The Corrections (to '74 UALR lab report)

What I’m doing here is correcting some of my old mistakes.  In my first-ever physics lab report (see “Car Speed Measurements UALR ’74"), I mistakenly rewrote 4/100 as .004 instead of 0.04, and I used .004 to calculate the error or uncertainty in the measured speed of a car.  Well, of six cars.  So I came out with an underestimate of the error in the measurements of speed.

 

As discussed in the lab handout, the formula for “error propagation” in this case is

 σ_v/v  =  σ_t/t  +  σ_d/d,

where v is the velocity, calculated earlier from the measured time, t, and measured distance, d.  The little sigmas represent the calculated or estimated uncertainties in the speed, distance and time.  We use t, d, and v and our estimates of σ_t and σ_d to find σ_v.  This formula simply says that the relative error in speed is the sum of the relative errors in time and distance.

The values for σ_t and σ_d are, respectively, 0.42 sec and 4 meters.  The distance d is always 100 meters, measured along the roadside before the experiment began.  One person estimates when the rear bumper of a car passes the starting point, signaling to the person with the stopwatch standing at the 100 meter point to start timing.  Then the 100 meter person stops timing when the rear bumper passes the 100 meter point.  That gives t.  Another person writes down the license number of the car.  The speed is then 100 meters divided by the measured number of seconds.

So the first thing is to check my calculated speeds. They’re all right! The speed uncertainties can now be recalculated.

Case 1.  σ_v  =  (σ_t/t  +  σ_d/d)v  =  (0.42s/5.5s  +  0.04)(18.2 m/s)   =  2.12 m/s 

Case 2.   …      (0.42/6.2  +  0.04) (16.1)  =  1.73 m/s

 

Case 3.  …    (0.42/7.2  +  0.04) (13.9)  =  1.37 m/s

Case 4.  …    (0.42/4.8  +  0.04)(20.8)  =  2.65 m/s

Case 5.  …    (0.42/6.0  +  0.04)(16.7)  =  1.84 m/s

Case 6.  …  (0.42/7.0  +  0.04)(14.3)  =  1.43 m/s

By license number, the speeds and their uncertainties are:

CCH 255                     18.2  ±  2.12 m/s                    40.8  ±  4.75 mph

CIL 164                       16.1  ±  1.73 m/s                    36.1  ±  3.88 mph

AAW 197                   13.9  ±  1.37 m/s                    31.1  ±  3.07 mph

DSY 611                     20.8  ±   2.65 m/s                   46.6  ±  5.94 mph

AAV 637                    16.7  ±  1.84 m/s                    37.4  ±  4.12 mph

MWM 646 (TX)        14.3  ±  1.43 m/s                    32.0  ±  1.43 mph

The Texan wasn’t living up to his or her reputation (in Arkansas) of being a speeder.  Could’ve been an Arkie driving a Texas car, or many other possibilities. 

Oh, you can give marriage a whirl

If you’ve got some cash in your purse

But don’t marry no one but a Texas girl

‘Cause no matter what happens, she’s seen worse

--a “courting song” introduced and sung by Pete Seeger (he was strumming a banjo also) on a folk music set of albums I have, given to my brother Steven and me by our dad in 1974.

Uncertainty principle, time reversal, hidden symmetry

About 11 p.m., August 17, 2013.  Played piano a little tonight after not playing for months and months.  I hope to get back to it, and then get better at it, soon!

What a normal person would say upon hearing that the uncertainty principle prohibits an arbitrarily precise measurement of both position and velocity is, “well, something that’s moving doesn’t have a position associated with it, because it’s moving!  So of course it’s moved on from where it was when you started to measure its position!”

I was just reading in “Quarks:  The Stuff of Matter” by Harald Fritzsch, p. 9, when that thought occurred to me.  Fritzsch says, “In a certain sense, the notions of velocity and location are complementary.”

Now the extremes of location and velocity in combination that we can imagine are:  a particle at rest (all location, no velocity) and a particle with, somehow, velocity but no location. 

 

The idea of a "particle" is the idea of a dot, like the period at the end of this sentence.  So in this extreme case--imagining a definite location and zero velocity--the uncertainty principle’s prohibition on knowing the particle’s velocity is exactly zero calls into question the very idea of a particle.

The other extreme case would be a moving particle, and the more rapidly moving it is, the more extreme it is.  Experimentally speaking, you can still locate it at a particular time by bouncing a quick flash of light off it, but in the quantum realm, it recoils slightly when you do.  Theoretically speaking, if you know its equation of motion and its initial location and velocity, a graph can be drawn of its position (location, location, and later location) as a function of time.  But the uncertainty principle prohibits knowing with arbitrary or unlimited precision both it's initial location and velocity (or momentum, really).

