Intro to Phys Sci I Mechanics Heat Instructor Carlos Perez Physical Quantities and Units 1 Physical Quantities A physical quantity is any number that is used to describe a physical property or phenomenon quantitatively Example your weight and your height The main point is that any property can be a physical quantity if we can assign it a numerical value and have a way to measure it In mechanics the three fundamental quantities are length time and mass Other examples of physical quantities are temperature electric charge energy momentum speed spin magnetic eld strength and many more 2 Units and Dimensional Analysis Units tell us what physical value our numbers represent To make accurate reliable measurements we need units of measurement that do not change and that can be duplicated by observers in various locations Measurement standards used by di erent people in di erent places must yield the same result Two major systems of measurement prevail in the world today The United States Customary System formerly called the Brithish system of units The Syst eme International SI also known as the International system of units and the metric system For PS 303 we will mainly use the SI Here are the standard units for this system Physical Quantity Unit Symbol Length Time Mass Temperature Electric current Luminous intensity Amount of substance meter metre second kilogram kelvin amp ere candela mole m s kg K A cd mol If you ever need to convert units you can use online resources or Google for that 2 1 Pre xes and Scienti c Notation Additional units can be derived from the seven base units of the SI by adding a pre x to the name of the fundamental unit For example the pre x kilo abbreviated k always means a unit larger by a factor of 1000 To see the usefulness of pre xes we will need to utilize scienti c notation Any real number can be written in the form number 10n in many ways where n is an integer number i e 3 2 1 0 1 2 3 4 Scienti c notation is the use of powers of 10 to conveniently write numbers that are too large or too small Let s see how this works 1 000 10 10 10 103 so 35 000 35 103 Let s try to make more sense of it The number 35 can be written as 35 0 why You ll see The positive power of 10 tells us how many positions to the right we need to move the decimal point For each position we add a zero 1 So 35 103 can be understood as Move the decimal point of 35 0 three positions to the right and add a zero for each position That means that 35 103 35 000 How does this work with negative powers of ten We move the decimal point to the left a number of positions equal to the power of ten For instance 62 4 10 3 means Move the decimal point three positions to the left If there is no digit in one of the positions add a zero In this case we have 62 4 10 3 0 0624 0 0624 62 4 10 3 10 3 0 001 so 1 10 1 10 1 10 Using scienti c notation it is easy to use the unit pre xes Each pre x has an associated power of ten to it Here is a table with some of the most common pre xes Pre x Symbol Power of 10 Decimal giga mega kilo deci centi mili micro nano G M k d c m n 109 106 103 10 1 10 2 10 3 10 6 10 9 1 000 000 000 1 000 000 1 000 0 1 0 01 0 001 0 000 001 0 000 000 001 For example two megagrams or 2 5 M g equals 2 5 106 g 2 500 000 g 2 2 Operating Units Mathematical rules dicate how our units combine when we do things like multiplication division addition and subtraction with di erent measurements Multiplication Division and Exponentiation You can put units into fractions and they ll follow all of the same rules for these three operations outlined in the previous section Note that the phrase unit a per unit b just means that your units are Take the following example where we re working with seconds s kilograms kg and meters m We can also bring some numbers into the fold Example and say we wanted to multiply this measured value by another like 8 seconds unit a unit b s m m kg m s m kg kg s m m s kg m m s kg m2 1 m 3 s 1 m 3 s m s 1 3 8 s 1 3 m s s 8 1 m s 3 s 1 8 m 1 1 3 s 1 8 m s 3 1 8 m 3 s 2 You are also free to do the number part separate from the units part if you d like For instance consider the product 4 kg 9 8 m s2 The number part is done as follows 4 9 8 39 2 and the unit part multiplies in the following way s2 Putting these together gives us the answer 39 2 kg m s2 kg m kg m s2 Addition and Subtraction You are only allowed to add or subtract units which are exactly the same This should match your intuition Let s look at a couple of examples to understand why this is important which works because you are adding a measurement of area to another measurement of area However CORRECT 3 m2 7 m2 10 m2 NONSENSE 3 m2 7 m3 The former operation does not make sense because it represents adding a measurement of area to a measure ment of volume What does that even mean In general whenever you are doing operations with measurements and physical quantities that are accompa nied by units you need to make sure the units are consistent to add or subtract them If two or more terms do not have exactly the same units then you cannot add or subtract them The result would be meaningless For instance 3 m2 s 7 m s meaningless since the units di er by one power of meters m The nice thing is you can make this work for you If you re working a problem and it calls for you to add kilograms to seconds just stop Either the concepts you re using are wrong or you ve made a mistake Either way to continue would be a waste of time 2 3 Dimensional Analysis In simple words dimensional analysis means keeping tabs on your units If you are using a quantity or measurement of length say in centimeters you can always convert it to use any other unit of length as long as you follow the right process For this we use the conversion factors Now conversion factors are interesting they are mathematical and physical equalities Let s look at the following 2 …