease in temperature causes some of the Methylene Chloride vapor in the head to condense into a liquid and fill up the neck a little bit. Since the liquid phase takes up much less space than the vapor phase, there is less vapor in the head to fill up practically the same volume. This means that the pressure in the head will decrease, causing a difference in pressure between the head and the base of the bird. As we saw in my earlier post, a difference in pressure results in a net force from the higher pressure area to the lower pressure area. This means that the little bit of vapor in the base of the bird forces the liquid up the neck and into the head. This gives the bird a heavy head, and forces it to dip. Once it dips, the liquid moves out of the way, letting the warmer vapor in the bottom move to the top warming the head a bit and starting the cycle all over again.

Let’s look a bit closer at the water evaporating from the head. For the readers who are less experienced with physics, you might not have really thought about the reason water evaporates even at room temperature. Shouldn’t liquid water only turn into a gas at 100 degrees C (at standard pressure)? Well, water of course does become a gas at less than its boiling point (look at the steam from your coffee cup for evidence). To see why this is you need to remember that a liquid is really just a collection of molecules undergoing random collisions with each other. The molecules are going to have an overall average velocity that increases with the liquid’s temperature, but overall, the molecules will have different velocities. Near the surface of the liquid (the liquid-air boundary), some of the molecules will have a high enough velocity to “escape” the liquid and, instead, mix with the air. The reverse is also true; some water molecules already in the air will have low enough velocities to “stick” to the liquid water. The reverse process, however, is much less likely if the concentration of water in the surrounding air is low enough. When the air is saturated with water then that means there is enough water in the air that the rate of evaporation and rate of condensation are equal. So the frequency of the Drinking Bird’s sips will depend on the humidity of the surrounding air (here’s a video showing that).

But I still haven’t answered my original question; where does the bird get its energy? As you may have guessed, it gets it from the surrounding air. Even though higher energy water molecules are being lost to the atmosphere from the head, this creates a temperature difference which ultimately drives the motion. The base of the bird is continually being warmed by the air, so when the bird dips, the warm Methylene Chloride in the base carries the thermal energy originally absorbed from the surroundings to the head. That’s what keeps it drinking.

What do you think would happen if instead of giving the bird a glass of water, you gave it a glass of alcohol? Would it be more, or less enthusiastic about drinking the alcohol? Why?

An inverse glass of water

I have another neat Do-It-Yourself physics experiment for you to try.

Here’s what you need:

• Uniform drinking glass (in other words, not a funky shaped glass).
• Piece of sturdy, flat, smooth cardboard (that won’t curl up if wet). Big enough to completely cover the opening of the glass.
• Water.

The process is easy but it’s best to do it over a sink, just in case. Fill the glass with some water then place the piece of cardboard on top of the glass so that it covers it completely. Using your hand to keep the cardboard in place, quickly flip the glass and the cardboard upside down. Hopefully at this point you are not covered in water and no water is leaking out. Make sure the glass is completely upturned so that the cardboard is parallel to the ground. At this point, if all goes well, you should be able to remove your hand from the cardboard and the water should stay in!

Some of you have probably figured out that air pressure has something to do with this, and you are entirely correct. But there are a few subtle things I would like to point out. The rest of you who aren’t familiar with this kind of air pressure acrobatics are probably thinking…

…but why doesn’t the water fall out of the glass?

Well, first you need to realize that the water is not the only fluid substance in the glass, there is also air (which actually reminds me of a shirt I have). If the water were to fall out, then air would have to take its place, otherwise the same amount of air currently in the glass would occupy a larger volume than it did before, causing a huge pressure difference between the room and the glass.

Realistically, there must be a pressure difference between the room and the inside of the glass because something needs to cancel out the downward force of gravity of the water. A difference in pressure will create a net force upwards which cancels out the weight of the water. So the water is supported by the cardboard, and the cardboard is supported by thin air!

…but wait a second. Where did this pressure difference come from, you ask?

What’s going on here is that gravity is pulling the water down slightly, which increases the volume of the air inside the glass. If you increase the volume of a gas (without changing the temperature or number of molecules) you will decrease the pressure of the gas. So that means: lower pressure inside, higher pressure outside, giving a net force inwards which means a force upwards on the cardboard.

…but if gravity pulls the water down slightly then there must be a gap between the cardboard and the rim of the cup, so why doesn’t the water spill out from the gap?

Well, that’s an excellent question. The answer to this is: surface tension. Have you ever tried to float a paperclip on water? Despite the fact that the paperclip is denser than water, it is still possible. This is because the interface (boundary) between the water and the air can very loosely be thought of as an elastic band. It will resist distortion up to a certain point. This happens because nature likes to minimize energy, and in this case, that corresponds to minimizing the surface area of the interface between the air and the water. If you push it, that will put it into a  higher energy configuration, so the surface will push back. (For more, see wikipedia).

The gap between the glass and the cardboard is tiny (less than a millimeter). This is too small a gap for the air to break the surface tension. The air can’t bubble up through the gap into the glass while the water flows out so the water and the card just stay there, suspended. Not just suspended in mid air… but by mid air.

Edit: For another post about air pressure that really sucks, check out the “Everyday Physics: suction cups” post on Shores of the Dirac Sea. (…I just had to make that joke…)

Our elevators are more awesome…

I’m a fan of the University of Toronto physics department’s elevators.

…is it because they’re fast?

…do they telepathically know which floor you want before pressing a button?

