Introductory Laboratory 0: Buffon's Needle

Transcription

1 Introductory Laboratory 0: Buffon's Needle Introduction: In geometry, the circumference of a circle divided by its diameter is a fundamental constant, denoted by the symbol ππ. Curiously, this number has no simple form. Expressed as a decimal, the digits of ππ, ππ = continue forever without settling into a regular pattern. This mysterious contrast between the simple geometric explanation of the value of ππ, and the complexity of the numerical value has inspired books, movies, and even cults. Because no simple mathematical form exists for π, we must rely on methods which are able to to calculate ππ iteratively where the approximation to its value becomes better and better with each iteration. The first such estimate was made by Archimedes roughly 00 years BC by calculating the circumference of polygons which fit just inside and just outside of a circle. Currently, the world's best estimate of ππ contains.4 triion decimal places. In this introductory lab we wi explore an experimental method to calculate ππ introduced by the French naturalist Georges Buffon in 733. Method: Buffon's method for estimating the value of ππ relies on the foowing observation. Suppose you mark a surface with parael lines separated by a distance. Now imagine throwing a straight object (a needle, or a toothpick) of length randomly onto that surface. It can be shown that in the case where <, the probability that the needle lands such that it crosses a line is pp =. ππ We can compute the probability pp experimentay by repeatedly throwing a needle on to the grid. If we throw the needle times and record nn hits and mm misses ( = nn + mm), we can approximate pp as pp = nn/. We can then approximate ππ as ππ = or ππ =. The more throws of the needle we make the better our approximation to ππ becomes.

2 Questions: [] Can you derive the expression for the probability pp? [Hint: draw a graph with the distance the needle lands from a line on one axis LLLLLLLL and the angle the needle lands with on the other axis. Which points yy on this graph produce a hit?] Let s do this calculation explicitly (please look at the answer only yy after you have tried doing the derivation yourself): Consider a toothpick which lands between two lines. The center of the θθ toothpick must be a distance yy from the nearer of the two lines. Ca the angle of the toothpick with respect to the horizontal axis θθ. xx ππ θθ < ππ. Projecting in the horizontal direction, the length of the toothpick from its center towards the nearer vertical line is xx = cos(θθ). If cos(θθ) = xx yy, the toothpick crosses the line, otherwise it doesn t. Consider the simple extreme cases: If θθ = 0, xx = so the toothpick crosses the nearest line if yy. sstt LLLLLLLL LLLLLLLL LLLLLLLL yy xx = θθ = 0 LLLLLLLL LLLLLLLL yy θθ = ππ xx = 0 If θθ = ππ, xx = 0 so the toothpick only crosses the nearest line if yy = 0 In between, the toothpick crosses the line if cos(θθ) yy.

3 Let s plot the phase space for the problem (whether the toothpick crosses the line as a function of yy and θθ. A four quadrants are the same (0 θθ ππ vs ππ θθ 0 and 0 yy and yy 0) so let s just consider the quadrant where both θθ and yy are positive. θθ In this quadrant, if toothpick crosses the line. cos(θθ) yy then the θθ = ππ TTTTTTTThpppppppp nnnnnn cccccccccc We can draw this result as shown. To the left of the curved blue line, the toothpick crosses the nearest line. To the right it does not cross. Then the probability that the toothpick crosses the nearest line is just the area of the region to the left of the blue line divided by the total area of the rectangle bounded by the dashed and dotted red lines: The area of the rectangle is: θθ = 0 yy = 0 TTTTTTTThpppppppp cccccccccccccc yy = yy = yy AA RRRRRRRRRRRRRRRRRR = ππ = ππππ 4. The area to the left of the blue line is: AA cccccccccccccccc = ππ = = ( sin(θθ)) 0 So the probability of crossing is: pp = AA cccccccccccccccc = AA RRRRRRRRRRRRRRRRRR ππππ 4. If we rearrange we can calculate ππ from pp:. (eq. ) ππ 0 cos(θθ) ππ = cos(θθ) sin ππ sin(0) =. (eq. ) 0 = 4 ππππ =. (eq. 3) ππππ pp = ππ =, (eq. 4) ππππ pppp Which was what we wanted to show.

