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Physics 621 Math Review SCIENTIFIC NOTATION Scientific Notation is based on exponential notation (where decimal places are expressed as a power of 10). The numerical part of the measurement is expressed as a number between 1 and 10 multiplied by a whole-number power of 10. M * 10n , 1≤ M < 10, where n is an integer (+ or - #). Standard Notation 2000 Standard Notation → 2 * 103 180 g → 1.8 * 102 g or 1.8 * 10-1 kg SIGNIFICANT FIGURES Significant Figures - The number of digits is rough but useful indication of a measurements precision. Each digit obtained as a result of measurement is a significant figure. The last digit of each measured quantity is always estimated. The zeros in a number warrant special attention. A zero that is the result of a measurement is significant, but zeros that serve only to mark a decimal point are not significant. Example: A) 65 ml (2 sig figs) B) 173.4 (4 sig figs) C) 13.2 g (3 sig figs) D) 5 ml (1 sig figs) Rules for Significant Figures 1. Non-zero digits are always significant. Ex. A) 234.7 L 2. A zero between other SF is significant. Ex. A) 1.05 (4 sig figs) (3 sig figs) B) 1.921 kg B) 2001 m (4 sig figs) (4 sig figs) 3. Final zeros to the right of the decimal point are significant Ex. A) 6.30 g (3 sig figs) B) 10.00 ml (4 sig figs) 4. Initial zeros are not significant and serve only to show place of decimal. Ex. A) 0.069 km (2 sig figs) B) 0.0107 (3 sig figs) 5.Final zeros in numbers with no decimal point may or may not be significant. Example: A) 20 marbles *count, exact, infinite* B) 2000 m *1* C) 20 lbs *1* precise from 15 - 24.99.... D) 2.0 x 10¹ lbs *2* precise from 19.5 - 20.499.... E) 2.00 x 10¹ lbs *3* precise from 19.95 - 20.0499... F) 1 km = 1000 m *definition, infinite* Interpretations of Significant Figures 200 lbs *1* significant figures ... better written 2 * 102 lbs 200 lbs *2* significant figures ... better written 2.0 * 102 lbs 200 lbs *3* significant figures ... better written 2.00 * 102 lbs COUNTS, CONSTANTS, DEFINITIONS All have an infinite number of significant figures.(∞) COUNT - Ex. 10 marbles, 3 people .... Exact CONSTANTS - Ex. Consider a + 2b = c . The number 2 is a constant. DEFINITIONS - Ex. 1 km = 1000 m, 12 = 1 dozen SIGNIFICANT FIGURE CALCULATIONS The result of any mathematical calculation involving measurements cannot be more precise than the least precise measurement. (Assume all the numbers below are from measuring) Addition and Subtraction When adding or subtracting measured quantities, the answer should be expressed to the same number of decimal places as the least precise quantity used in the calculation. ( If needed use a LINE OF SIGNIFICANCE to aid in solving these.) Example: A) B) C) D) E) F) 94.02 + 61.1 + 3.1416 = 158.2616 = 158.3 4.01 - 2.30642 = 1.70357 = 1.70 6500 + 730 = 7230 = 7230 98 + 9 = 107 1107 - 107 = 1000 = 1000 1100 -51 = 1049 = 1049 Multiplication, Division, and Square Root When multiplying, dividing, or finding the square root of measured quantities, the answer should have the same number of significant digits as the least precise quantity used in the calculation. Example: A) 5.6432 * 0.020 = 0.112864 = 0.11 B) 2500 *2 = 5000 = 5*103 C) 26.3 * 35 = 920.5 = 920 ROUNDING When completing calculations, do not round any of the intermediate answers on your way to finding a solution to a problem. The only rounding that should occur is in the final answer that is being reported. Example: 16.75 2.4 40.2 1.2078601...... 