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Ch. 1: Introduction, Measurement, Estimating

Ch. 1: Introduction, Measurement, Estimating. Chapter 1 Outline. 1. The Nature of Science 2. Models, Theories, & Laws 3. Measurement & Uncertainty Significant Figures 4. Units, Standards, & the SI System 5. Converting Units 6. Order of Magnitude: Rapid Estimating

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Ch. 1: Introduction, Measurement, Estimating

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  1. Ch. 1: Introduction, Measurement, Estimating

  2. Chapter 1 Outline 1. The Nature of Science 2. Models, Theories, & Laws 3. Measurement & Uncertainty Significant Figures 4. Units, Standards, & the SI System 5. Converting Units 6. Order of Magnitude: Rapid Estimating 7. Dimensions & Dimensional Analysis

  3. Physics:The most basic of all sciences! • Physics: The “Parent” of all sciences! • Physics= The study of the behavior of and the structure of matter and energy and of the interaction between matter and energy.

  4. Sub Areas of Physics • This course(Phys. 1408,the Physics of the 16th & 17th Centuries): • Motion (MECHANICS) (most of our time!) • Fluids & Waves • Next course(Phys. 2401,the Physics of 18th & the 19th Centuries): • Electricity & magnetism • Light & optics • Advanced courses (Phys. 2402 & others. The Physics of the 20th Century!): • Relativity, atomic structure, condensed matter, nuclear physics, …. These are the most interesting Physics topics & the topics which are the most relevant to modern technology!

  5. Mechanics:“Classical” Mechanics

  6. Mechanics:“Classical” Mechanics • “Classical” Physics: “Classical”   Before the 20th Century The foundation of pure & applied macroscopic physics & engineering! • Newton’s Laws+ Boltzmann’sStatistical Mechanics(& Thermodynamics):  Describe most of macroscopic world! • However, at high speeds (v ~ c) we need Special Relativity:(Early 20th Century: 1905) Ch. 14 of M&T • Also, for small sizes(atomic & smaller) we need Quantum Mechanics:(1900 through ~ 1930) Physics 4307! “Classical” Mechanics:(17th & 18th Centuries) Still useful today!

  7. The mechanics in this course is limited to macroscopic objects moving at speeds v much, much smaller than the speed of light c = 3  108 m/s. As long as v << c, our discussion will be valid. “Classical” Mechanics So, we will work exclusively in the gray region in the figure.

  8. FYI: The Structure of Physics

  9. Mechanics • The science of HOW objects move (behave) under given forces. • (Usually) Does not deal with the sourcesof forces. Answers the question: “Given the forces, how do objects move”?

  10. Physics: General Discussion • The Goal of Physics(& all of science):To quantitatively and qualitatively describe the “world around us”. • PhysicsIS NOTmerely a collection of facts & formulas! • PhysicsISa creative activity! • Physics Observation Explanation. • Requires IMAGINATION!!

  11. Physics & Its Relation to Other Fields • The “Parent” of all Sciences! • The foundation for and is connected to ALL branches of science and engineering. • Also useful in everyday life and in MANYprofessions • Chemistry • Life Sciences (Medicine also!!) • Architecture • Engineering • Various technological fields

  12. Physics Principles are used in many practical applications, including construction. Communication between Architects & Engineers is essential if disaster is to be avoided.

  13. The Nature of Science • Physics is an EXPERIMENTAL science! • Experiments & Observations: • Important first steps toward scientific theory. • It requires imagination to tell what is important • Theories: • Created to explain experiments & observations. Will also make predictions • Experiments & Observations: • Will tell if predictions are accurate. • No theorycan be absolutely verified • But a theory CAN be proven false!!!

  14. Theory • Quantitative (mathematical) descriptionof experimental observations. • Not justWHATis observed butWHYit is observed as it is andHOWit works the way it does. • Tests of theories: • Experimental observations: More experiments, more observation. • Predictions: Made before observations & experiments.

  15. Model, Theory, Law • Model: An analogy of a physical phenomenon to something we are familiar with. • Theory: More detailed than a model. Puts the model into mathematical language. • Law: Concise & general statement about how nature behaves. Must be verified by many, many experiments! Only a few laws. • Not comparable to laws of government!

  16. How does a new theoryget accepted? • Predictionsagree better with data than old theory • Explainsa greater range of phenomena than old theory • Example: • Aristotle believed that objects would return to a state of rest once put in motion. • Galileorealized that an object put in motion would stay in motion until some force stopped it.

  17. Measurement & Uncertainty.Significant Figures No measurement is exact; there is always someuncertaintydue to limited instrument accuracy and difficulty reading results. The photograph to the left illustrates this – it would be difficult to measure the width of this 2  4 to better than amillimeter.

