BASIC MATH FOR PHYSICS Unit I-Part A

SCIENCE is… the search for relationships that explain and predict the behavior of the universe.

PHYSICS is… the science concerned with relationships between matter, energy, and its transformations.

There is no such thing as absolute certainty of a scientific claim. The validity of a scientific conclusion is always limited by: the experiment design, equipment, etc... the experimenter human error, interpretation, etc... our limited knowledge ignorance, future discoveries, etc...

an experimentally confirmed explanation Scientific Law a statement describing a natural event Scientific Theory an experimentally confirmed explanation for a natural event Scientific Hypothesis an educated guess (experimentally untested)

perception of “Communist” system natural resistance to change developed in France in 1795 a.k.a. “SI” - International System of Units The U.S. was (and still is) reluctant to “go metric.” very costly to change perception of “Communist” system natural resistance to change American pride

The SI unit of: length is the meter, m time is the second, s mass is the kilogram, kg. electric charge is the Coulomb, C temperature is the degree Kelvin, K an amount of a substance is the mole, mol luminous intensity is the candle, cd

“Derived units” are combinations of these “fundamental units” Examples include speed in m/s, area in m2, force in kg.m/s2, acceleration in m/s2, volume in m3, energy in kg.m2/s2

1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101 deka da 10-18 atto a 10-15 femto f 10-12 pico p 10-9 nano n 10-6 micro m 10-3 milli m 10-2 centi c 10-1 deci d

Accuracy % error = x 100% accepted - observed accepted All measurements have some degree of uncertainty. Precision single measurement - exactness, definiteness group of measurements - agreement, closeness together Accuracy closeness to the accepted value accepted - observed accepted % error = x 100%

Example of the differences between precision and accuracy for a set of measurements: Four student lab groups performed data collection activities in order to determine the resistance of some unknown resistor (you will do this later in the course). Data from 4 trials are displayed below. Suppose the accepted value for the resistance is 500. Then we would classify each groups’ trials as: Group 1: neither precise nor accurate Group 2: precise, but not accurate Group 3: accurate, but not precise Group 4: both precise and accurate Group Trial 1 Trial 2 Trial 3 Trial 4 Avg 1 34 612 78 126 2 127 128 3 20 500 62 980 4 502 501 503 498

ALGEBRA & EQUATIONS THE USE OF BASIC ALGEBRA REQUIRES ONLY A FEW FUNDAMENTAL RULES WHICH ARE USED OVER AND OVER TO REARRANGE AND SOLVE EQUATIONS. (1) ANY NUMBER DIVIDED BY ITSELF IS EQUAL TO ONE. (2) WHAT IS EVER DONE TO ONE SIDE OF AN EQUATION MUST BE DONE EQUALLY TO THE OTHER SIDE. (3) ADDITIONS OR SUBTRACTIONS WHICH ARE ENCLOSED IN PARENTHESES ARE GENERALLY CARRIED OUT FIRST. (4) WHEN VALUES IN PARENTHESES ARE MULTIPLIED OR DIVIDED BY A COMMON TERM EACH CAN BE MULTIPLIED OR DIVIDED SEPARATELY BEFORE ADDING OR SUBTRACTING THE GROUPED TERMS.

10/10 =1 X /X = 1 Y /Y =1 10 + X + 5 = Y + 10 X + 15 = Y + 10 5 x ( ALGEBRA & EQUATIONS RULE 1 – A VALUE DIVIDED BY ITSELF EQUALS 1 10/10 =1 X /X = 1 Y /Y =1 RULE 2 – OPERATE ON BOTH SIDES EQUALLY 10 + X + 5 = Y + 10 X + 15 = Y + 10 IF WE ADD 10 TO THE LEFT SIDE WE MUST ADD 10 TO THE RIGHT 5 x ( X + 5 = Y ) 5 x X + 5 = Y IF WE MULTIPLY THE LEFT SIDE BY 5 WE MUST MULTIPLY THE RIGHT BY 5 5X + 25 = 5Y

RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST ALGEBRA & EQUATIONS RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST Y = ( 5 + 4 ) ( X + 2) Y = 9 ( X + 2 ) THE PARENTHESES TERMS (5 + 5) ARE ADDED FIRST Y = 9 X + 18 2 S = 15 T 2 S = T ( 22 - 7 ) THE PARENTHESES TERMS (22 – 7) ARE SUBTRACTED FIRST S = 225 T

RULE 4 – VALUES CAN BE DISTRIBUTED THROUGH TERMS IN PARENTHESES ALGEBRA & EQUATIONS RULE 4 – VALUES CAN BE DISTRIBUTED THROUGH TERMS IN PARENTHESES Y = 4 ( T + 15 ) Y = 4 T + 60 EACH TERM IN THE PARENTHESES MUST BE MULTIPLIED BY 4 Y = ( R x R ) - 3 R + 2 R - 6 Y = ( R + 2 ) ( R - 3 ) Y = R - 1R - 6 2 ALL TERMS MUST BE MULTIPLIED BY EACH OTHER THEN ADDED Y = R - R - 6 2

SOLVING ALGEBRAIC EQUATIONS SOLVING AN ALGEBRAIC EQUATION REQUIRES THAT THE UNKNOWN VARIABLE BE ISOLATED ON THE LEFT SIDE OF THE EQUAL SIGN IN THE NUMERATOR POSITION AND ALL OTHER TERMS BE PLACED ON THE RIGHT SIDE OF THE EQUAL SIGN. THIS MOVEMENT OF TERMS FROM LEFT TO RIGHT AND FROM NUMERATOR TO DENOMINATION AND BACK, IS ACCOMPLISHED USING THE BASIC RULES OF ALGEBRA WHICH WERE PREVIOUSLY DISCUSSED.

SOLVING ALGEBRAIC EQUATIONS THESE RULES CAN BE IMPLEMENTED PRACTICALLY USING SIMPLIFIED PROCEDURES (KEEP IN MIND THE REASON THAT THESE PROCEDURES WORK IS BECAUSE OF THE ALGEBRAIC RULES). PROCEDURE 1 – WHEN A TERM WITH A PLUS OR MINUS SIGN IS MOVED FROM ONE SIDE OF THE EQUATION TO THE OTHER, THE SIGN IS CHANGED. Y + 5 = 3X - 5 N - 4 = 6 M + 4

SOLVING ALGEBRAIC EQUATIONS PROCEDURE 2 – WHEN A TERM IS MOVED FROM THE DENOMINATOR ACROSS AN EQUAL SIGN TO THE OTHER SIDE OF THE EQUATION IT IS PLACED IN THE NUMERATOR. LIKEWISE, WHEN A TERM IS MOVED FROM NUMERATOR ON ONE SIDE IT IS PLACED IN THE DENOMINATOR ON THE OTHER SIDE. A C B D ----- = ------ ----- = --- F x K G M N M B K = ---------- F G x M N x K A C D = ----------- x B

TRIGNOMETRY TRIGNOMETRIC RELATIONSHIPS ARE BASED ON THE RIGHT TRIANGLE (A TRIANGLE CONTAINING A 900 ANGLE). THE MOST FUNDAMENTAL CONCEPT IS THE PYTHAGOREAN THEOREM (A2 + B2 = C2) WHERE A AND B ARE THE SHORTER SIDES (THE LEGS) OF THE TRIANGLE AND C IS THE LONGEST SIDE CALLED THE HYPOTENUSE. RATIOS OF THE SIDES OF THE RIGHT TRIANGLE ARE GIVEN NAMES SUCH AS SINE, COSINE AND TANGENT. DEPENDING ON THE ANGLE BETWEEN A LEG (ONE OF THE SHORTER SIDES) AND THE HYPOTENUSE (THE LONGEST SIDE), THE RATIO OF SIDES FOR A PARTICULAR ANGLE ALWAYS HAS THE SAME VALUE NO MATTER WHAT SIZE THE TRIANGLE.

