Units and Conversions - Geneva 304



Units and Conversions

➢ American (English) system - yards, pounds, tons, seconds

➢ Metric system (SI)

o mks system => meter (m), kilogram (kg), second (s)

o Metric prefixes for each power of 10

Prefix Abbreviation Value Prefix Abbreviation Value

deci d 10-1 decka da 101

centi c 10-2 hecto h 102

milli m 10-3 kilo k 103

micro m 10-6 mega M 106

nano n 10-9 giga G 109

pico p 10-12 tera T 1012

femto f 10-15

atto a 10-18

▪ American to SI conversions:

1 m = 39.37 in

1 km = 0.6214 mi

1 kg = 2.205 lb

1 L = 0.2642 gal

o How to convert

▪ We can multiply anything by 1 in an equation

▪ Conversion factors can be written as fractions that equal 1

➢ For Example: 7 = 7 , so 7 / 7 = 1

1 m = 39.37 in ,

…SO

1 m / 39.37 in = 1

OR…

39.37 in / 1 m = 1

➢ Example: Express 170 pounds in units of kg.

1 kg 170 x 1 lbs x kg

170 lbs x ----------- = ----------- x ------------- = 77.1 kg

2.205 lbs 2.205 lbs

o Try: Express 170 m/s in units of miles/hour.

Scientific Notation

➢ Convenient way to write huge numbers that deal with the macrocosmic and the microcosmic in Astronomy

o Write in powers of 10 => 101 = 10 & 10-2 = 1/100

o The exponent tells you how many times to multiply or divide by ten

o Written as:

▪ a.00 x 10n , where 'a' and 'n' are whole numbers

▪ 100 = 1

▪ 100 is written as 102, or 1 x 102

➢ Converting from normal to scientific notation

o Decimal moves right => negative exponent => 0.0304 = 3.04 x 10-2

o Decimal moves left => positive exponent => 3040 = 3.04 x 103

➢ Converting from scientific to normal notation

o Positive exponent => decimal moves right => 3.04 x 103 = 3040

o Negative exponent => decimal moves left => 3.04 x 10-2 = 0.0304

o Try: Convert 2,398,000 to scientific notation.

o Try: Convert 6.549 x 104 to normal notation.

➢ Adding and Subtracting with Scientific Notation

o Convert the numbers so the exponents are equal

o Then add or subtract the numbers in the front part of the notation

(6.7 x 109) + (4.2 x 109) = (6.7 + 4.2) x 109 = 10.9 x 109 = 1.09 x 1010

(4 x 108) - (3 x 106) = (4 x 108) - (0.03 x 108) = (4 - 0.03) x 108 = 3.97 x 108

o Try: (5.5 x 109) + (1.55 x 1010)

o Try: (6.2 x 108) - (2.0 x 107)

➢ Multiplication and Division with Scientific Notation

o Rearrange so you can multiply or divide the numbers in front of the powers of ten, and add (for multiplication) or subtract (for division) the exponents

(6 x 102) x (4 x 10-5) = (6 x 4) x (102 x 10-5) = 24 x 102-5 = 24 x 10-3 = 2.4 x 10-2

(9 x 108)

(9 x 108) / (3 x 106) = ----------- = (9/3) x (108/106) = 3 x 108-6 = 3 x 102 (3 x 106)

o Try: (8 x 107) x (5 x 10-5)

o Try: (4 x 108) / (6 x 108)

Significant Digits

➢ Numbers should be given to the greatest accuracy that they are known

➢ Write answers with as many decimal places as the number in the problem with the least amount of decimal places

➢ Rules for Significant Digits

o All non-zero numbers are significant. 943 => 3

o Trailing zeroes after a decimal point are significant.

▪ 9430 => 3 943.0 => 4 943.10 => 5

o Captive zeroes are significant. 4903 => 4

o Leading zeroes are not significant. 0001 => 1 0.005 => 1

o All measurements or any number with units must abide by these rules.

o Any counting number or number with no units is significant.

▪ 5 = 5.000… 5 apples = 5.000…

o Examples: 2.3 + 4.71 = 7.0 (NOT 7.01)

Number Sig. Digits Number Sig. Digits

2998 4 .1 1

10 1 .0100 3

190 2 190. 3

0.001 1 190.0 4

0.001000 4 1459 4

0.010 2 459.000 6

Basic Geometry

➢ Dimensions of Circles and Spheres

o The circumference of a circle with radius R is 2πR

o The area of a circle of radius R equals πR2

o The surface area of a sphere of radius R is given by 4πR2

o The volume of a sphere of radius R is (4/3)πR3

➢ Measuring Angles - Degrees

o 3600 in a circle

o 60' (minutes) in one degree

o 60" (seconds) in one minute

Basic Trigonometry

➢ In a right triangle:

o The longest side, opposite the right angle is the hypotenuse (c)

o The side adjacent to the angle labeled θ is side (b)

o The side opposite to the angle labeled θ is side (a)

o Pythagorean Theorem: a2 + b2 = c2

o Trig functions:

▪ sin θ = a / c

▪ cos θ = b / c

▪ tan θ = a / b

Basic Algebra

➢ You can move variables (unknowns) across an equals sign in an equation just by moving them from the top on one side, to the bottom on the other

o Example: d = distance; v = velocity; t = time; a = acceleration

▪ d = vt , can be changed to => t = d / v OR v = d / t

▪ v = at , can be changed to => t = v / a OR a = v / t

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