 

This extreme case of no location is the more difficult to imagine, but is approximated by a light wave, which is said to have zero rest mass because it only exists in its traveling-at-the-speed-of-light state.   Actually, this is one reason we should be more precise in talking about the uncertainty principle & discuss location & momentum, rather than location & velocity, because the uncertainty principle DOES apply to light (photons), in spite of the fact that we can measure the velocity of light with a precision not prohibited by the uncertainty principle.  When we measure “c” (speed of light), we aren’t measuring “p” (momentum), which is, for electromagnetic waves,

p = E/c = hν/λν = h/λ.

 

E = energy, h = Planck’s constant, ν = frequency, λ = wavelength

 

12:10 p.m. Sat. 26 Oct. 2013   Reverse time so that Big Bang is Big Collapse—this is a commonplace thought experiment.  But how do we know how gravity would really behave?  This thought occurred to me after I read Sean Carroll’s sentence “Now let’s run the movie backward,” on page 49 in his From Eternity to Here book.  But there is really nothing in physics that corresponds to running a movie backwards, because the observer is still going forward in time while he watches the movie going backwards.  He or she.

But, a few pages later—surprise—this is pretty much what Sean says, that it isn’t going to show (in the movie) what we would expect, which would be galaxies dissolving into some other, more homogeneous, matter.  Instead, he says, “Growth of structure is an irreversible process that naturally happens toward the future, whether the universe is expanding or contracting.  It represents an increase in entropy.”

Before he says that, he says this about the contraction:  “Imagine that we lived in a universe much like our current one, with the same kind of distribution of galaxies and clusters, but that was contracting rather than expanding.”  So, my mistake!  This is on page 53, and isn’t the same as the “movie run backwards” scenario-ho-ho of p. 49, which was actually his way of getting back to the Big Bang singularity.  The same old analogy!  But he more or less makes my point, and gives it credence, by saying a contracting universe that is like ours would experience an entropy increase.  That is really all we can imagine when we talk about “reversing time” from here, from our current (initial, for collapse) conditions.

(End of Journal – next page contains the end of my eight pages of notes about Dallas trip in June, written backwards from the last page in the journal, a case of writing going backwards in the book, but not of “writing backwards.”)

Sunday 9:15 a.m.  24 NOV 2013 About 30° & overcast. This is from my other journal, the non-scientific one, Sept 10 or 11 2012:

Hidden Symmetry

I got up in the middle of the night on Sunday—early Monday mornin’—to write down the words above. Then later on in the day—wait, it must have been Sat-Sun middle of night, Sept 8^th and 9^th—later on I opened the Subatomic Physics text, and I encountered those exact words! They are on p. 370 in the section on Massive Gauge Bosons. Hidden symmetry is just one way to refer to the case where the Hamiltonian for the interaction “retains the full symmetry” and only the ground state breaks it. It can occur when “the ground state of the Hamiltonian is degenerate; the choice of a particular state among the degenerate ones then breaks the symmetry.” This stuff ought to be written in my other journal, my scientific one.

(Done.  "Hidden symmetry" in theoretical physics, by the way, is already as common as pig tracks, so this was no great revelation of mine.  Usually instead of calling it hidden, physicists call it broken.  My idea of hidden symmetry would be that the current physics idea of symmetry is what is broken.)

 

Saturday 30 November 2013, 10:15 a.m., sunny with the mercury at about 52° (as they used to say).  To continue where the last entry in my previous journal (physics journal, October 26 of this year) left off, where I said “all we can imagine when we talk about ‘reversing time’. . .”.  Well, I have to correct that to say we can imagine whatever we want, but from the point of view of the whole universe, watching a movie run backwards as a way modeling the reversal of time is nonsense.  The observer of the movie is still going forward in time.  Watching a segment of a movie where someone walks backwards would not tell the observer whether that someone was really walking backwards or the movie was going backwards (let’s assume the feet, which might be a giveaway, aren’t visible in the frame).  But the movie observer knows the movie is being run backwards when an egg is unscrambled and returned to its shell, or a diver comes feet first out of a swimming pool and returns to the diving board.  Thought processes, neurons firing, biological processes and time itself—all of these are still going forward as usual for the movie watcher.  That’s all I’m saying.  And that’s all I can manage to think of because it all seems beyond my rapport with the subject at the moment.

