A physical quantity is any quantity that can be measured with a certain mathematical precision. Example: Force

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1 1 Unit Systems Conversions Powers of Physical Quantities Dimensions Dimensional Analysis Scientific Notation Computer Notation Calculator Notation Significant Figures Math Using Significant Figures Order of Magnitude Estimation Fermi Problems What is a physical quantity? A physical quantity is any quantity that can be measured with a certain mathematical precision. Example: Force What is a dimension? The product or quotient of fundamental physical quantities, raised to the appropriate powers, to form a derived physical quantity. Example: mass x length / time 2 (ML/T 2 ) What is a unit? 2 A precisely defined (standard) value of physical quantity against which any measurements of that quantity can be compared. Example: Newton = kilogram x meters / second 2

2 Sytème International length meters (m) time seconds (s) mass kilograms (kg) temperature Kelvin (K) current Amperes (A) amount of substance Mole (mol) luminous intensity candela (cd) 3 US Customary System length inches (in, ") time seconds (s) mass pounds (lb) temperature Fahrenheit (F) current Amperes (A) amount of substance Mole (mol) luminous intensity candela (cd) 4

3 Other units length Feet (ft, '), mile (m, mi), furlong, hand time minute (m, min), hour (hr), second (s), fortnight, while force Newton (N) temperature Celsius (C) energy Joule (J) power Watt (W) pressure Pascal (P) 5 magnetic field Tesla (T), Gauss (G) Common conversion factors length 1 in = 2.54 cm time 60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, days= 1 yr force 1 N = lb energy 1 J = 7 erg, 1 ev = 1.602x -19 J power 1 hp = 746 W pressure 1 atm = 1.3 kpa 6 magnetic field 1 T = 4 G

4 How to convert units 1 day = 1day 24 hr 1day 1 day = sec 1 day = 6, 400 sec 60 min 1hr 60 sec 1min 7 Most commonly used prefixes for powers of tera T 1,000,000,000, giga G 1,000,000,000 9 mega M 1,000,000 6 kilo k 00 3 centi c milli m micro µ nano n

5 Common physical quantities Mass Distance Time Speed / Velocity Acceleration Force 9 Dimensions of common physical quantities Mass M Distance L Time T Speed / Velocity L / T Acceleration L / T 2 Force M L / T 2

6 What is the equation that relates force and mass? Force M L / T 2 Force = M (L / T 2 ) = M (L/T) 2 / L = M (L / T 2 ) + M (L/T) 2 / L Possible Equations F = ma 2 v F = ma + m l F F 2 v = m l v = 4 ma +16m l 2 So how did Isaac Newton know which is correct? 11 What is scientific notation? 1,562, x x -3 12

7 Using scientific notation in formulas Addition and Subtraction set both numbers to the same exponent add or subtract the decimal numbers the exponent of the sum is the same as that of the numbers being added Multiplication multiply the decimal numbers add the exponents 13 Division divide the decimal numbers subtract the exponents How do computer s write scientific notation? 1,562, e e-3 14

8 Using scientific notation in calculators When putting numbers in a calculator, it is best to covert them to non-prefixed units first (e.g. 1mm 1x -3 m) and then convert back to the desired units when the problem is complete. When using a calculator, it is also better to put in the full number (e.g. 1350x -3 m instead of 1.35 m for 1350 mm). In this way you will avoid many of the decimal place errors so common in this class. 15 Significant figures defined Significant figures in a number indicate the certainty to which a number is known. (For example 1350 mm is known to within ~0.5 mm.) Other than leading zeros, all digits in a number are significant. (For example has significant figures.) 16 Numbers are rounded up or down to the nearest significant figure.

9 Significant figures are used because any further digits added to your number have no physical meaning. Any physically measured number (including physical constants) will be written with the correct number of significant figures unless otherwise noted. 17 Numerical constants, such as the 4 or the π in the equation V sphere 4 = π r 3 have an infinite number of significant figures because they are NOT measured. 3 Using significant figures When multiplying or dividing numbers, the calculated number has the same number of significant figures as the number with the least significant figures used in the calculation. 9 ( ) = =

10 Using significant figures 19 When adding or subtracting numbers, the number of significant figures in the calculated number must be such that the decimal place of the result is not beyond the least decimal number in the numbers used in the calculation. ( ) = = Examples using significant figures = ( 0.07 ) 5.6 = =

11 Why do we use order of magnitude? Order of magnitude is used to make estimates. For example: How many professors are there in the U.S.? (These kinds of questions are named for Enrico Fermi, who first proposed them.) Order of magnitude is also used to check calculations. 21 Order of magnitude defined To determine the order of magnitude of a number, you must put the number in scientific notation using one digit before the decimal. (e.g x x -3 ) If the decimal number is less than five, the order of magnitude is then the exponent. If it is greater than or equal to five, the order of magnitude is the exponent plus one has order of magnitude has order of magnitude -2

12 Estimation Estimation is not the same as calculation. However, it will almost always be within one exponent of the calculated answer. An estimate is a fast way to check a number or choose between two numbers given as an answer Actual = Estimate 3 ( 1 ) = 1 An Example Fermi Problem To within an order of magnitude how many bars of soap are sold in the United States each year? There are about 1x people in the U.S. (the actual number is closer to 3x ). 2. Each person lives in a family of about 1 person (the average is really closer to 3). 3. Each family uses about 1 bar of soap each week (a better number would be 0.5). 4. There are about 0 weeks in a year (the number is actually 52). ( 1 ) = 1 Compare this to the actual answer ( 3 ) = 2.34