Notes: Dimensional Analysis / Conversions

Transcription

1 Wat is a unit system? A unit system is a metod of taking a measurement. Simple as tat. We ave units for distance, time, temperature, pressure, energy, mass, and many more. Wy is it important to ave a standard? It is important to ave a standard because oterwise we would ave many complications. If Saq O Neil tells me to build im a ouse wit te dimensions of a foot, and I use my foot instead of is, well, e may ave a muc arder time getting troug and around is ouse because my foot is significantly smaller. Along te same concept, if I tell you te directions to my ouse are 1200 Billy leaps East of I-95 on Beac Blvd, you will ave no idea ow to get to my ome. Wat are some standards tat te world uses? Tere are two specific systems tat te world uses, te metric system (Rest of te world uses), and te Imperial/Englis system. Te metric system is a system tat is muc easier to use because everyting is capable of being multiplied or divided by 10. Te imperial system is far more complex because you ave 12 inces in a foot, 3 foot in a yard, 5280 feet in a mile, ow ever many pints in a gallon, and so on. Terefore, it is preferable to use te metric and SI system. Random fact: Te United States actually was one of te original 17 countries to adopt te metric system. Sortly after te American Civil War, Congress autorized te use of te metric system in te Metric Act of 1866 [3] and supplied eac state wit a set of standard metric weigts and measures. In 1875, te United States solidified its commitment to te development of te internationally recognized metric system by becoming one of te original seventeen signatory nations to te Metre Convention or te Treaty of te Metre. I guess te problem was actually implementing it. Note: Tere are differences between te metric system and te SI system (International Systems of Units). Suc as te measurement for temperature in te SI system is te Kelvin, and te measurement for temperature in te metric system is te Celsius. However, tey are often intercangeable (at least wen taking te difference) in problems, but not always. Example: How many centimeters are in 76 meters? Answer: 76 m ( 100 cm 1 m ) = 7600 cm

2 Example: How many inces are in 76 yards? Answer: 76 yd ( 3 ft 1 yd in ) (12 1 ft ) = 2,736 in. Te point of te last two examples was tat it very muc easier to do te conversion for meter to centimeter in your ead tan it is yards to inces. Te metric system is just an easier system wit less converting numbers. Wat are te SI base units? Also, a cart of te different levels of te scale. Please learn tat 1 kilo = 1000.

3 DIMENSIONAL ANALYSIS AND CONVERSIONS: Wat do I mean by dimensional analysis? Dimensional analysis is essentially just keeping track of your units. It is important because wen you are multiplying, dividing, and adding many tings togeter in problems wit equations ten you need to make sure your problem comes out to te rigt dimensions. For instance, te units of acceleration are m s 2. Well, if you end up wit m s tan tat means you ave a velocity, and not an acceleration. Tere could be two reasons for te error: not keeping track of your units, or you forgot to divide by time a second time so tat you ave te s 2. It is especially important to keep track of te dimensions wit converting. You don t want your nurse to forget to confuse er units and accidently give you 2.5 liters of someting instead of 2.5 milliliters of it, do you? Putting 2.5 liters of anyting in you migt kill you (wo knows se may be new and very scared so casually remind er tat is far too muc liquid to put into you). Wen converting units, in general, you want to divide out te unit tat you were originally in, and make sure tat it is replaced by te unit tat you want it to be in. Suc as: Given unit x Desired unit Given unit = Desired unit If tat leaves you confused tan maybe tis will better explain it. Given unit 1 x Desired unit Given unit So wen you multiply tem togeter, you get: (Given unit)(desired unit) (Given unit) Well, just like two divided by 2 is equal to 1 ( 2/2 = 1 ), te given units cancel out to make 1, leaving you only wit te desired unit.

