Math Skills Physics Introductory Unit • There are several skills, some of which you have already learned, that you will need to use extensively in Physics. • These include the following: d – Algebra (manipulation of formulas) v = t – Scientific Notation (very lg/sm numbers) Math – Significant Digits 0.07034 Rules! – Unit Conversions 1m = 100cm ~The Mathematical Background~ Algebra Algebra (Sample) • Numerous times while studying Physics, you will be required to use algebra to solve equations. • This involves the manipulation (moving around) of variables with the goal being to solve the equation for the unknown variable (i.e. to isolate the variable). • Isolating the variable involves the use of inverse (opposite) operations to move the other variables. – Addition(+) and Subtraction(-) are inverse operations. – Multiplication(× or ·) and Division(÷) are inverse operations. – Squaring(2) and square rooting(√) are inverse operations. Other Algebra Samples • Given the equation: d v= t • Solve for t. vt = d d t= v • Given the equation: v2 2 = v12 + 2a ( d 2 − d1 ) • Solve for v2. v2 = ± 5.2 × 10−4 (v 2 1 + 2a ( d 2 − d1 ) ) Note: When you take the square root, a ± symbol must be included in front of the radical. • • • • Consider the formula shown. Solve the equation in terms of d. To do this, we must move t. What operation is t associated with? Division • What is the inverse operation? Multiplication • Perform the operation to solve for d. • Some other problems may involve more than one step. v= v ⋅t = d t d ⋅t t d = v ⋅t Scientific Notation • Scientific notation relies on exponential powers of ten (10x) to simplify extremely large and small numbers. Standard Notation Scientific Notation 4, 673, 000 = 4.673 × 106 Power of Ten Coefficient • In all cases, numbers written in scientific notation have a single digit in the ones place followed by the remaining digits placed to the right of the decimal point. This is called the coefficient. • A multiplied power of ten is indicated afterwards. Scientific Notation (Cont.) Scientific Notation (Multiplication) • Large numbers correspond to positive powers of ten. • At times, numbers in scientific notation will be multiplied as shown below. 14, 000 = 1.4 ×104 ( 4.2 ×10 )( 3.1×10 ) 6 • Small numbers correspond to negative powers of ten. 0.00034 = 3.4 ×10 −4 • Figuring out the power on the ten relates to how many places you need to move the decimal point from its initial position. 12080 = 1.208 × 10 4 0.037 = 3.7 ×10 −2 2 Moves 4 Moves Scientific Notation (Division) • At times, numbers in scientific notation will be divided as shown below. (8.4 ×10 ) (1.4 ×10 ) 3 2 • As before, you need to combine terms. The exponent rule changes to subtraction when division is involved. 3 8.4 10 2 1.4 10 6.0 × 101 Significant Digits • Significant digits (sometimes called significant figures) are those digits that are considered important in a given number. • In order to determine which digits are significant, one must look to the following rules. 2 • The trick is to combine the powers of ten with each other and the non-exponent terms with each other. Then simplify. ( 4.2 × 3.1) (106 ×102 ) 13.02 ×108 1.302 ×109 Note: Remember that exponents add when like bases are multiplied. Scientific Notation (10x) • Numbers that are simply powers of ten can be written in a shorter form without a coefficient. • Consider the example dealing with 100,000. 100, 000 = 1.0 ×105 • In simplified form it can be written as follows: 100,000 = 105 • The same holds true for small numbers. 0.001 = 10 −3 Significant Digits (Special Cases) • A bar can be placed over zeros that are not normally significant in order to make them significant. 1 Significant Digit 400 vs. 400 3 Significant Digits – All nonzero digits are significant. 370 or 0.056 – Final zeros after the decimal point are significant. 43.0 or 0.0560 – Zeros between other significant digits are significant. 306 or 0.705 – Zeros used solely for spacing are not significant. 