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PHYSICS 231 Lecture 1: Units, Measurement, Dimension “This could be the discovery of the century, depending of course, on far it goes down” PHY 231 1 Why? Make No Mistake: Medical Errors Can Be Deadly Serious The American Hospital Association lists these as some common types of medication errors: •incomplete patient information •unavailable drug information •miscommunication of drug orders, which can involve poor handwriting, confusion between drugs with similar names, misuse of zeroes and decimal points, confusion of metric and other dosing units, and inappropriate abbreviations PHY 231 2 example • Medication errors in US: 44000-98000 deaths/year • 4th-6th leading cause of death in US • 3000-6800 are due to medication math errors • Unit conversion (e.g. grams into milligrams) • Translation language into equations (e.g. a solution of 50 1-mg drops per liter water into 1 mg/20 ml) • Unit conversions are also important in many other fields (packaging, architecture, construction etc etc) PHY 231 3 What is a meter? • Early 18th Century: •Length of pendulum with half length of 1s •Distance from N-pole to Equator via Paris /107 • 1791: second was chosen and 1874 Alloy was made • 1889:more precise by using Platinum/Iridium Alloy • 1960:Using wavelength by using Krypton86 radiation • 1983: distance traveled by light in vacuum in 1/299792458 s PHY 231 4 What is a second? • Originally 1/86400 of a mean solar day • 1960: based on a tropical year • 1967: 9192631770 periods of radiation corresponding to the transition between two hyperfine states of the ground state of 133Cesium PHY 231 5 What is a kilogram? • End of 18th century: 1 dm3 of water • 1889: defined to a Platinum-Iridium weight PHY 231 6 Système Internationale (SI) 7 Standard Units: We will use them!! Quantity Name unit length meter m mass kilogram kg time second electrical Current Ampere A temperature Kelvin s K amount of substance mole mol Luminous intensity cd candela Examples: Speed: m/s Acceleration: m/s2 Force: kg·m/s2 (N) NIST: http://physics.nist.gov/cuu/index.html PHY 231 7 Dimensional Analysis Dimension should be treated as algebraic quantities! x=x0+(at2)/2 Correct dimensionally? [m]=[m]+([m]/[s]2)[s]2… YES! Think about unit conversions! X,X0 in ft. a in m/s2 [ft]=[ft]+([m]/[s]2)[s]2… ???? 1 m = 3.281 ft [ft]=[ft]+([ft]/[s]2)[s]2… YES! GOOD WAY TO CHECK IN EXAMS & LIFE!! PHY 231 8 scales PHY 231 9 Very large & very small numbers 0.000000000000001=10-15=1E-15 femto (f) 0.000000000001=10-12=1E-12 pico (p) 0.000000001=10-9=1E-9 nano (n) 0.000001=10-6=1E-6 micro (μ) 0.001=10-3=1E-3 milli (m) 0.01=10-2=1E-2 centi (c) 0.1=10-1=1E-1 deci (d) 1=100=1E+0 10=101=1E+1 deca (da) 100=102=1E+2 hecto (h) 1000=103=1E+3 kilo (k) 1000000=106=1E+6 mega (M) 1000000000=109=1E+9 giga (G) 10000000000=1012=1E+12 tera (T) PHY 231 10 Greek alphabet We will use some of those… PHY 231 11 Uncertainty & Significance 1.2411 trillion digits known!! π=3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196... PHY 231 12 Significance & Uncertainty Circumference=2πR=2·3.1415926... ·4.2 = Calculator: 26.38937… m Right answer: 26 m. R=4.2 m 4.2 means that the true value lies between 4.1 and 4.3: 2π4.1=25.761…=26 m 2π4.3=27.0176…=27 m so: Right Answer with error: 26±1 The number of significant figures for a result of a division or multiplication is the least accurate of the quantities being divided or multiplied. PHY 231 13 Significance & Uncertainty For addition and subtraction the number of decimal places should be equal to the smallest number of decimal places of any term in the sum: 3.0001 + 0.0025 = 3.0026 NUMBER OF DECIMAL PLACES IS NOT THE SAME AS THE NUMBER OF SIGNIFICANT FIGURES! 0.0025: 2 significant figures 4 decimal places Scientific notation. For example 7·107 or 7E+07 PHY 231 14 2D Geometrical objects object square surface circumference Length2 4xlength triangle ½x base x height Side1+side2+side3 circle πx radius2 2 x π x radius Radius=diameter/2 rectangle oval Length x height 2xlength + 2xheight πxr1xr2 - PHY 231 15 3D geometrical objects object surface volume 6xlength2 length3 2hw+2lh+2wh hlw sphere 4πr2 4/3 x πr3 cylinder 2πrh πr2h cube rectangle PHY 231 16 Coordinate Systems r b Cartesian Coordinates: (x,y)=(a,b) Plane polar coordinates: (r,θ) θ a r = a +b 2 Frame transformation: 2 tan(θ ) = b / a PHY 231 17 TRIGONOMETRY SOH-CAH-TOA: sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent Pythagorean theorem: c = a +b 2 2 2 Note that sin,cos,tan are dimensionless. 2π radians corresponds to 360o PHY 231 18 Be careful… l β h α h=l x sinα =l x cosβ Check carefully which angle is given PHY 231 19 Solving quadratic equations ax 2 + bx + c = 0 a≠0 − b ± b 2 − 4ac x= 2a In general there are 2 solutions. In physics problems, One of them is often not realistic and is thrown out. PHY 231 20 How to solve a problem? There are no general rules but here are some pointers: 1) READ problem carefully! 2) Summarize (throw away unnecessary info) 3) Visualize (drawing can often help) 4) Convert units (consistency) 5) Set up equations: • Plug in numbers if not comfortable with solving sets of equations • If confidents, plug in numbers at last moment 6) Check whether answers make sense 7) In exams: once you have solved a problem, check calculations one more time at the end of exam. Especially important if you tend to make small mistakes PHY 231 21 Problem:The diameter of the orbit of the earth around the sun is 4x1011 m. (1) What is the distance traveled by the earth in 1 year? (2) in the polar coord. System with the sun in the center, over what angle does the earth travel in 0.30 year? (1) a) 1x1012 m (2) a) 108 degrees b) 2x1012 m b) 120 degrees c) 2.5x1012 m c) 1.88 radians d) 1.3x1023 m d) 1.1x102 degrees PHY 231 22 Calculate the length of the shorter of two sides of a rectangle, which has an area of 24 m2 and a perimeter (circumference) of 22 m. PHY 231 23 PHY 231 24