A summary of the topics in the following notes: Fundamental quantities are time, length and mass. Every other definition we will make this semester will be a combination of these. An expressed quantity must include units to have physical meaning. There is only one designated SI unit for each concept we will define. There are many non-standard i.e. not SI) unites for each concept. A conversion factor can be used to express a quantity in different units. Scientific notation and unit prefixes are convenient ways to express very large or small numbers. Fundamental Quantities Physics, and science in general, is quantitative. That is, we measure or quantify) things. Most of what we will do this semester is make definitions of the things we measure in physics, and see how those definitions and measurements relate to the other definitions and their measurements. Measurements require two things: a numerical value and corresponding units. A number can only represent something physical i.e. a measurement) if it is accompanied by units. Our first three definitions are categorized as fundamental quantities: time distance or length mass These three quantities are fundamental because they are the basic building blocks that we will use to make all of our other definitions. In this sense, physics is pretty simple: our entire semester is really about studying time, distance and mass. SI Units There are many units we can use to measure time: seconds, minutes, hours, days, years. The ability to use different units is important because it allows us to express very large or very small quantities with numbers that are reasonable. For example, your age in years is a relative reasonable number around 20?), but expressing your age in seconds around 700 billion) would be unreasonable. A range of units can be useful for keeping our numbers reasonable, it creates serious problems when we try to combine measurements in calculations. For this reason the standard unit system, formally referred to as systeme internationale or SI was established in the mid-19 th century. The strength of the Page 1 of 6
SI units is that each definition has one designated standard or SI unit, and when SI units are combined in a calculation, the resulting combination is by definition) the SI unit of the result. SI units combine to form other SI units. We will use this principle when creating units for new definitions and for calculating units for the answers at the end of each problem i.e. if all of the factors are expressed in SI units, we know the answer will also be in SI units.) For our three fundamental concepts of time, length or distance) and mass: Concept: time length mass SI Unit is the: second meter kilogram Any unit of measurement that is not an SI unit is referred to as a non-standard unit. Non-standard units can be very useful, especially if they allow us to express quantities with smaller numbers i.e. it might be more convenient to use miles instead of meters to express large distances.) When should you or can you use SI units versus non-standard units? Always use SI units if your calculation is complicated. Use non-standard units to keep the numerical values reasonable. What is a reasonable number? It s no coincidence that we use a base-10 number system and we have ten fingers. From the day we re born we are taxed with trying to figure out what is going on our poor baby brains!) And what do we see more of than anything else? Ten little things sticking out of our little hands. The quantity of ten is etched into our brains. But our brains also quickly learn spatial relationships and we are very good at being able to see things up to 10 times 10, or 100. Our brains have a difficult time conceptualizing numbers beyond that, especially thousands or beyond. Think about this: if you have to estimate the number of people in our classroom, you can do it at a glance; it s easy for us to accurately see about 30 or 40 people. In contrast, if you re in a crowded stadium, you know there are thousands of people with you... but our brains don t even try to look and quantify the number of people. It could be 5000, 10,000 or a hundred thousand. To us it just looks like a lot of people. For this reason, we can claim that reasonable numbers are those that our brains are comfortable with, that our brains can see. This would be numbers a few hundred or less. Scientific Notation While non-standard units can be used to help us express our values with reasonable numbers, we can also achieve this by using scientific notation. Scientific notation allows us to express unreasonable numbers in a way that appears reasonable. For example, if you had to express your age in SI units of seconds, 6.23 x 10 9 seconds seems a little easier to look at than 623,000,000 seconds. Page 2 of 6
Scientific notation is relatively simple: express your value in decimal form with three significant figures i.e. two decimal places) followed by x 10?? where?? is replaced with the appropriate number for your value. For example: 325,000,000 seconds becomes 3.25 x 10 8 s 19,400,000,000 meters becomes 1.94 x 10 10 m Scientific notation can also be used for very small numbers, i.e. numbers much less than one. In this case, the exponent on the 10 will be negative. For example: 0.00000825 kilogram becomes 8.25 x 10-6 kg The advantage to using scientific notation is that it allows us to express very large or very small numbers in a convenient, compact form. For this reason, you should avoid using scientific notation for numbers that would result in an exponent between 3 and 3 i.e. 10 3 to 10-3 ). There is nothing wrong with using scientific notation within this range, but it does not help write your value in a simpler way. Prefixes Another way to express very large numbers in a simpler, more compact way is through the use of prefixes. A prefix is simply a letter attached to the front of the unit notation. The letter is shorthand notation for some number of factors of ten. In essence, prefixes are a shorthand way of writing a number in scientific notation. The prefixes you will use most often are: Prefix name Prefix abbreviation Prefix meaning Giga G x 10 9 Mega M x 10 6 kilo k x 10 3 centi c x 10-2 milli m x 10-3 micro µ x 10-6 nano n x 10-9 This is not a complete list of available prefixes; additional prefixes are defined for numbers greater than 10 9 and smaller than 10-6, but we will not use those prefixes this semester. Prefixes and scientific notation are interchangeable; that is, scientific notation can be replaced with a prefix, and prefixes can be replaced with their equivalent scientific notation. For example: Page 3 of 6
