December 19, 2010

R wins 364-331

Renae won today. I got all the high value tiles, but she got the only bingo: refront.

R wins 364-331

Renae won today. I got all the high value tiles, but she got the only bingo: refront.

December 15, 2010

Running track lengths

Yesterday, Rhett at Dot Physics had a comment about last week's puzzler on Car Talk.  He had an alternate solution to the question about how a pair of people could walk side by side for an hour and cover different distances.  His solution was that they were walking on a circular track. (Read his solution for a full explanation.)

At the end of the post he said:

It doesn’t even have to be a circle – it can be just curved at the ends and straight in the middle. Of course in this case, the husband would have to slow down on the straight parts (or the wife would have to speed up) in order for them to stay side by side. But it could be done.

There was something about this that didn't seem right.  I remember the track I used to jog on had one lane that was 8 laps for a mile.  Let's use that as the lane for the husband in our example.

Here's the layout of the track:
Let's use Rhett's values for the inner and outer radii: 4 m and 5 m, respectively. If the outer track is an 1/8th of a mile, then how long are the straight sections?

1/8th of a mile is about 200 m. The circumference of the curved part of the outer track is d = 2×π×5 m = 31.4 m. So the length of each straight section for the outer track is 84.9 m. 

The circumference of the curved part for the inner track is d = 2×π×4 m = 25.1 m. But the length of the straight sections is the same.  So the total length of the inner track is 25.1 m + 2 × 84.9 m = 194.9 m.

The wife has to walk 33 laps to go 4 miles. If the husband and wife are walking side-by-side the whole way, he also walks 33 laps.  His lane was 1/8th of a mile, so he has gone 4 miles, plus an extra 1/8th.

I still like Dot Physics.

December 14, 2010

Results are in

Over the weekend, I was watching the Mythbusters episode which was testing the idiom "It's like taking candy from a baby."  The phrase is used to imply that something is as easy as, well, taking candy from a baby. But long ago, I realized that phrase makes no sense at all. Anyone who's every been around babies knows that if a baby wants something, then even if you can easily take it from them, what you're really going to have is whatever you took away plus an unhappy baby. If the something you are taking away is candy, then you're likely to get tears from the baby, too. So taking away candy from a baby is just mean.  I think that's a better representation of the idiom.  What said you on the poll?

Clearly, most people went with the traditional meaning.  The three "other" responses all included meanness or cruelty in some way.  Still, most people (3:1 from this non-scientific poll) seem to think it is easy and not that mean or cruel to take candy from a baby. 

December 13, 2010

What can you do with this?

I just wrote this up over at the home.drewsday blog, but the real reason for taking the photos was to use this in class next quarter. We start with thermodynamics; one of the first topics is thermal expansion and contraction.

I regularly read Dan Meyer's blog. He was (is?) a math teacher, but I'm consistently inspired by his ideas. One of his regular features is something he calls "What can you do with this?" [WCYDWT]. The idea is that he finds an example of something in the world which illustrates a math concept and brings it into the classroom.

When I saw the contraction of the vinyl siding on my house, I knew I could bring it into the class next quarter.  The question is: What can you do with it?

I want to present this to class, so I have to think of the questions that would be appropriate.  Usually, when I start a problem in class I make a list of everything I know and everything I don't know (or want to know).

What I know

  • temperature outside today was 12° F. ( T)
  • nominal length of the siding was 12 feet. ( L)
  • change of length (on one side) was 1/8 inch. ( ΔL )
What I don't know (would like to know)

  • temperature when the siding was painted ( T)
  • coefficient of linear expansion for the vinyl siding (α)
Relevant equation

  • ΔL = αL0ΔT
The problem is that I have two unknowns (ΔT and α).

I have no idea what the temperature was when the house was painted. I didn't even know the house existed when it was painted.  I suppose I could come up with a reasonable estimate, but realistically, there is a pretty wide range of temperatures to work with. Conservatively, I would guess that the painting could have been done when the siding was anywhere between 60° F and 90° F. That is a pretty wide range, so I'm not sure I want to try to estimate that and use it to find α.  So let's make the initial temperature something we want to find. 

This means that I need to figure out what the coefficient of linear expansion is for the siding. In class we discuss how the thermal expansion process is approximately linear over a certain range. I have no idea if the expansion of vinyl siding is linear or not.  I'm going to cross my fingers and assume it's linear-ish. 