First entry in my new scientific ledger

Just bought this "record book" at Staples, 11 a.m. approximately, Thursday, Nov. 21, 2013.  A regular journal costs about $10 to $15, but I didn't see any of those I really liked.  While I was standing there an employee kindly asked if I needed any help, and I asked him, "Do you still have ledgers?"  He thought for a second then took me to a different aisle and I found this, priced at about $45.  But it's really what I was looking for, so I'm glad he (Darryl, I believe) asked if I needed help.  This ledger book has 300 numbered pages, but is otherwise like a normal blank journal with ruled paper, except maybe has a better quality binding.

 

Yep, tomorrow is the 50th anniversary of JFK's assassination, which is (tomorrow is) also a Friday like it was in 1963.

 

This morning I was half asleep, probably around 4 o'clock, and found myself wondering if there's a type of average where the order of the numbers--increasing or decreasing--matters.  I turned on the light and wrote the thought down.  Now I'm back to thinking about it, here at 11:45 a.m. (still 21 Nov.), sitting in my dining room, having escaped work to run errands and eat lunch.  It's not really the increasing versus decreasing (well, maybe it is, who knows?) as much as just changing the order of the numbers that are summed in doing a calculation of the average or mean value.

 

For just regular numbers, rational, or real, or complex, it doesn't matter.  Are there situations--as usual, I'm thinking of physical situations and not just math--where either the summing order matters or the--what?  All that's left is to divide by the number of numbers that were summed, right?  But there are two steps involved, the summing and the dividing.

 

The property of noncommutation in the realm of multiplication is well known.  Matrices, for example, don't commute under multiplication, or however you're supposed to say it.  So is there a mathematical operation, based on a physical situation, where noncommutation under addition occurs?

 

You all could be laughing like hyenas out there in the future, but I don't know if that's because I'm overlooking or forgetting some example of addition not commuting, or because the idea is an absurdity.  But laughing is good for the soul, so long as it isn't at someone else's expense.  And you're not bothering me one bit, so don't worry about it.

 

I'ts a dark, cool but not cold, and not yet rainy, day today.  I'm writing in the semi-darkness (no lights on) of the dining room at 807 W. 12th Ave, old Pine Bluff.  Jessie (the dog) is asleep on the couch in the adjacent living room.  I just called David Peyton at work to tell him that with running errands, eating lunch, and having to meet the heating and air guy at 1:30 so he can check the furnace and replace my air filter, it's unlikely I'll be back until about closing time.  Then I can get some paperwork done.

 



7th grade report card May '67

This report card of mine is from the same time period as A Serious Man.  I was about the same age as Danny is in the movie.  A thought just occurred to me on writing the name of the movie.  Maybe there's a clue to the unseriousness of the movie in the title itself.  Think about "symmetry" and its opposite, "asymmetry".  Thus you can also have serious and a-serious.  I was a-serious about school when I was in the 7th grade, definitely not a bit serious about getting A's. 

 

For an explanation of what the V's and X's are about, see my Ink on His Face blog post of 27 June 2012, which shows the front and back of my 8th grade Dial Jr High report card.

Who's afraid of the standard deviation?

The lab report in my previous post doesn’t show any actual standard deviation calculations. So I recently followed the instructions of the “People Are Not Perfect” lab handout (after nearly forty years) and timed a metronome.  I used a digital stopwatch that measures to the nearest hundreth of a second to time the metronome on my Yamaha electronic keyboard through four beats.  I repeated the measurement ten times.  These are the results, with the units being seconds:

20.52
20.41
20.49
20.36
20.18
20.49
20.29
20.49
20.46
20.47

Here is the average of these measurements:

<t> = (t₁ + t₂ + t₃ + … + t_N )/N  = Σ t_i/N =  (204.16)/10 = 20.416 seconds.

The deviation of each value from the average is t_i - <t>.  Here they all are

0.104
-0.006
0.074
-0.056
-0.236
0.074
-0.126
0.074
0.044
0.054

If you add up all the deviations you get zero (try it), because the below-the-average deviations are negative numbers, and they cancel the above-the-average deviations.  So, as you know by now, you can’t find a useful “average deviation” by adding up the deviations and dividing by N.  Instead, you square each deviation (the squares are all positive, see?), then add up all these squared deviations and divide by N. Well, for a set of data (a "sample" as opposed to a “population") we divide by N-1.  I wrote about the reason for the use of N-1 instead of N in my post called “Degrees of freedom.”  Finally, we take the square root and have the standard deviation.  (In other situations, this process of squaring, finding the mean, then taking the square root goes by the name of “root mean square,” or RMS, which I’ll come back to later, if the need arises.) The deviations squared are

0.010816

0.000036

0.000030

0.003136

0.055696

0.005476

0.015876

0.005476

0.001936

0.002916

The sum of these is 0.101394.  Lower case sigma is a common symbol for the standard deviation, so we’ll call the standard deviation in time “little-sigma-sub-t.”  