The Right Triangle C = the hypotenuse A A & B = the legs  B C A 900  A & B = the legs Pythagorean Theorem A + B = C 2 B C = A + B 2 A RIGHT TRIANGLE A = C - B 2   900 + = 1800 B = C - A 2

TRIG FUNCTIONS THE RATIO OF THE SIDE OPPOSITE THE ANGLE AND THE HYPOTENUSE IS CALLED THE SINE OF THE ANGLE. THE SINE OF 30 0 FOR EXAMPLE IS ALWAYS ½ NO MATTER HOW LARGE OR SMALL THE TRIANGLE. THIS MEANS THAT THE OPPOSITE SIDE IS ALWAYS HALF AS LONG AS THE HYPOTENUSE IF THE ANGLE IS 30 0. (30 0 COORESPONSES TO 1/12 OF A CIRCLE OR ONE SLICE OF A 12 SLICE PIZZA!) THE RATIO OF THE SIDE ADJACENT TO THE ANGLE AND THE HYPOTENUSE IS CALLED THE COSINE. THE COSINE OF 60 0 IS ALWAYS ½ WHICH MEANS THIS TIME THE ADJACENT SIDE IS HALF AS LONG AS THE HYPOTENUSE. (60 0 REPRESENTS 1/6 OF A COMPLETE CIRCLE, ONE SLICE OF A 6 SLICE PIZZA) THE RATIO OF THE SIDE ADJACENT TO THE ANGLE AND THE SIDE OPPOSITE THE ANGLE IS CALLED THE TANGENT. IF THE ADJACENT AND THE OPPOSITE SIDES ARE EQUAL, THE RATIO (TANGENT VALUE) IS 1.0 AND THE ANGLE IS 45 0 ( 45 0 IS 1/8 OF A FULL CIRCLE)

Fundamental Trigonometry (SIDE RATIOS) Sin = A / C  C C C A A A Cos = B / C   Tan  = A / B B B B A RIGHT TRIANGLE

(the number of atoms in a drop of water) Scientific Numbers In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier 5,010,000,000,000,000,000,000 a very large number (the number of atoms in a drop of water) 0.000000000000000000000327 a very small number (mass of a gold atom in grams)

Scientific numbers use powers of 10 2 100 = 10 x 10 = 10 3 1000 = 10 x 10 x 10 = 10 1 -1 0.10 = 1 / 10 = 10 -2 2 0.01 = 1 / 100 = 1 / 10 = 10 2 523 = 5.23 x 100 = 5.23 x 10 -2 2 0.0523 = 5.23/100 = 5.23/10 = 5.23 x 10

Scientific Numbers RULE 1 As the decimal is moved to the left The power of 10 increases one value for each decimal place moved Any number to the Zero power = 1 450,000,000 = 450,000,000. x 10 8 2 3 1 450,000,000 = 450,000,000. x 10 8 4.5 x 10

Scientific Numbers 0.0000072 = 0.0000072 x 10 -6 -2 -3 -1 RULE 2 As the decimal is moved to the right The power of 10 decreases one value for each decimal place moved Any number to the Zero power = 1 0.0000072 = 0.0000072 x 10 -6 -2 -3 -1 0.0000072 = 0.0000072 x 10 -6 7.2 x 10

When scientific numbers are multiplied RULE 3 When scientific numbers are multiplied The powers of 10 are added 100 x 1000 = 100,000 2 100 = 10 1000 = 10 3 2 3 (2 + 3) 5 10 x 10 = 10 = 10 = 100,000 2 3 5 (3 x 10 ) x ( 2 x 10 ) = 6 x 10

When scientific numbers are divided The powers of 10 are subtracted RULE 4 When scientific numbers are divided The powers of 10 are subtracted 10000 / 100 = 100 4 10000 = 10 2 100 = 10 4 2 (4 - 2) 2 10 / 10 = 10 = 10 = 100 4 2 2 (5 x 10 ) / (2 x 10 ) = 2.5 x 10