  18. Measurement & Uncertainty • Physics is an EXPERIMENTALscience! • Finds relations between physical quantities. • Expresses those relations in the language of mathematics. (LAWS & THEORIES) • Experiments are NEVER 100% accurate. • Alwayshave uncertainty in final result. • Experimental error. • Common to state this precision (when known).

  19. Consider a simple measurement of the width of a board. Find 23.2 cm. • However, measurement is only accurate to 0.1 cm (estimated).  Write width as (23.2  0.1) cm  0.1 cm  Experimental uncertainty • Percent Uncertainty:  (0.1/23.2)  100   0.4%

  20. Significant Figures Significant figures (“sig figs”): The number of significant figuresis the number of reliably known digits in a number.It is usually possible to tell the number of significant figures by the way the number is written: 23.21 cm has 4 significant figures 0.062 cm has 2 significant figures (initial zeroes don’t count) 80 km is ambiguous: it could have 1 or 2 significant figures. If it has 3, it should be written 80.0 km.

  21. Calculations Involving Several Numbers When multiplying or dividing numbers: The number of sig figs in the result  the same number of sig figs as the number used in the calculation with the fewest sig figs. When adding or subtracting numbers: The answer is no more accurate than the least accurate number used in the calculation.

  22. Example: (Not to scale!) • Area of board, dimensions 11.3 cm  6.8 cm • Area = (11.3)  (6.8) = 76.84 cm2 11.3 has 3 sig figs , 6.8 has 2 sig figs  76.84 has too many sig figs! • Proper number of sig figs in answer = 2  Round off 76.84 & keep only 2 sig figs  Reliable answer for area = 77 cm2

  23. Sig Figs • General Rule: The final result of a multiplication or division should have only as many sig figs as the number used in the calculation which has the with least number of sig figs. • NOTE!!!! All digits on your calculator are NOT significant!!

  24. Calculators will not give you the right number of significant figures; they usually give too many, but sometimes give too few (especially if there are trailing zeroes after a decimal point). The top calculator shows the result of 2.0 / 3.0. The bottom calculator shows the result of 2.5 x 3.2.

  25. Conceptual Example 1-1: Significant figures • Using a protractor, you measure an angle of 30°. (a) How many significant figures should you quote in this measurement? (b) Use a calculator to find the cosine of the angle you measured. • (a) Precision ~ 1° (not 0.1°). So 2 sig figs & angle is 30° (not 30.0°). • (b)Calculator: cos(30°) = 0.866025403. But angle precision is 2 sig figs so answer should also be 2 sig figs. So cos(30°) = 0.87

  26. Powers of 10(Scientific Notation) • READAppendices A-2 & A-3 • Common to express very large or very small numbers using power of 10 notation. • Examples: 39,600 = 3.96  104 (moved decimal 4 places to left) 0.0021 = 2.1  10-3 (moved decimal 3 places to right) PLEASE USE SCIENTIFIC NOTATION!!

  27. Powers of 10(Scientific Notation) PLEASE USE SCIENTIFIC NOTATION!! • This is more than a request!!I’m making it a requirement!!I want to see powers of 10 notation on exams!! For example: • For large numbers, like 39,600, I want to see3.96  104 & NOT39,600!! • For small numbers, like 0.0021, I want to see2.1  10-3 & NOT0.0021!! • On the exams, you will lose points if you don’t do this!!

  28. Accuracy vs. Precision • Accuracy is how close a measurement comes to the accepted (true) value. • Precision is the repeatability of the measurement using the same instrument & getting the same result! It is possible to be accurate without being precise and to be precise without being accurate!

  29. Units, Standards, SI System • All measured physical quantities have units. • Units are VITAL in physics!! • In this course (and in most of the modern world, except the USA!) we will use (almost) exclusively the SIsystem of units. SI = “Systéme International” (French) More commonly called the “MKS system” (meter-kilogram-second) or more simply, “the metric system”

  30. SI or MKS System • Defined in terms of standards for length, mass, and time. • Length unit:Meter (m)(kilometer = km = 1000 m) • Standard Meter. Newest definition in terms of speed of light  Length of path traveled by light in vacuum in (1/299,792,458) of a second! • Time unit:Second (s) • Standard Second. Newest definition  time required for 9,192,631,770 oscillations of radiation emitted by cesium atoms! • Mass unit:Kilogram (kg) • Standard Kilogram Mass of a specific platinum-iridium alloy cylinder kept at Intl Bureau of Weights & Measures in France

  31. Larger & Smaller Units are Defined from SI standards by Powers of 10 & Greek Prefixes __ These are the standard SI prefixes for indicating powers of 10. Many (k, c, m, μ) are familiar; Others (Y, Z, E, h, da, a, z, y) are rarely used.  __  __  __ 

  32. Typical Lengths (approx.)  

  33. Typical Times (approx.)  

  34. Typical Masses (approx.)  

  35. Units, Standards, and the SI System We will work (almost) exclusively in the SI System, where the basic units are kilograms, meters, & seconds.