4 PROBLEMS: (SAME ONES WE DID IN CLASS) 1. Convert 1.34 kg (kilograms) to g (grams)? Remember tat kilo means 1000, so tat means tat kilo grams is equal to one tousand grams ( 1 kg = 1000 g ). To start off, set up your problem as below: 1.34 kg ( ) Or 1.34 kg ) 1 Now ask yourself were does my kilogram (kg) go? Te top or te bottom? Well, we are trying to get rid of it correct? So it goes in te bottom (te denominator) since te original kg is on te top (in te numerator) kg xxxx g 1 xxxx kg ) You see were te grams went? It went in te top since it is replacing te kg. So, wat fills te xxxx s? Remember our conversion factor is 1 kg = 1000 g. Tat means te 1000 always stays wit te gram, and te 1 stays wit te kg kg 1000 g 1 1 kg ) So, now we multiply: 1340 kg g 1 kg

5 Now, we divide out because kg/kg is just 1. = 1340 g 2. Convert 240 km to miles. (Conversion factor: 1 mile = km) (I will not work tis one out like te above because it is te same concept) (240 km) ( xxxx xxxx ) 240 km xxxx 1 xxxx ) We want our km to be cancelled out so in te xxxx parenteses, our km sould be in te bottom. 240 km 1 mi km ) = 149 mi 3. Convert 60 mp to kp (mi/ to km/). (Conversion factor: 1 mile = km) Multiply togeter: Te miles cancel out, and you are left wit: 60 mi km ) 1 mi mi km 1 mi = km

6 4. Convert 43 km/ to m/s. So tis one is a bit more tricky. You ave to convert two different units tat will not cancel eac oter out. Te best metod for dealing wit tis is to convert one unit at a time instead trying to do tem bot at te same time. Let s start wit te km to meter and ten worry about te our to second. 43 km ( ) (xxxx xxxx ) Since I am wanting to convert my km to meter first, were does my km need to go in te xxxx bracket? In te denominator so tat it cancels out rigt? 43 km ( ) ( xxxx xxxx km ) Tat means tat my meter needs to go in te numerator (top). Wic makes since because I need my meter to replace my km, correct? 43 km xxxx m xxxx km ) Don t forget tat kilo means a tousand, tus my 1 km = 1000 m. Multiply tem togeter. 43 km m km ) 43,000 km m km My kilometer cancels out and I am left wit meter per our (m/). Or 43, 000 m

7 So now we ave converted our kilometer to meter. But remember tat we wanted it in meter per second (m/s), so we must now replace te our. xxxx xxxx ) So, were does my our in te xxxx bracket need to go? Remember it needs to cancel te our in te ( ). Te our in te ( our to be in te top of te xxxx bracket. xxxx xxxx ) ) is on te bottom; terefore, I need Tat means my seconds must go in te bottom, and it makes sense because I want seconds to replace my our in te ( ). xxxx xxxx s ) Conversion factor: 1 = 3600 s because tere are 60 minutes in an our, and 60 seconds in a minutes tus 60*60 = Multiply out: s ) m s Te our on te top is divided by te our in te bottom and you are left wit m/s m s Now, divide by 3600 and your final answer is: = 11.9 m s

8 5. Convert 98 mi (mp) to in s. Conversion factor: 1 mile = 5280 ft and 1 ft = 12 in. Convert te miles to inces first. 98 mi ( ) (xxxx xxxx ) 98 mi ( ) ( xxxx xxxx mi ) 98 mi ft xxxx xxxx mi ) 98 mi ft 5280 ) 1 mi mi ft mi ft ft xxxx xxxx ) Still trying to get mile to inces. So far we ave acieved getting mile to foot. Conversion factor: 1 ft = 12 in ft xxxx xxxx ft ) ft xxxx in xxxx ft ) ft 12 in 1 ft ) 6,209,280 ft in ft

9 Te foot s cancel out and you end up wit: 6,209,280 in So we are in inc per our but we wanted inc per second, so we must now eliminate te our. 6,209,280 in xxxx xxxx ) 6,209,280 in xxxx xxxx ) 6,209,280 in xxxx xxxx s ) 6,209,280 in s ) 6,209, in s Te ours divide out. 6,209, in s So you ave got your units correct now and are in in/s. Now finis te problem up by dividing by 3600 and you get. = in s