24, 000 or 0.007 1 Significant Digit 0.003 vs. 0.003 2 Significant Digits • This usually occurs after some instances of rounding. Here a problem would specify to how many digits you must round. Significant Digits (Rounding) • Instead of rounding to a place, you round a number to a specified number of significant digits. This is done by rounding up or rounding off the number that would constitute an extra place. • Round the number 45.63 to 3 significant digits. Significant Digits (Mult/Div) • Keeping correct significant digits while multiplying and dividing relies on the same process. 45.6 • Round the number 6798 to three significant digits. 6800 Significant Digits (Add/Sub) – Align the addends (for addition) or the minuends and subtrahends (for subtraction) vertically. – Add or subtract the values. – Draw a vertical line down the least precise number (the one with least decimal places). – Round to the left of the vertical line. – Addition problems can have more than two addends. Addition 363.7 + 14.374 363.7 +14.734 378.434 378.4 × Multiply By Conversion Factor Fraction 41.2 −0.779 40.421 40.4 = Equals Term With New Units Dividing 4 1 7.261 ÷ 0.2 36.305 40 Significant Digits (Sci. Notation) Sig. Digit Counting Multiplication/Division 4 8.803 × 1014 ( 5.91×10 )(1.9 ×10 ) 2 4.5 × 10−144 Unit Conversions (Overview) Term With Original Units 3 • Only the coefficients in scientific notation numbers count towards significant digits. • Aside from this, all normal rules apply. • See the examples below. Subtraction • In physics you will encounter many different types of units. • At times, you will need to convert from one to another within the same type of measurement. (e.g. time or speed) • This is done by multiplying the original number by a fraction that cancels out the original units and replaces them with the new ones. • In order to do this, you need to know the conversion factor. (e.g. 1ft. = 12in.) 2 0.54 × 6.33 3.4182 3.4 – Count the number of significant digits in each of the numbers being multiplied or divided. – Calculate and round your answer to the number of significant digits found in the least significant input. – It is sometimes easier to write these problems horizontally. – How many significant digits does the number have? 4 – Which digit must be rounded? the 3 Round Off! – Round up or off? 45.63 • Adding and subtracting rely on similar processes when significant digits are being kept. Multiplying 3 2 −4 1.1229 ×10 6 3 1.1× 103 Unit Conversion (Metric Prefixes) • • Knowing metric prefixes is very important, as they are represented in many measurements that you will encounter. A complete table of these can be found in your book on page 9. The prefixes that occur most frequently are shown to the right. Prefix Symbol Multiple Nano n 10-9 Micro µ 10-6 Milli m 10-3 Centi c 10-2 Kilo k 103 Mega M 106 Giga G 109 Unit Conversions (Simple) • Simple conversions involve one unit change. • Example: Convert 1.34m into cm. – The conversion factor is as follows: 1m = 100cm – See the calculation below. Desired Unit (to incorporate) 1.34m × 100cm = 134cm 1m Undesired Unit (to cancel) A Second Look Now you try to convert 456cm into m. 456cm × 1m = 4.56m 100cm Unit Conversions (Square/Cubic) • Square and cubic unit conversions deal mostly with areas and volumes respectively (e.g. ft2 or cm3). • Here the linear conversion factor must be known and then taken to the desired power before being used in a fraction to make the conversion. • Convert 500ft2 to m2. What if it were cubic units? 2 3 2 1m 1m 500 ft × 500 ft 3 × 3.28 ft 3.28 ft 1m2 1m3 3 500 ft 2 × 500 ft × 10.7584 ft 2 35.2875 ft 3 46.5m2 14.2m3 Unit Conversions (Compound) • Compound conversions involve fractional units, such as speed or any other rate (e.g. mi/h). • Multiple units within the fraction may have to be converted here. • Convert 60mi/h into ft/s. 60 mih × 5280 ft 1h × = 88 1mi 3600 s This fraction cancels mi. ft s Conversion Factors 1mi = 5280 ft 1h = 3600 s This fraction cancels h. Conclusion • Physics is a math-based science course. • All four major skills will come into use during the course of the year, many as early as next section. • Don’t forget how to do: – Algebraic Manipulation – Scientific Notation – Significant Digits – Unit Conversions
© Copyright 2024