328 km is the same as 328 x 10 3 m or 3.28 x 10 5 m 21.4 µs is the same as 21.4 x 10-6 s or 2.14 x10-5 s Prefixes are very easy to use when expressing values because they can simply be attached to units. However, when you are calculating, all prefixes should be converted into their equivalent factors of ten. This ensures that all values are expressed in standard units with no prefixes. Unit Conversions If you encounter information in non-standard units, you might need to convert the values to standard units so that you can then use the values in your calculations. You also might want to convert values into non-standard units so that the values can be expressed in a simpler way. Converting units is simple, but the rules for the unit conversion process should always be followed: 1. Write the original value, with its units, in parentheses. 2. To the right of the original value, write a pair of parentheses for your unit conversion factor. 3. Write the unit conversion factor in the parentheses: a fraction in which the numerator and denominator are equivalent values in different units. The units of your conversion factor should be chosen so that the units of the original value can be algebraically eliminated and the new units can be introduced. Once your unit conversion is properly written, you can then calculate units, factors of ten and the number of your newly converted value in that order). Note that the first two of these three items should not involve a calculator. Only the last step, calculation of the number, might require a calculator. A few simple examples: 1609 m ) 3.16 miles 1 mile ) = 5080 m 3600 s ) 1.47 hr 1 hr ) = 5290 s 1609 m ) 13.4 ft 5280 ft ) = 4.08 m Page 4 of 6
Notice in the last example that in the first two examples the conversion factor included a 1. Many times you will be able to write the conversion factor this way, i.e. 1 mile is equal to 1609 meters, 1 hour is equal to 3600 seconds. But you are not restricted to this form. The only requirement for a conversion factor is that the numerator and denominator are equivalent. In the third conversion factor I chose to use the fact that one mile is equivalent to both 1609 meters and 5280 feet. Since a conversion factor is equivalent to one, you can multiply by multiple conversion factors while still retaining the original value although in different units). This can be useful if there is not an obvious direct equivalence between the given units and those to which you would like to convert i.e. converting years into seconds) or if the original units are compound units i.e. a combination of fundamental units) and you must convert the components of the units separately. For example, when converting years into seconds, you might choose to first convert into days and hours: 365 day 24 hr 3600 s ) 4.23 yr 1 yr ) 1 day ) 1 hr ) = 1.33 x 108 s Or you might need to convert a speed from miles per hour to meters per second: 1609 m 1 hr ) 100 mile/hr 1 mile ) 3600 s ) = 44.7 m/s Be very careful when converting units that have an exponent! You will usually see these units when you are working with volumes and areas. For example, a volume might be expressed in m 3 or in 3 or cm 3. Unfortunately, sometimes these units are written cubic meters, or cu in or cc. It is very important to recognize that when you are working with these units, whether written explicitly with an exponent or not, the unit actually expresses multiple factors of a basic unit. That is, m 3 really means meter times meter times meter. If you want to convert a unit with an exponent to another unit with an exponent, you must convert each factor of the unit. For example, to convert inches cubed to meters cubed: 8.76 in3 1 m 1 m 1 m ) 39.38 in ) 39.38 in ) 39.38 in ) = 1.43 x 10-4 m 3 Notice that the three conversion factors are identical. You can write them, using standard algebraic definitions, as one conversion factor cubed : 8.76 in3 1 m 3 ) 39.38 in ) = 1.43 x 10-4 m 3 Page 5 of 6
If you choose this notation, be careful: the exponent on the conversion factor must be applied to everything with the conversion factor. That is, you have to cube the m, cube the in and cube the 39.38. Be very careful of prefixes on units with an exponent! Based on the description of prefixes above, you might assume that a value of 20.0 cc, which means 20.0 cm 3, is the same as 20.0 x 10-2 m 3. Notice that I simply rewrote the c as x 10-2. But this is WRONG! Unfortunately, we get a bit lazy sometimes when we get comfortable with the notation we are using, and our use of units like cm 3 is a prime example. cm 3 actually means cm) 3 or x 10-2 ) 3 m 3 or x 10-6 m 3 The cube on the cm refers to cubing a centimeter, i.e. it is one cm by one cm by one cm. You can verify this connection by formally converting one cc to meter cubed: 1.00 cm3 1 m 1 m 1 m ) 100 cm ) 100 cm ) 100 cm ) = 1.00 x 10-6 m 3 Be very careful when converting a unit that includes a prefix and an exponent. Finally, when converting areas and volumes take extra care when using units that are direct units of volumes and areas. For example, gallons and liters are units of volumes. Acres and hectares are units of area. You can convert directly from meters cubed to gallons, since both are units of volume; i.e. you do not need multiple conversion factors. 2.36 m3 ) 1 m 3 ) 264.1 gal = 623 gal Page 6 of 6