With a bit of googling, I learned that vinyl siding is made of unplasticized polyvinyl chloride (uPVC). I also found the coefficient of linear expansion is listed as 50 ×10-6 °C-1 for PVC. I couldn't find uPVC thermal coefficient, but every listed coefficient I could find was close to this value. 

Using that thermal expansion coefficient, and solving for the initial temperature, I found that the initial temperature was 23.6° C, which is 74.5° F.  (Right in the middle of my estimated range, hmmm...)

I think that the value for the coefficient is reasonable.  I plan to work this example in class, then extend it by asking the class to calculate what the maximum expansion/contraction range would be for a 12' length of siding in Illinois weather.  I'll be looking for other ways to extend this. 

Side notes:

  1. The units specified online for the coefficient of thermal expansion are often given as [L]/[L][T] (e.g. m/mK). Of course, the length units cancel and all that remains is inverse temperature units, which is how our textbook lists the units.
  2. Coefficients of linear expansion for common materials varied quite a bit (sorry, not scientifically quantified) from one table to another. Surprisingly, the PVC coefficient seemed to always be within 2% of 50 ×10-6 °C-1.

December 03, 2010

Loudness and sound pressure levels

This morning the blogs and twitter feeds were linking all over the place to a passive amplifier for your iPhone. (See it here: Science & Sons OS v1.0)

The idea behind a passive amplifier is that it requires no electrical power to increase the amplitude of the sound wave which is presented to your ear. Just pop your iphone into the cradle and crank up the tunes. As long as the battery in the iphone is charged, you have amplified sound.

It sounded cool, so I checked it out. This is what caught my eye:
The Phonofone III is an elegantly designed passive amplifier crafted from ceramic and designed explicitly for iPhone. This clever device amplifies the volume emited from an iPhone internal speaker roughly 4x (approx. 60 decibels).
Wait. What? I get a 60 dB gain out of a device with no power?  Let's see if that passes the sanity check. I don't have an iphone, but I have heard the iphone playing audio from its internal speaker. Unamplified, at a moderate level, I would say the sound level observed from an ipod (at an average distance) is about the same as the sound level of a typical conversation.  Let's look at what the approximate sound level would be:

(I found this chart on OSHA's website. I assume the image is public domain, like most government images.)  From the chart you can see that conversation at 1 m is approximately 60 dB(A).  (The "A" means A-weighted, which is a weighting factor used to approximate the frequency response of the human ear.)   Let's be conservative and estimate the iphone unamplified level to be about 55 dB. That means that if I drop the iphone into the passive device the sound level should be 55 dB + 60 dB = 115 dB.  That's louder than the "Discotheque" rating in the chart (when/where was this chart generated?) which is also about 20 dB louder than a jackhammer at 15 meters.  Somehow I really doubt that the single horn + iphone is going to be able to compete with the speakers and amplifier of a dance club.

What about the other claim?  They say the sound will be about 4x louder.  Here's where the claim might hold some water. The basic idea of a horn is to provide better impedance matching  between the driver and the sound field and to control directivity of the sound radiation.  Assuming the horn is pointing at an observer, the sound level at the observers ears should be higher with the horn than without the horn.

What if their claim that using the horn causes a perceived 4x increase in loudness is true?  (Loudness is a perceptual quantity, where sound level - either sound pressure level or sound intensity level - is a measured quantity.)    Then, the only mistake that they made is equating a 4x increase in loudness with a 60 dB gain in sound level.  Here's a graph from a lab we do in my "Sound and Acoustics" class:

In this lab students hear a broadband tone which they assign an arbitrary loudness level rating. In this class, we all agreed to call the reference tone (relative sound level = 0 dB) a loudness level of 100.  Note, there are no units, since it's an arbitrary scale we made up for the lab. Then they hear several other tones where the level has been increased or decreased randomly and they are asked to rate the loudness of the tone with respect to the loudness of the reference tone.  

What I've plotted above is the class average of loudness level vs the actual relative sound level (in dB) for all trials.  Each sample was presented twice (not in sequential order) so the scatter is a sort of approximation to the uncertainty in the measurement.  The solid line represents a model that is what we would expect to see for a larger sample of the population.  For such a small class, the trend is pretty close to the "expected" behavior.

Note that a 4x increase in loudness, from either 25 to 100 or 100 to 400, corresponds to a relative gain in sound level of 20 dB, not 60 dB.

Under ideal testing circumstances, I could believe that a passive amplifier like the horn amp would give a 20 dB gain right in front of the horn.  But someone should have caught the 4x = 60 dB nonsense.