3.698 9 33.282 Now, at the very end round to 1 sig fig Answer is “1” INVERSELY AND DIRECTLY PROPORTIONAL When considering what effect changing one or more variables has on another variable in mathematical relationships inversely and directly proportional relationships are used. AB C D C ∝ A, C ↑ A↑, or C↓ A↓, if C doubles then A doubles C ∝ 1/D D ↑ C↓, or D↓ C↑ , if D doubles then C becomes half Directly Proportional Quantities Quantities that are directly proportional to one another increase or decrease by the same factor. Quantities that are directly proportional to one another occupy the same position on opposite sides of the proportion sign (either both located in the numerator position or both in the denominator position). AX = ZP Z & X are directly proportional to each other. ( Z α X ) Inversely Proportional Quantities Quantities that are inversely proportional to one another change by the reciprocal of one another (or 1/x of one another). In a proportion, quantities that are inversely proportional to one another occupy opposite positions on opposite sides of the proportion sign. Z N X M Z & M are inversely proportional to each other. ( Z α 1/M) Example: Given the following formula, what would happen to v if r is doubled and T is tripled? 2 r v T Answer: v would change by a factor of, 2 v 3 Example: Given the following formula, what would happen to mc if T is changed by a factor of 2 and G by a factor of ½? r 3 Gmc 2 2 T 4 Answer: v would change by a factor of, 1 mc 2 Unit Analysis Often we need to change the units in which a physical quantity is expressed. For example we may need to change seconds and minutes, hours, days or even years, to do this we use conversion factors. Example: 60 sec 1 1min and 1min 1 60 sec When a quantity is multiplied by conversion factor it does not change the amount of quantity just the units the quantity is measured in. When a conversion factor is evaluated its value is equal to 1. Any number multiplied by 1 remains unchanged. Example: 180sec1 1min 180sec 60sec Second are cancelled which leaves units of minutes Answer: (3 min) Example: How many centimetres are in 1 km? 1000m 100cm 1km ? 1km 1m 1*105 cm Example: How many seconds are in one day? 24hr 60 min 60sec 1day ? 1day 1hr 1min 86400 sec Example: Convert 2.4 km/hr to m/s km 1hr 1min 1000m 2.4 ? hr 60 min 60sec 1km 0.666… m/s The metric system Prefix Symbol Factor 1 tera T 1 000 000 000 000 10 12 1 giga G 1 000 000 000 10 9 1 mega M 1 000 000 10 6 1 kilo k 1 000 10 3 1 hecto h 100 10 2 1 deca da 10 10 1 base unit base unit 1 10 0 1 deci d 0.1 10 -1 1 centi c 0.01 10 -2 1 milli m 0.001 10 -3 1 micro 0.000 001 10 -6 1 nano n 0.000 000 001 10 -9 1 pico p 0.000 000 000 001 10 -12 1 femto f 0.000 000 000 000 001 10 -15 1 atto a 0.000 000 000 000 000 001 10 -18 Re-arranging formulas Given the following formula, q Ek 2 r Solve for q: Er q k 2 Given the following formula, 4 2 r ac 2 T Solve for T: 4 r T ac Given the following formula, d1 v 2 f vi t Solve for vf: 2d vf vi t Trigonometry Trigonometry is the study and solution of Triangles. Solving a triangle means finding the value of each of its sides and angles. B c a ө C b Side Opposite Hypothenuse Side Adjacent Cos θ= Hypothenuse Side Opposite Tan θ= Side Adjacent Sinθ= A = a c b c = a b = Find the angles of the following triangle. B α 6 10 β C θ A 8 Opp 6 sin 0.6 Hyp 10 sin 1 sin sin 0.6 1 39.6 40 β is a right angle therefore it is 90° and because all the angles of a triangle add up to 180° ά must be 50° The measurements have changed. Find side BA and side AC B Sin34=2/BA 2 α 0.559=2/BA 0.559BA=2 β 34º C A BA=2/0.559 BA~3.578 The Pythagorean theorem when used in this triangle states that… (BC)2+(AC)2=(AB)2 (AC)2=(AB)2-(BC)2 (AC)2=12.802-4=8.802 (AC)~3