  36. Other Systems of Units • CGS(centimeter-gram-second) system • Centimeter= 0.01 meter • Gram = 0.001 kilogram • British (Engineering) System (foot-pound-second; or US Customary system) • “Everyday life” system of units • Only used by USA & some third world countries. Rest of world (including Britain!) uses SI system. We will not use the British System! • Conversions exist between the British & SI systems. We will not use them in this course!

  37. In this class, we willNOTdo unit conversions! We will workexclusivelyin SI (MKS) units!

  38. Basic & Derived Quantities • Basic Quantity  Must be defined in terms of a standard (meter, kilogram, second). • Derived Quantity  Defined in terms of combinations of basic quantities • Unit of speed (v = distance/time) = meter/second = m/s • Unit of density (ρ = m/V) = kg/m3

  39. Units and Equations • In dealing with equations, remember that the units must be the same on both sides of an equation (otherwise, it is not an equation)! • Example: You go 90 km/hr for 40 minutes. How far did you go?

  40. Units and Equations • In dealing with equations, remember that the units must be the same on both sides of an equation (otherwise, it is not an equation)! • Example: You go 90 km/hr for 40 minutes. How far did you go? • Equation from Ch. 2: x = vt, v = 90 km/hr, t = 40 min. To use this equation, first convert t to hours: t = (⅔)hr so,x = (90 km/hr)  [(⅔)hr] = 60 km The hour unit (hr) has (literally) cancelled out in the numerator & denominator!

  41. Converting Units • As in the example, units in the numerator & the denominator can cancel out (as in algebra) • Illustration:Convert 80 km/hr to m/s Conversions: 1 km = 1000 m; 1hr = 3600 s

  42. Converting Units • As in the example, units in the numerator & the denominator can cancel out (as in algebra) • Illustration:Convert 80 km/hr to m/s Conversions: 1 km = 1000 m; 1hr = 3600 s  80 km/hr = (80 km/hr) (1000 m/km) (1hr/3600 s) (Cancel units!) 80 km/hr  22 m/s (22.222…m/s) • Useful conversions: 1 m/s  3.6 km/hr; 1 km/hr  (1/3.6) m/s

  43. Order of Magnitude; Rapid Estimating • Sometimes, we are interested in only an approximate value for a quantity. We are interested in obtaining rough or order of magnitude estimates. • Order of magnitude estimates: Made by rounding off all numbers in a calculation to 1 sig fig, along with power of 10. • Can be accurate to within a factor of 10 (often better)

  44. Example:V = πr2d • Example: Estimate! Estimate how much water there is in a particular lake, which is roughly circular, about 1 km across, & you guess it has an average depth of about 10 m.

  45. Example 1-6: Thickness of a page. Estimate the thickness of a page of your textbook. Hint:You don’t need one of these!

  46. Example 1-7:Height by triangulation. Estimate the height of the building shown by “triangulation,” with the help of a bus-stop pole and a friend.(See how useful the diagram is!)

  47. Example 1-8:Estimate the Earth radius. If you have ever been on the shore of a large lake, you may have noticed that you cannot see the beaches, piers, or rocks at water level across the lake on the opposite shore. The lake seems to bulge out between you and the opposite shore—a good clue that the Earth is round. Suppose you climb a stepladder and discover that when your eyes are 10 ft (3.0 m) above the water, you can just see the rocks at water level on the opposite shore. On a map, you estimate the distance to the opposite shore as d≈ 6.1 km. Use h = 3.0 m to estimate the radius R of the Earth.

  48. Dimensions & Dimensional Analysis The dimensions of a quantity are the base units that make it up; generally written using square brackets. Example: Speed = distance/time Dimensions of speed: [L/T] Quantities that are being added or subtracted must have the same dimensions. In addition, a quantity calculated as the solution to a problem should have the correct dimensions.

  49. Dimensional Analysis • If the formula for a physical quantity is known  The correct units can easily be found! • Examples:Volume:V = L3 Volume unit = m3 Cube with L =1 mmV = 1 mm3= 10-9 m3 Density: ρ = m/V Density unit = kg/m3 ρ = 5.3 kg/m3 = 5.3 10-6 g/mm3 • If the units of a physical quantity are known  The correct formula can be “guessed”! • Examples: Velocity: Car velocity is 60 km/h Velocity unit = km/h Formula:v = d/t(d = distance, t = time) Acceleration: Car acceleration is 5 m/s2 Acceleration unit = m/s2 Formula:a = v/t(v = velocity, t = time)

  50. Dimensional analysis is the checking of dimensions of all quantities in an equation to ensure that those which are added, subtracted, or equated have the same dimensions. Example: Is this the correct equation for velocity? Check the dimensions: Wrong!

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