2053Lecture1 - Scientific Method & Experiments...

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Unformatted text preview: Scientific Method & Experiments Experiments Topics of Discussion Topics Scientific Method Experiments System of Units, Unit Conversion & System Significant Figures Significant Dimensional Analysis Observations & Statistics The Language of Physics The Galileo Galilei (1564-1642) remarked that “Philosophy (science and nature) is written in a great book which lies before us – I mean the universe – but we cannot understand it if we do not first learn the language.” Facts & Fiction Facts For more than two thousand years, Aristotle claimed that a rock drops at a faster rate than a feather. Galileo showed using a very cleaver experiment that a rock and a feather drop at the same rate! Hypotheses, Models & Laws Hypotheses, An hypothesis is a limited statement An hypothesis regarding cause and effect in specific situations; it also refers to our state of knowledge before experimental work has been performed and perhaps even before new phenomena have been predicted. new The word model is reserved for situations The model when it is known that the hypothesis has at least limited validity. at A scientific theory or law represents an scientific law Hypotheses, Models & Laws, cont. cont. hypothesis, or a group of related hypotheses, which has been confirmed through repeated experimental tests. Scientific Method Scientific The scientific method is the process by The scientific which scientists, collectively and over time, endeavor to construct an accurate (reliable, consistent and non-arbitrary) representation of the world. The scientific method attempts to minimize the influence of bias or prejudice in the experimenter when testing an hypothesis or a theory. Scientific Method, cont. Scientific We summarize the scientific method We in the following four steps. in Observation and description of a Observation phenomenon or group of phenomena. phenomenon Formulation of an hypothesis to explain Formulation the phenomena. In physics, the hypothesis often takes the form of a causal mechanism or a mathematical relation. relation. Scientific Method, cont. Scientific Use of the hypothesis to predict the Use existence of other phenomena, or to predict quantitatively the results of new observations. observations. Performance of experimental tests of Performance the predictions by several independent experimenters and properly performed experiments. experiments. Common Mistakes in Applying the Scientific Method The most fundamental error is to mistake the The hypothesis for an explanation of a phenomenon, without performing experiments to confirm or disprove the hypothesis. Another common mistake is to ignore or rule out Another data that does not support the hypothesis. data Another common mistake arises from the failure Another to estimate quantitatively systematic errors (and all errors). all Experimental Error Experimental Error in experiments have several Error sources. Two types of experimental error are systemic and random. systemic random A non-random or systematic error is due to non-random systematic factors which bias the result in one direction. Hence, no measurement, and therefore no experiment, can be perfectly precise. experiment, Random error is error intrinsic to instruments of measurement. Because this type of error has equal probability of producing a measurement higher or lower numerically than the "true" value, it is called random error. than Experimental Error, cont. Experimental Therefore, the result of any measurement Therefore, has two essential components: a numerical value (in a specified system of numerical units) giving the best estimate possible of the quantity measured, and the degree of uncertainty associated with this estimated uncertainty value. Thus, we write the length of a cylinder 5.35cm with an uncertainty of 5.35 0.05cm as 5.35cm ± 0.05cm, which which 0.05 5.35cm means that the length of the cylinder lies between 5.3cm and 5.4cm. 5.3 5.4 Experimental Error, cont. Experimental In science, we have standard ways of In estimating and in some cases reducing errors. Thus, it is important to determine the accuracy of a particular measurement and, when stating quantitative results, to quote the measurement error. A measurement without a quoted error is meaningless. The comparison between experiment and theory is made within the context of experimental errors. context Experimental Error, cont. Experimental Scientists ask, how close are the results Scientists from the theoretical prediction? Have all sources of systematic and random errors been properly identified and estimated? been Systems of Units Systems Some examples for unit systems are: MKS, SI or “Le Systeme International MKS, d’Unites” – meter (m), kilogram (kg) and and second (s), second CGS or Gaussian system – centimeter (cm), CGS ), gram (g) and second (s), and gram BE or “British Engineering” – foot (ft), slug (sl) BE and second (s). and FPS or “foot, pound and second” system – FPS foot (ft), pound (lb) and second (s). foot Unit Conversion Unit The conversion algorithm is a process of The conversion changing a number in one unit system to another unit system. another Write the necessary conversion factors. Write an equation beginning with the number Write to be converted followed by the appropriate conversion factors. conversion Compute the result. An Example of Unit Conversion An Suppose we wish to convert 39m/hr2 to Suppose 39 cm/min2. Note that we need the following cm/min Note conversion factors: 100cm=1m and 1hr=60min. 100 Thus, we have 39m/hr2×(100cm/1m)×(1hr/60min)2=1.1cm/min2. Dimensional Analysis Dimensional The dimensional analysis algorithm is a process The dimensional algorithm used to establish the dimensional consistency of an equation. an The labels [L], [M], and [T] are assigned to The and length, mass and