σ_t = sqrt(0.101394/9) = sqrt(0.011266) = 0.10614 seconds,

or, rounding it to the fraction of a second shown on the stopwatch (one hundredth of a second), the sample standard deviation is 0.11 sec.

Well, I know this isn’t the exciting part, sorry.  It just has to be done for some reason I can’t explain.  I hope to get to a more exciting part soon.

If we use N =10 instead of N-1 = 9, what do we get?

sqrt(0.101394/10) = sqrt(0.0101394) = 0.10069 seconds,

or if we round it to hundredths , 0.10 sec.

Now, what about that formula from A Serious Man? I said earlier I wanted to compare it with the UALR lab’s formula, but I later figured out it represents the population or theoretical standard deviation, whereas the lab uses the sample standard deviation. So now I just want to compare it to the "divide by N=10" case and see if they are equal.  Here it is:

σ_t = (<t² >- <t>²)^1/2          (population or theoretical std. dev.)

Recall that <t²> is the average of t-squared, while <t>²is the square of the average of t.  You would or could read the former as "tee-squared bracket" and the latter as "bracket-tee squared."  

(Compare how, in the movie, Larry writes these and reads them off the board, where it's "p" instead of "t":  he writes both as <p>² and says "bracket pee squared minus bracket pee squared,"  so he has written something minus itself, which is identically zero, which is another case of the Coen brothers messing with our sense of reality.  This time they mess with our math anxiety, too.)  

What the formula for σ_t says is “subtract the square of the average of t from the average of the square of t and take the square root.”  Putting in the numbers gives

… oh, no, I haven’t yet squared all the t_i’s in order to find <t²>.  Later!

Okay, now it’s later.  Here's how <t²> is found. I squared each measurement and got

421.0704
416.5681
419.8401
414.5296
407.2324
419.8401
411.6841
419.8401
418.6116
419.0209

We want the average of these.  The sum is 4168.237, and dividing by 10 gives:

<t² > = 416.8237 = "the average of the square of t."

At the beginning, above, I found the average of the t measurements to be 20.416.  So the square of the average  is

<t>² = (20.416)² = 416.813056.

And now…

σ_t = (<t² > - <t>²)^1/2  = (416.8237 - 416.813056)^1/2

     = (0.010644)^1/2

   = 0.1032

When rounded to the nearest hundredth second, this is equal to 0.10 from above.  To make the comparison more precise, we can use four digits and do a percent difference: Take the difference of the two and divide it by the average of the two, then multiply by 100.  I did this and the percent difference is 2.54%.

 

Now finally I will compare the 0.10 sec result, the "wrong" way of doing this data-based standard deviation, with our calculated sample standard deviation, 0.10614 sec, which rounds off to 0.11 sec.   The percent difference is  [0.01/(0.21/2)] x 100  =  9.52%, or about 10%, as maybe you could tell by simple comparison of 0.11 and 0.10?  In a way, this is comparing apples and oranges, because the "N" associated with a population isn't going to be the "N" used in the N-1 associated with the sample.  In the Box, Hunter and Hunter text, this is made clear by their use of n for the number of sample values collected and N for the population number.

 

So that's it, friends, you can draw your own  conclusions.  If you conclude that I don't work with actual numbers very often, you'd be right. 

Next time I'm going to use this 0.11 sec standard deviation to recalculate the numbers for the Cars and Speed Limits lab,  and also convert the speeds from meters per second to miles per hour. The conversion factor to change from m/s to mph is 2.24, so as one quick example from my 1974 lab report, the car measured to have a speed of 18.2 m/s would have been traveling at 40.8 miles an hour.

 A final thought:  these speeds are measurements of average speed over 100 meters rather than measurements of instantaneous speed. (Your speedometer tells you what your instantaneous speed is.)  What if you were speeding down University Avenue in Little Rock on a certain day in early June 1974 and you saw some kids along the side of the road ahead of you, apparently college students, taking some kind of measurements?  You'd likely slow down.  You might even be still slowing down in the 100 meter interval over which they were timing you, and even could have been going, say, 55 mph at the beginning of the 100 meters and 25 mph at the end, to take a very extreme possibility.  Your average speed in that case is (55 + 25)/2 = 40 mph even though you were speeding at first.  (Yeh, the speed limit on that four-lane boulevard is 40 mph, and yes, this average found by a different method should equal the measured average.  But from now on I'm going to be more wary of things that should turn out to be equal...)