PHYS 1712 Physics Laboratory I Purpose of this course : Why you (as a physics major) need to learn to do experiment? Because physics is an experimental science! That is the only way we can find out how Nature behaves In this course you will : 1. Learn how to use basic equipment such as conventional and digital oscilloscopes, power supplies, signal generators, digital multimeters, etc. to do experiment and make measurements. 2. Learn how to collect, analyze, handle data and estimate their errors by using Excel. 3. Learn how to describe your results to other people by writing a short report. A brief outline of this course • You’ll need to do 6 experiments in this term. • You are required to write a short report on each experiment. You should have received hardcopies of the Lab manual and Short notes on error analysis Other useful materials relating to this course is available in the course webpage : http://www.phy.cuhk.edu.hk/course/2013-2014/2/phys1712 • When you come to each lab session, do not come unprepared! • Study the lab manual thoroughly before you come • Write the first page of the lab report before you come • Here is an e.g. of the first page. • In the first page of the report you should state the objective of the experiment, and describe how the experiment will be performed. • Use your own words, don’t merely copy from the manual! • Lab opening hours : 2:15 pm – 6:30 pm • You may come earlier and familiarize yourself with the equipment, but do not start the experiment before 2:30 pm • Marks will be deducted if you come after 2:30 pm • After you arrived in the lab : • Sign the attendance sheet • Hand in the first page of the lab report of the present experiment to the TA. The TA will give it back to you after checking • Hand in the lab report of the previous experiment • Marks will be deducted for late submission • The graded report will be returned to you in the next lab session Marking scheme for the lab report : (Also available from the course webpage) • Before you come to the lab, download the data sheet from the course webpage into a memory stick • You should be able to finish the measurements of each experiment within 2 hours. Try to finish the calculation and data analysis in the lab, so that in case you have questions you may ask the TA. Save you data in your memory stick. • The TA will stay until 6:30 pm, after which the lab will be closed • If you wish, you may stay in the lab until 6:30 pm to write the lab report, or you can complete the lab report at home. You are not expected to remain in the lab for any other business than that related to the experiment • Before you leave the lab, remember to upload your data, clean up the table and the computer The lab report : • Clear, concise, informative and no-nonsense. About 4 pages excluding the appendices should be adequate • Hand-written is okay, typing is not necessary. The hand writing must be clear • Draw your own diagram (hand-drawn is okay), do not copy and paste the diagram from the manual or other sources from the web • Try to use your own words and avoid copying the writings from the manual. Use past tense and passive voice. Lab regulations : • You may discuss the experiment with your classmates, but please speak softly • No eating or drinking • The lab is strictly for the purpose of doing experiment, not for social gathering. Please do not hang around in the lab after you finished your measurement if you are not writing your report Course assessment • Preparation Performance in lab Lab reports 15 % 15 % 70 % • Different Sessions of the class are graded independently • You’ll fail the course if you • Absent without good reasons in 3 labs, or • Failed to hand in 3 lab reports • If you absent in an experiment with compelling reasons, a make-up experiment can be arranged Any questions? Taking measurements and analyzing data • We all make mistakes, especially when we are exploring the unknown in which no one knows the ‘correct’ answer • To avoid making careless mistake, use your physicist intuition from time to time to judge if your data or result makes any sense (e.g. use an order-of-magnitude estimation) • Provided all the careless mistakes are avoided, there could still be differences between your measurements and the ‘true’ values – experimental error • By error we don’t mean a mistake, it means how much we can trust about a measurement – that is, the uncertainty of the measurement • We always present a measurement or result together with its uncertainty. The magnitude of the uncertainty is dependent on the method and nature of the measurement. e.g. If I am given a stick, I can judge by my eyes that the stick has a length somewhat between 0.8 and 0.9 m. I’ll present my ‘measurement’ as 0.85 ± 0.05 m If instead I used a measuring tape, with a marking down to the millimeter scale, and find the length to be between the 0.832 and 0.833 m marks. I could present my measurement as 0.8325 ± 0.0005 m • The error should have one or at most two non-zero digits. The no. of significant digits of the data should be consistent with the size of the error The following are incorrect presentations of data : 21.23556 ± 0.2 11.23 ± 0.337543 2.03 ± 0.00032 The correct format should be : 21.2 ± 0.2 11.23 ± 0.34 2.03000 ± 0.00032 e.g. the elementary charge e has been very accurately determined to be e = (1.602176487 ± 0.000000040) × 10-19 C or, in concise form, e = 1.602176487(40) × 10-19 C • However, in most of our experiment, we usually