…do they transport you by moving in more than one dimension?

Nope. Better. Let’s take a look inside…

Hmm… looks pretty standard so fa — but wait! What’s that in the far left corner? It looks like a glass cylinder. Let’s go in for a closer look…

<ghasp>It’s a force meter attached to a 750g weight!</ghasp>

A force meter is a type of measuring instrument that enables you to measure the amount of force acting on the object it is attached to — which in this case is a 750 gram weight.

If you don’t understand why this interesting, you need to understand the following: physicists like solving problems, and at least once in every physicist’s life (s)he wonders “what is the acceleration of this elevator, and what g-force am I experiencing?“. This question, of course, bothers us for the whole 30 second elevator ride, and we wish we had a measuring apparatus to figure it out. Eventually the curiosity subsides and we carry on with our daily lives… but now in the U of T physics department, we don’t have to.

Let me show you how it works. The top of the force meter is attached to the elevator and the bottom is attached to a hanging weight. The force meter will measure the force between the elevator and the weight. If the elevator is not accelerating (even if it is moving at a constant speed) the force between the elevator and the weight will be that of gravity. As the elevator speeds up to move to a higher floor it will have to pull on the weight with a greater force so that it not only counteracts gravity, but also pulls the weight upwards. Newton tells us that the net force acting on an object is its mass times its acceleration. So if we know the force acting on the weight and what its mass is, then we can find out the acceleration. I took a quick reading and noticed that the difference between the force when the elevator was not moving and when the elevator was accelerating upwards was approximately 1 Newton. So we can take this, divide it by 750 grams (using google calculator) and find that the result is an acceleration of: ~ 1.33 m / s².

Great! Now let’s get some context on this. Comparing it to gravity (which is 9.81 m / s²) we can say that an elevator accelerating upwards is equal to a g-force of: 1.14, which is small considering that fighter pilots can withstand a g-force of 9. Taking the elevator downwards will initially give you a g-force of: 0.86, which is roughly comparable to standing on Venus (0.904).

So, yes… very awesome. It makes me wonder how many other universities’ physics departments have little things like this. What kinds of neat publicly accessible physics toys are in your physics department?

Measuring the speed of light with chocolate and a microwave oven

Here’s a great excuse to eat a lot of chocolate in the name of science.

First of all, you need to understand that microwaves are just electromagnetic waves with a certain frequency and wavelength just like visible light. Wavelength is the length between consecutive peaks of the wave. It’s a very intuitive name. You can see it labeled in the picture to the right as the Greek letter “lambda” (λ). Since waves propagate (move), we can also define a quantity called the frequency. Frequency is the number of peaks of a wave that pass a certain fixed point per second. Wavelength is a measure of distance, and frequency is a measure of one divided by time. So to find the speed (“c“) of the wave, you just need to multiply these two quantities together:

c := speed = (distance) x (1/time) = (frequency) x (wavelength)

…but why the chocolate and the microwave, you ask?

What you need to do is use a microwave oven and a piece of chocolate to measure the frequency and wavelength of microwaves. Then you can find the speed of light! Fortunately, microwaves usually have the operating frequency written on the back. Check the back of your microwave. Mine says the frequency is 2450 MHz ( = 2,450,000,000/1 second).

Now that we have the frequency, all we need is the wavelength; this is where the chocolate comes in handy… You might have a microwave with a spinning dish inside. You can probably guess what that’s for. It’s to help heat things up evenly (like stirring a pot of soup on the stove). Whereas on a stove the heat is concentrated on the bottom of a pot, the energy (and thus heat) that microwaves give to food is concentrated at the peaks of the microwaves (which are standing waves in a microwave oven). If we take out the rotating dish then we can find these peaks, measure the distance between them, and find the wavelength. So we just need to heat the chocolate up a bit, find some soft spots (where the peaks of the microwave standing wave are) and measure the distance between them with a ruler.

Here’s what you’ll need:

• Large chocolate bar (bigger than 5 inches)
• Ruler (to measure distances)
• Microwave oven (with rotating dish removed)
• Coffee (optional… it goes well with chocolate)

Place the chocolate bar (unwrapped) in the microwave oven and heat it up (without moving it) until you can see soft spots forming. If I were to hazard a guess for the timing, I’d say heat about thirty seconds… but that’s a guess. It really depends how powerful your microwave oven is.

When you have at least two soft spots forming in the chocolate, take it out and try to measure the distance between them with a ruler. (I had to prod the chocolate lightly with a spoon to find the soft spots). My chocolate didn’t turn out very nicely, but I was able to make a very rough measurement of about 4.5 inches between the centers of the soft spots. Edit: when I made this measurement I forgot that I was measuring peaks of a standing wave which are half the wavelength of the microwave. So really, you should find the distance between the soft spots and multiply by 2 to get the wavelength. Thanks, Lord Axil. Somehow I must have missed a soft spot when measuring, which automatically corrected this factor of two.

Now we can use the wonders of Google to do the calculations for us. I can just type the following right in Google and it will calculate the speed in the proper units: “2450MHz*4.5inches“.

Google says:

(2450 megahertz) * 4.5 inches = 280 035 000 m / s

…but how close are we to the real answer, you ask?

We can check our answer with Google again. Just type “c” into google and it will give you the speed of light!

Google says:

the speed of light = 299 792 458 m / s

For a measurement made with a ruler and a chocolate bar, it’s not too far off. It works!

But don’t take my word for it, listen to this amateur scientist explain the experiment.