4 [] If the distance and are measured with a precision σσ and σσ, and the probability PP is known with uncertainty σσ PP, what is the uncertainty σσ ππ in the result to ππ? Remember our basic relationship for error propagation: σσ ff(xx,xx,xx 3, ) = xx xx,xx ss, σσ xx + σσ xx xx xx,xx ss, +. (eq. 5) In this case, the function ππ(pp., ) = ff(pp,, ) =, (eq. 6) pppp so = pp = ππ pp, Plug in to eq. 5: = = ππ, = pp = ππ (eq. 7) σσ ππ(pp.,) = ππ pp σσ pp + ππ σσ + ππ,,,, σσ, (eq. 8),, We can simplify this equation by looking at the relative error rather than the absolute error. Divide through by ππ and eliminate extra signs: σσ ππ(pp.,) ππ Put σσ inside,, : σσ ππ(pp.,) ππ = σσ pp + σσ + = σσ pp + σσ + σσ σσ, (eq. 9). (eq. 0) This equation for propagation of relative error is a true as long as the individual errors are uncorrelated with each other (for the more complicated cases, see It is also the reason that working with relative error is easier than working with absolute error. [3] Inferred vs. Relative Error: From your measurements, you can now calculate both the measured relative error and the theoretical or inferred relative error for ππ: Given your measured values of {pp, pp, pp } calculate = pp ii, and σσ pp = (pp ii ), similarly calculate, σσ and, σσ for your measured values of {,, MM }

5 and,, QQ. First look at σσ pp, σσ pp and σσ : Is one of these much larger than the others? If so it is the dominant pp pp source of error (and you can essentiay neglect the others). In your lab write of give these individual values, discuss their relative significance and why they have the relationship they do. You should make and present this analysis in every experiment you do in this course! Now calculate the theoretical or inferred relative error using eq. 0: σσ ππ ππ = σσ pp What is this value? + σσ + σσ. (eq. 0) Separately, you wi calculate the measured error as foows: From {pp, pp, pp }, and, calculate {ππ, ππ, ππ }, using eq. 6 ππ ii = value of ππ = ππ ii. The measured relative error is now: σσ ππ pp ii. Your estimated. (eq. ) ππ = ππ ππ( ) (ππ ii ππ). (eq. ) You expect that the inferred and measured relative errors should be about the same (they need not be exactly equal). Compare the two values and their difference. Is this difference significant? Why or why not? If one relative error is much bigger than the other, you should think carefuy about why and explain this discrepancy in your report. [4] Which uncertainties in your experiment would you label as systematic? Which as statistical? [Hint look up the concept of counting error in Wikipedia or your statistics notes and textbook. Also consider measurement errors like the accuracy and precision of your ruler, or the possibility of interaction between your toothpicks (what would the effect of clumping toothpicks be on your result]? [5] If you make throws of a single toothpick {pp, pp,, pp } resulting in nn hits and mm misses, what is the uncertainty in pp? Consider each throw to be a measurement of the value of pp which results in either a hit (pp ii = ) or a miss (pp ii = 0) and apply the formulas for estimating the error on the mean of a distribution from repeated measurements (see statistics handouts from first class) to show that:

