Decimal Scientific Decimal Scientific

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1 Experiment 00 - Numerical Review Name: 1. Scientific Notation Describing the universe requires some very big (and some very small) numbers. Such numbers are tough to write in long decimal notation, so we ll be using scientific notation. Scientific notation is written as a power of 10 in the form m x 10 e where m is the mantissa and e is the exponent. The mantissa is a decimal number between 1.0 and and the exponent is an integer. To write numbers in scientific notation, move the decimal until only one digit appears to the left of the decimal. Count the number of places the decimal was moved and place that number in the exponent. For example, 540,000 = 5.4x10 5 or, in many calculators and computer programs this is written: 5.4E5 meaning 5.4 with the decimal moved 5 places to the right. Similarly = x10 2 and = 4.2x10-4 and 234.5x10 2 = 2.345x10 4 You get the idea. Now try it. Convert the following to scientific notation. Decimal Scientific Decimal Scientific ,000,000, x x Arithmetic in Scientific Notation To multiply numbers in scientific notation, first multiply the mantissas and then add the exponents. For example, 2.5x10 6 x 2.0x10 4 = (2.5x2.0) x = 5.0x To divide, divide the mantissas and then subtract the exponents. For example, 6.4x10 5 / 3.2x10 2 = (6.4 / 3.2) x = 2.0x10 3 Now try the following. 4.52x10 12 x 1.5x10 16 = 9.9x10 7 x 8.0x10 2 = 1.5x10-3 x 1.5x10 2 = 8.1x10-5 x 1.5x10-6 = 1.5x10 32 / 3.0x10 2 = 8.0x10-5 / 2.0x10-6 = Be careful if you need to add or subtract numbers in scientific notation. 4.0x x10 5 = 4.2x10 6 since 4.0x10 6 = 4,000, x10 5 = + 200, x10 6 = 4,200,000 Practice: Estimate how many shoes there are in the world. Use scientific notation, and some basic roughguess numbers to produce an estimate.

2 3. Converting Units Often we make a measurement in one unit (such as meters) but some other unit is desired for a computation or answer (such as kilometers). You can use the tables in the Appendix of your textbook to find handy conversion factors from one unit to another. Example: You have meters. How many kilometers is this? There are 1000 m/km. Because kilometers are larger than meters, we need fewer of them to specify the same distance, so divide the number of meters by the number of meters per kilometer and notice how the units cancel out and leave you with the desired result. 2.34x10 12 m = km Another way to think about this operation, is that you want fewer km than m, so 1000 m km just move the decimal place three to the left since there are 10 3 m per kilometer. Or, if the new desired unit is smaller, and you expect more of them, then multiply. For example, how many cm are there in 42 km? 42 km 10 5 cm km = cm Use the information in the appendix of your text to convert the following. 2 year = s 1000 feet= m 50 km = m 3x10 6 m = cm 52,600,000,000 km = m 3450 seconds = minute 6.0x10 18 m = mm 600 hours = days 5.2x10 12 kg = g 365 days = s 99 minutes = hr 1200 days = yr

3 4. Angles and Trigonometry Science and engineering is filled with examples where we need to use trig functions to determine angles or sides of triangles, or to compute the projection of one vector onto another. Solve for the unknown side or angle in the following triangles as review and practice. oppopsite side sin θ= hypoentuse adjacent side cosθ= hypoentuse oppopsite side tan θ= adjacent side 3.5 m =41 y= 2.1 m =60 z= =70 x= t= 5. Vector Addition Vectors allow us to specify directions in two or three dimensions by expressing a direction as the sum or direction along two or more axes. In two dimensions, we let ^i be the unit vector in the X-direction and ^j is the unit vector in Y-direction, and then r is the vector sum of the X- and Y-components. See the first example below for an instance of vector addition, and then complete the two vector addition problems, drawing the individual vectors and the total vector in each case, following the example. a=3 ^i +1 ^j b=2 ^i +4 ^j c= a + b c=(3+2) ^i +(1+4)^j=5 ^i +5 ^j a=1 ^i 4 ^j b=3 ^i +2 ^j c= a+ b a= 3 ^i +1 ^j b= 2^i 4 ^j c= a+ b c b a