Scientific Numbers 2 (100) = 10,000 2 100 = 10 4 2 2 (2 x 2) RULE 5 When scientific numbers are raised to powers The powers of 10 are multiplied 2 (100) = 10,000 2 100 = 10 4 2 2 (2 x 2) (10 ) = 10 = 10 = 10,000 2 (3000) = 9,000,000 3 2 2 (2 x 3) 6 (3 x 10 ) = 3 x 10 = 9 x 10

Scientific Numbers square root = 1/2 power cube root = 1/3 power 1/2 RULE 6 Roots of scientific numbers are treated as fractional powers. The powers of 10 are multiplied square root = 1/2 power cube root = 1/3 power 1/2 10,000 = (10,000) 4 10,000 = 10 1/2 (1/2 x 4) 4 1/2 (10,000) = (10 ) = 10 = 100 1/2 1/2 6 1/2 3 6 (9 x 10 ) = 9 x (10 ) = 3 x 10

Scientific Numbers 2 2.34 x 10 3 + 4.24 x 10 --------- 3 2 RULE 7 When scientific numbers are added or subtracted The powers of 10 must be the same for each term. 2 Powers of 10 are Different. Values Cannot be added ! 2.34 x 10 3 + 4.24 x 10 --------- 3 Move the decimal And change the power Of 10 2 2.34 x 10 = 0.234 x 10 3 0.234 x 10 Power are now the Same and values Can be added. 3 + 4.24 x 10 --------- 4.47 x 10 3

Order Of Magnitude Estimation Is Equivalent To Rounding Off Your Estimate To The Nearest Power Of 10

Order Of Magnitude Examples The meter is the standard of measurement in this class A Football Field Is About 102m Long (100m) It’s not 10m, and it’s not 1000m A Credit Card is About 10-3m thick (1mm) It’s not 0.1mm (the thickness of a sheet of copy paper), and it’s not 10mm (width of your pinky) The Period of a Backyard Swing Is About 100s It’s not 0.1s, and it’s not 10s

Units NO UNIT == NO MEANING Almost always, our measurements are in some physical unit THEY ARE NOT JUST NUMBERS!!!! THEY HAVE PHYSICAL MEANING NO UNIT == NO MEANING

These are Called “Derived Units” In Addition, We Use Combinations of Basic Units These are Called “Derived Units” meters per second kilograms per meter3

Unit Conversion Using The Unit Factor Method “Unit” Has Two Meanings: “The” Unit (kg, s, m) or “Unit == ONE” A Way To Convert Values From One Unit To Another It Works Because If You Multiply Something Times 1 It Does Not Change The Value Also called “Factor-Label” method

Unit Factors Unit Factors Are Created From Conversion Factors Like 2.54 cm = 1 inch We Can Make 2 Unit Factors From This

Using Unit Factors Multiply The Value You Want To Convert By The Appropriate Unit Factor Cancel Out Similar Units, Leaving The Desired Unit Multiply Values and You Are Done!! Example : How Many Inches Are In 25.4 Centimeters?

How Many Centimeters Are In Six Inches? Using Unit Factors In Your Notes How Many Centimeters Are In Six Inches?

Using Unit Factors In Your Notes Express 24 cm in inches

Multiple Conversion Steps You Can Also String Multiple Unit Factors Together In Your Notes How Many Seconds Are In 2 Years?

Dimensional Analysis “Analyzing The Dimensions” Of An Equation To Be Sure The Units Match Up And Make Sense Start With A Fancy, Schmanzy Equation Replace Each Unknown With Its “Dimension”

Dimensional Analysis d = distance in meters = [m] v = velocity in meters / second = [m]/[s] a = acceleration in meters / second2 = [m]/[s]2 t = time in seconds = [s] or [s]2

Dimensional Analysis [m] = [m]/[s] X [s] + ½ [m]/[s]2 X [s]2 Cancel Common Dimensions [m] = [m] + ½ [m] Ignore The ½ Factor [meters] = [meters] This Equation Is Dimensionally Consistent