6 σσ pp = pp( pp). (eq. 3) As above, I present this derivation (please look at the answer only after you have tried it yourself): and = pp ii, (eq. 4) σσ ppii = (pp ii ). (eq. 5) Now, expand the square on the right-hand side of equation 5: σσ ppii = Since pp ii εε{0,}, pp ii = pp ii, so: σσ ppii = pp ii pp ii +. (eq. 6) pp ii pp ii +. (eq. 7) Expand pp ii = pp ii pp ii : σσ ppii = pp ii pp ii pp ii +. (eq. 8) Factor out pp ii in the first two terms: σσ ppii = pp ii( ) pp ii +. (eq. 9) is a constant, so pu, and out of the summation: σσ ppii = ( ) pp ii pp ii +. (eq. 0) pp ii = and =, so: σσ ppii = {( ) + }. (eq. ) But =, so the second and third terms cancel, and: σσ ppii = {( )}. (eq. ) Remember that the uncertainty of the mean squared σσ is times the uncertainty of the individual measurements squared σσ ppii, so: σσ = σσ pp ii = ( ) ( ) =. (eq. 3)

7 [6] If we assume that and are perfectly we known (i.e., σσ = 0, σσ = 0), how many throws (N) do you need to measure ππ with % uncertainty (i.e. for σσ ππ = 0.0)? Ca the uncertainty δδ σσ ππ ππ ππ As above, I present this derivation (please look at the answer only after you have tried it yourself): From equation 0, δδ = σσ ππ ππ = σσ pp + σσ Since we assume σσ = 0, σσ = 0: + σσ. (eq. 4) δδ = σσ pp. (eq. 5) From equation 3, σσ = ( ), so: ( δδ ) = ( ) =. (eq. 6) ( ) Solve for δδ: So: = δδ, (eq. 7) = δδ +. (eq. 8) [7] If we assume that l and d are both measured to be 4.0 ± 0.cm, how many throws (what value of ) does it take to make the uncertainty resulting from your measurement of PP equal to the uncertainty from the measurements of and? [Hint, see point above and eq. 8]. At this point we refer to the measurement as systematics-limited since the statistical error is no longer the largest contribution to the relative error. Repeating the measurement further makes little sense until the systematic errors can be reduced via improved methods and instruments. [8] In the experiment, you can choose to make the distance which separates the lines either nearly equal to or much larger than. What are the advantages and disadvantages of each? Which choice of gives you the smaest relative error for a given number of repetitions ()? [Hint, what value of pp gives the smaest possible value for in equation 8? Why is this the optimum?what ratio of gives this value of pp (use equation 3)?]

8 Procedure: Count out 0 toothpicks and measure their length. Compute the average length of the toothpicks in your sample and estimate the uncertainty in this length. Next draw a set of parael lines on a piece of paper at a distance d apart. Be sure to choose a distance such that >. A good choice is to pick a value for d that wi give you a large hit probability, do pick d as close to l as you can while ensuring that <. Estimate your uncertainty in d. Devise a way to throw your toothpicks such that they land with random positions and orientations on the paper. Toss any toothpicks that do not land on the paper again. After a 0 sticks have been thrown, record the number of hits. Let's ca this set of 0 toothpicks a trial. Repeat this process as many times as you can. Finish at least = 50 trials. Analysis: [] Using the estimate pp = nn, compute your estimate of ππ after each trial (include a preceding trials in the calculation). Plot the value of ππ as a function of the trial number. Do you see the trend you expect? [] Make a histogram which shows the number of times a trial yielded,,, 3,..., 0 hits on a line. Calculate the mean and standard deviation of this distribution. For a counting experiment like this one, the standard deviation of the distribution is roughly equal to the square-root of the mean. How does this expectation compare with your measurements? Calculate the uncertainty in the mean of this distribution. Based on this calculation what do you estimate for and for the error in (remember that σσ = σσ pp )? What value of ππ does this value of yield? What is the uncertainty in ππ, σσ ππ from this measurement? [3] Use the results derived in Question 4 to estimate pp and its uncertainty. How does this compare to the result in part [] above? What result do you get for ππ using this method? [4] Exchange your data with as many lab groups as you can. Using a of the data available to you, what is your estimate for ππ? What is the uncertainty in your experimental calculation? Which contributions to the uncertainties matter most? The limited statistics? The measurement of??