4 6. Measurement and Uncertainties Very few measurements are direct measurements. Length, perhaps, is a direct measurement, when one uses a well-calibrated comparison tool of a standard length. Most measurement, such as mass and temperature are indirect...they depend on intermediate measurements and apparatus and a subsequent calculation. For many students it comes as a surprise that absolutely exact measurements are impossible. If we weigh a small piece of material on a balance, a typical result could be grams. This is, however, only an approximation to the true weight, just as the value is only an approximation to the number. A more sensitive balance would give a more accurate number. This is true of all measurements. Measurements always are imprecise...that is, there is some inherent uncertainty (we use the word uncertainty rather than error in most cases, as error implies a mistake) in the measurement, no matter how careful we try to be. In any kind of science or engineering, getting the right answer is usually the easy part...calculating how certain you are of that answer, i.e., what is the uncertainty on your answer, is the hard part...and an important part. The uncertainties reflect both the precision of the measurement/measurer and the accuracy of the instrument. The degree of precision with which an observer can read a given linear scale depends upon the definiteness of the marks on the scale and the skill with which the observer can estimate fractional parts of scale division. In many instruments of precision, the linear scale is provided with some sort of vernier, which is a mechanical substitute for the estimation of fractional parts of scale divisions. Its use requires skill and judgment. The degree of accuracy is determined by how close we can expect to be to the true or actual value. For instance, when we measure the length of a small object, we should expect that a meter stick will give a less accurate answer than a micrometer, provided that both instruments have been calibrated well. A common way of increasing the accuracy of a measurement done with an instrument of a given precision is to repeat a measurement many times under identical circumstances and then build an experimental average.

5 6a Experimental Averages The first step in quantifying and evaluating an experimental result is to establish a way to reduce random error by building an average of repeated experimental readings. The purpose of the averaging is to improve the knowledge about the actual quantity. Thus, we expect that the average is a better approximation of the actual (true) value than a single measurement. We express that confidence by rounding it to a better precision (more digits) provided that we do have a statistic that allows for that improvement. Find the arithmetic average (or mean) velocity and acceleration for the following sets of data. Mean vel [m/s] Acc [m/s 2 ] vel [m/s] Arithmetic average(mean)of x x= 1 N x i Where N is the number of measurements. 6b. Uncertainties and Weighted Means Sources of uncertainty (or error) are many, but they are divided into two classes: accidental (random) and systematic error. By using precise instruments, the accuracy of the value we extract can be increased. It is our task to determine the most accurate value of a quantity and to work out its actual accuracy. The difference between the observed value of any physical property and the unknown exact value is called the error of observation. Random Errors are disordered in their incidence and variable in their magnitude, changing from positive to negative values in no ascertainable sequence. They are usually due to limitations on the part of the observer or the instrument, or the conditions under which the measurements are made, even when the observer is very careful. One (somewhat silly) example is if you are trying to weigh yourself on a scale but the building itself is vibrating due to an earthquake, leading to a great variety of results. Random errors may be partially sorted out by repeated observations. Sometimes the measurement is too large, sometime it is too small, but on average, it approximates the actual value. Systematic errors may arise from the observer or the instrument. They are usually the more troublesome, for repeated measurements do not necessarily reveal them. Even when known they can be difficult to eliminate. Unlike random errors, systematic errors almost always shift the observed value away from the actual value. In other words they can add an offset to the measurements. One example of a systematic error is if you are trying to weigh yourself, but you are wearing clothes, so the results is systematically larger than your actual weight. Or perhaps the scale is calibrated too high or too low. There are all kinds of systematic errors. As another example, let s take a look at a hypothetical sequence of values made for the gravitational acceleration on earth: 9.78, 9.81, 9.81, 9.79, 12.5, 9.80 [m/s 2 ]. It seems quite possible that some mistake was made in recording 12.5 and it is reasonable to exclude that value from further analysis. It represents an obvious systematic error. There is no absolute limit for which we may assume that the above is the case. For our undergraduate lab we want to keep records of all data and exclude outliers only if they are off the average of the remaining data by 100% or more and only if we have just one outlier.