time, respectively. length, Replace each symbol of an equation with the Replace above convention. above Reduce the equation. Given the equation F = –2πrlvη/R, where where F = [ML/T2] is the force, r = [L] is the radius, l = [L] is the length, v = [L/T] is the speed and R = [L] is the distance. What are the dimensions and SI unit of the viscosity η? Solving for the viscosity η, the dimensions Solving the are η = FR / 2πrlv = [ M/LT ], and the rlv M/LT and proper SI unit is kg/ms. kg/ms A Dimensional Analysis Example Example Significant Figures & Examples Examples The number of significant figures in a The significant number is the number of digits whose values are known with certainty. values Any digit that is not zero is significant. Thus, Any 549 has three significant figures and 1.892 549 1.892 has four significant figures. has significant Zeros between non–zero digits are significant. zero Thus, 4023 has four significant figures. four 4023 Significant Figures & Examples, cont. Examples, Zeros to the left of the first non–zero digit are zero not significant. Thus, 0.000034 has only two has significant figures, which can be more easily seen if it is written as 3.4×10–5. 3.4 For numbers with decimal points, zeros to the For right of a non zero digit are significant. Thus 2.00 has three significant figures and 0.050 2.00 0.050 has two significant figures. For this reason it is important to keep the trailing zeros to indicate the actual number of significant figures. the Significant Figures & Examples, cont. Examples, For numbers without decimal points, trailing For zeros may or may not be significant. Thus, 400 indicates only one significant figure. To indicate that the trailing zeros are significant a decimal point must be added. For example, 700. has three significant figures, and 70 has 700. 70 one significant figure. one Significant Figures & Examples, cont. Examples, Exact numbers have an infinite number of Exact significant digits. For example, if there are two oranges on a table, then the number of oranges is 2.000.... Defined numbers are 2.000... Defined also have this property. For example, the number of centimeters per inch (2.54) has an number has infinite number of significant digits, as does the speed of light (299792458 m/s). the 299792458 m/s Significant Figures & Examples, cont. Examples, After addition or subtraction, the result is After significant only to the place determined by the place largest last significant place in the original numbers. For example, 89.332 + 1.1 = 90.432 89.332 should be rounded to get 90.4 since 1.1 has 90.4 1.1 significance in the tenths place. Significant Figures & Examples, cont. Examples, After multiplication or division, the number of After number significant figures in the result is determined by the original number with the smallest number of significant figures. For example, (2.80) (4.5039) = 12.61092 (2.80) should be rounded off to 12.6 since 2.80 has 12.6 2.80 three significant figures. three Measurement & Experimental Error Experimental In the laboratory, we collect measurements to In test a hypothesis or to confirm a theory. test All measurements have errors that are either All systematic or random in nature. systematic Experimental error is captured by statistical Experimental methods, such as a percent difference, standard deviation or a linear regression. Basic Definitions of Statistics Basic A measurement, denoted by xi, iis an s measurement denoted observation that is qualitative (categorical) or quantitative (numerical) description of a phenomenon. phenomenon. The set of all measurements, denoted by {xi}, The is called the population. population A sample population, also denoted by {xi}, iis s sample also a selected (finite) subset of the population. selected We represent the experiment with a Venn We diagram expresses a relationship between diagram sets. From a population of all possible values, we measure a sample population {xi}. Population Sample Venn Diagrams Venn x1, x2, x3,…, xN Statistics of Experimental Error Statistics Recall that there are two types of errors: Recall systematic and random. Many times you will find results quoted with two errors. The first error quoted is usually the random error, and the second is called the systematic error. If only one error is quoted, then the errors from all sources are added together. We will use statistical tools, such as the mean, standard deviation or linear regression, to quantify the experimental error. experimental The Mean & Standard Deviation The Measurements tend to cluster around a Measurements certain numerical value called the (arithmetic) certain mean, and is given by mean and . The “closeness” of a set of measurements to The the mean is called the sample standard deviation, and is given by deviation, i= 1 1 x= N ∑ N xi σ= ∑ ( x − x) i= 1 i N 2 N−1 . The Mean & Standard Deviation, cont. The Mean & Standard Deviation, cont. cont. Given a set of measurements {xi}, we can Given we determine the mean and standard deviation, which we write as x ± σ . which Linear Regression Linear During an experiment, suppose we collect two During sets of measurements {xi} and {yi}. If we If graph the measurements along the x– and y– axis, respectively, then one can use a linear axis, regression to determine the “best” fitting line that relates these two measurements to one another. relates Linear Regression Linear Using a method called a linear regression, we Using linear we can fit a line, of the form y=mx+b, to a sample y=mx+b to population of points {(xi, yi)}, where {( m= ∑ N i= 1 1 N N xi y i − ∑ xi ∑ y i N i= 1 i= 1 ∑ and N i= 1 1 x− ∑ xi N i= 1 2 i N 2 b = y − mx . ...
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This note was uploaded on 02/10/2010 for the course PHY 2053 taught by Professor Hardy during the Spring '10 term at University of Southern Maine.

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