have accuracy up to 1 or 2 digits only . • Note: a common mistake is to copy all the digits displayed in a calculator or when using a spreadsheet (e.g. Excel) Digital equipments are generally more preferable than analog equipments(e.g. a measuring tape), but can we trust all the digits as displayed by the equipment? The manufacturer of the equipment usually states the accuracy of the equipment in its specification e.g. According to the specification of the model GDM-8135 digital multimeter, the error of the DC voltage reading is “(0.1% of reading + 1 digit)”. Suppose the displayed is 18.5 mV. The uncertainty is 18.5 × 0.1% + 0.1 = 0.12 ≈ 0.1, so the data is (18.5 ± 0.1) mV. Suppose the displayed is 118.5 mV. The uncertainty is 118.5 × 0.1% + 0.1 ≈ 0.2, so the data is (118.5 ± 0.2) mV. • In many situations, the error appears in a random fashion. e.g. Random fluctuation of the equipment due to temperature, humidity, power supply instability, etc. Randomness is also an intrinsic nature of some physical processes, e.g. emissive decay of radioisotopes. • Random errors can be reduced by repeated measurements. However, repeated measurements could not help to reduce instrumental or systematic errors, e.g. parallax error in reading a scale, mis-calibrated equipment, etc. • If the error is random, repeated measurements gives us the • Best estimation of the ‘true’ value – the mean • Best estimation of the uncertainty – the standard error Suppose we repeat a measurement n times and obtain the data: x1, x2, x3 … xn. We can count the occurrence, or frequency of the measured value that lies within selected ranges. If we display the data in the form of a histogram we’ll get sometime like : 40 30 20 10 What can we say about our data? First of all, we can calculate the mean x of our data by n x  (  xi ) / n i 1 Usually, the mean is the best estimate of the ‘true’ value from our measurement We can also calculate the standard deviation  , which is the averaged deviation from the ‘true’ value. It represents the spread of our data : 1  n 1  n i 1 ( xi  x )2 The distribution of our measured values can be assumed to obey the normal distribution if the error is truly random : 0.4 0.3 f ( x) 0.2 0.1 0.0 -3 -2 -1 0 1 2 3 (x  x ) / 1  ( x  x ) 2 / 2 2 f ( x)  e  2 , where    f ( x )dx  1 The area under the curve in region x    x  x   is 0.68. This means if you take a measurement one more time, you have a 68% chance to get a measurement x falling within the interval x   We estimated the ‘true’ value by the mean x . How reliable is this estimation? In other word, if we take another set of n measurements and calculate the mean x , how would it differ from the previous x ? It turns out that the various x obtained also follow a normal distribution, with the standard deviation  x given by  n 1 2 x   ( x  x )  i n(n  1) i 1 n This  x is a good estimate of the probable error of the measurement. It is called the standard error. e.g. Suppose we did an experiment to measure the free-fall time t of an object over a distance. We obtained the values 5.43 , 5.14 , 5.41 , 5.82 , 5.58 , 5.05 We calculate the mean = Standard error = second 1 5.43 + 5.14 + 5.41 + 5.82 + 5.58 + 5.05 6 1 0.025 6(5) 2 + 0.265 2 + 0.005 2 + 0.415 2 + 0.175 2 = 5.405 + 0.355 = 0.115 Therefore, we experimentally obtained t = 5.4 ± 0.1 s This means that there is about 68 % chance that the ‘true’ t value lies between 5.3 and 5.5 s 2 Propagation of error Suppose we want to measure a quantity p, which is the sum of two independently measured quantities q and s. If we measured q = 2.8 ± 0.2 and s = 4.1 ± 0.3, what is p? Of course we’d have p = q + s = 6.9, but what is its error? One answer is 0.2 + 0.3 = 0.5, but it may overestimate the error because it assumed the maximum error occurred at the same time for both q and s. It turns out that a better formula to use is dp = (0.2)2  (0.3)2 = 0.36  0.4 So that p = 6.9 ± 0.4 There is a general rule to calculate the propagation of error, which is briefly outlined in the notes. At this stage, we shall simply employ the formulas for special cases: 1) If k  m c p i 1 then d k  i i , where ci are constant coefficients, m c d p  i i 1 2) If k  2 i m ( pi) i (with constant i 1 then dk  d pi   ( i )   k i 1  pi  m 2 2 i ), e.g. In a free-fall experiment we wish to find g by measuring the distance d and the time t and use the equation d = gt2/2. Suppose d = 1.095 ± 0.001 m and t = 0.472 ± 0.002 s, what is g from this experiment? 2(1.095) 2d  9.83 We have g  2 , so that g  2 (0.472) t We also have Hence 𝛿𝑔 = 𝑔 δg = 9.83 ∙ (1)2 0.001 1.095 We obtain g = 9.83 ± 0.08 2 𝛿𝑑 𝑑 + (−2)2 2 𝛿𝑡 𝑡 0.002 +4 0.472 2 2 = 0.084 Least square fit to a linear graph If x and y are related by a linear equation : y = ax + b and we wish to find a and b by measuring x and y. The best way is to vary x and correspondingly measure y to arrive with a set of data points (xi , yi) 14 12 Distance (cm) 10 8 6 4 2 0 0 2 4 6 8 Time (sec) 10 12 14 16 14 12 Distance (cm) 10 8 6 4 2 0 0 2 4 6 8 Time (sec) 10 12 14 16 Given the set of n data points: (xi,yi), i = 1, 2,…n. a and b can be obtained by minimizing the mean-square deviation S, with n 1 S   ( yi  axi  b)2 n i 1 where yi  axi  b is the deviation of the point (xi, yi) from the straight line Examples of using Excel for data analysis … • Calculate the mean and standard error … • Draw a histogram of the data … • Calculate the slope and y-intercept by linear least square fit • Plot the x-y data … • Insert error bars