6 Sometimes it is possible to estimate the uncertainty associated with each measurement. For example, If you try to count the number of shoppers that pass through the entrance to Wal Mart in any 10 minute interval, you'd be able to make a pretty accurate count if it's 1 am., and people are just trickling in. Your uncertainty would be quite small. On the other hand, if you try to count the shoppers at 6 am. the day after Thanksgiving, you're likely to make more counting mistakes and have a larger uncertainty on each count. So what's the average number? In this case, you want to compute an average that give more weight to data that are more reliable and less weight to data that is deemed to have larger uncertainties. The way to do this is to compute a weighted average. Most often, we use the inverse-square of the uncertainties as the weight. If the uncertainty on a measurement i is i, then the weight is w i =(1/ i ) 2. Compute the arithmetic average and the weighted average of the following set of measurements. weighted average of x x w = w i x i w i vel [m/s] Uncertainty [m/s] Weight w where x i are the individual measurements and w i are the weights on each measurement. Note that if all the uncertaintes are the same (or all the weights are the same) then the weighted average just reduces to the simple arithmetic average. What is the weighted average for these data? The simple arithmetic average? Describe in your own words the effect of using weights? Describe what would happen if all of the weights were identical:

7 6c. Significant Digits Often calculations will yield numbers with large numbers (perhaps infinitely many!) decimal places. Not all of these decimal places are significant in the sense that they communicate reliable information about the accuracy with which the quantity in question may actually be known. For example, if you take a board which you measure to be cm long and cut it into 3 pieces, you find that 121.3/3 yields centimeters. It makes no sense to quote the result to more than one decimal place since you only knew the length of the board to 1 decimal place (presumably plus or minus 0.1 cm) to begin with. The rule of thumb is: Multiplication & division: cite only as many significant figures as the measured number with the smallest number of significant figures. Addition & subtraction: cite as many decimal places as the measured number with the smallest number of decimal places. Each digit counts as a significant figure except leading zeros or trailing zeros without a decimal point. Number # of significant digits Calculation Result (in sig figs) 23 two 3.24 [3 sig figs] x 2.07 [3 sig figs] two 3.2 [two] x [four] three 5.55 [three] / 3.3 [two] two 1.05 [three] [four] 4510 three [ ] [ ] 4501 four [ ] [ ] 0.01 one 1005 [ ] x 231 [ ] 0.2 one 1000 [ ] x 40 [ ] 0.20 two [ ] x 40. [ ] three [ ] x 40.0 [ ]

8 6d. Experimental Error and Data Scatter Another step in quantifying and evaluating an experimental result is to establish a way to describe the scatter or dispersion in the data due to random error. The first way to build such a measure of data dispersion is called the standard deviation, defined as σ= (x i x) 2 N 1 where N is the number of data points, x is the arithmetic average, and x i is each of the individual data points. Find the mean and the standard deviation for the data set in the table below: vel [m/s] v =

9 6e. Comparing experimental results to theoretical expectations The goal of every physics or engineering experiment is to test the theory which predicts certain outcomes for the experiment. The way we achieve this is to build a reliable experimental value based on averaging data and to characterize it by an experimental error. This is then compared to the theoretical value. Consider the following experimental data set consisting of time measurements and velocity measurements for a particle traveling in a straight line. t [s] v [m/s] Dist. [m] Mean Distance: Standard Deviation of Distance: Suppose now that the theoretical distance is the theoretical speed (2.750 m/s) and time (0.110 sec) gives a distance: d (m) = v (m/s) x t (s) = (m/s) x (s) = m. The percentage error is defined as: Measured value theoretical value % error= theoretical value Compute the % error. Compare the theoretical value to the measured value. Are these two values within one standard deviation? If the uncertainties (errors) are distribution in a normal or Gaussian manner, we expect that 68% of the time (in other words, in 68% of such experiments if we repeated the whole experiment), the theoretical value and the measured value will differ by less than 1 standard deviation. 95% of the time the theoretical value and the measured value will differ by less than 2 standard deviations!