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Mr. C's Dimensional Analysis Page 1 of 5 Many problems in chemistry, math, physics, and engineering involve dimensional analysis. Dimensional analysis involves calculations where the UNITS CANCEL ONE ANOTHER OUT, leaving only the desired unit. It is best to approach each problem the same way, starting with a given, and proceeding in such a way that the unit in the first step cancels with the unit in the second step. You are already familiar with dimensional analysis although you might not have realized it. For example, if your teacher asks you to bring in 2-dozen marbles, what would you do? Chances are that you would find some marbles and count out 24 of them. So what did you do? What you did was dimensional analysis in your head. You knew that a dozen marbles is really just 12 marbles, and since you need 2-dozen, you multiplied 12 by 2. Easy, right!? Let’s look at a more systematic way to do this that can be applied to a number of science and engineering problems. The approach is to always start with the GIVEN. In our case, it is 2-dozen marbles. Then we will convert (switch from one thing to another). To do this, multiply by a fraction (really just both multiplication and division at the same time). In every step after the given, take the units from the previous step (in our case dozen) and PUT IT ON THE BOTTOM. Given: 2 doz. Next step, multiply by a fraction and make sure that the doz is on the bottom. 2 doz marbles x __________ doz marbles Now we need to find a CONVERSION FACTOR . This is just something that converts dozen to something else. The conversion factor we will use is 12 marbles / doz marbles. 2 doz marbles x 12 marbles doz marbles Notice what happened. Since we have the same thing on top and bottom, the units canceled. Note that the number 2 did not cancel, just the doz marbles. Now all we have to do is multiply 2 x 12 marbles to get 24 marbles. There is one more “trick” that we can use, and that is to remember that we cannot add apples to oranges, but we CAN ADD LIKE TERMS!! Example: You are 4 miles from school heading due north at a constant velocity of 15km/hr due south. How many meters from the school will you be (and which direction) after traveling for 45 minutes? Solution: We know that we must add meters to meters because the question specified meters for the answer. 4mi 5280ft 12in 2.54cm 1m –15km 1000m 1hr 45min 1mi 1ft 1in 100cm + 1hr 1km 60min =–4812.6m mi ft in cm 6437.4m km/hr m/hr m/min -11250m 4812m S Lets look at some ridiculous examples. Lets say that there are 22 boleighs in every 3 siefs. There are also 2 runts in every 13 siefs and 6 runts in every yaunts. So our conversion factors are: 22 boleighs 2 runts 6 runts Or, just the same 3 siefs 13 siefs 1 yaunt 3 siefs. 13 siefs 1 yaunt 22 boleighs 2 runts 6 runts Say we want to know how many yaunts are in 3300 boleighs. Lets start with this as the given and use our conversion factors making sure that the unit from the first step is in the bottom of the next. 3300 boleighs x 3 siefs x 2 runts x 1 yaunt 22 boleighs 13 siefs 6 runts Now lets make sure that all of the units cancel and leave us with what we are looking for. Mr. C's Dimensional Analysis Page 2 of 5 3300 boleighs x 3 siefs x 2 runts x 1 yaunt = 11.54 yaunts 22 boleighs 13 siefs 6 runts If there is more than one given: make sure the unit you are looking for ends up in the numerator. If not, just invert (“Flip”) your answer. Practice (show work and cancel units) Name:______________ For all of the following, use the format shown showing that units cancel when one is on top and another on bottom: 2.345km 1000m 1000mm 1cm 1in 1ft 1mi Conversions Given 1km 1m 10mm 2.54cm 12in 5280ft Value and unit for this step 2.345km 2345m 2345000mm 234500cm 92322.8in 7693.6 1.457mi Look up the conversion factors needed and work each of the following. Round to the 3 rd decimal. 1. How many oz are in 2.5 cups? 2. How many oz are in 2.5 gallons? 3. How many cm are in 3 feet? 4. How many feet are in 2.5 miles? 5. How many inches are in 2.5 miles? 6. How many cm are in 2.5 miles? 7. How many km are in 2.5 miles? 8. How many seconds are in 0.5 hours? 9. If you travel 2.5 miles in 0.5 hours, what is your speed in mi/hr? In km/hr? In m/s? 10. How many yaunts are in 4500 runts? 11. How many seifs? 12. How many boleighs? 13. How many boleighs are in 99 runts? 14. How many runts are in 99 boleighs? 15. How many yaunts are in 2110 siefs? Mr. C's Dimensional Analysis Page 3 of 5 16. How many siefs are in 2110 yaunts? 17. How many siefs are in 90 runts? 18. How many cups are in 12 quarts? 19. How many pints? 20. How many gallons are in 112 cups? 21. pints? 22. quarts? 23. oz.? The specific heat of liquid water is 4.184 J/g OC. 24. How many Joules (how much energy) are required to heat 4184g of water 20 OC? 25. If you can supply 3000 J to water at 40 OC, how many grams could you heat to 70OC? 26. The density of gold is 19.3 g/mL. If you have a ring that is 45g of pure gold, what’s the ring’s volume of gold? 27. If you have 0.055 L of pure gold, how many grams of gold do you have? How many kg? 28. We know from the periodic table that there are 197 grams of gold per mole of gold. If you have 0.80 moles of gold, what is its mass? 29. If you have 45 grams of pure gold, how many moles of gold do you have? Review Questions For any given dimensional analysis problem: 1 What do we always start with? 2 What do we do in each subsequent step? 3 When do we know when to stop? 4 What should happen to the units? How? 5 What units will the answer have? Mr. C's Dimensional Analysis Page 4 of 5 AP and Honors Physics ONLY The book cover provides many fundamental constants and derived units. The appendix may provide other useful information. In this exercise, you will use these quantities to develop relationships (formulas). Example: The units for acceleration are m/s2. Put in terms of a) length and time and b) velocity and time. 2 2 Solution: length is in meters, so it must go on top. Time is in seconds, so we need t on the bottom since s 2 2 is on the bottom. a = /t . In terms of velocity: v = /t, but we need /t so we just need another t on the bottom, so we end up with v/t. Use substitution to show the dimensional analysis: 2 Start with v/t. Now substitute v = /t giving v/t = (/t)/t which equals /t Fill in all boxes Derived Unit Expressed as Fundamental Unit Complete the equation In Terms of 2 2 Farad C s potential difference, C = Q/V (Capacitance) kg m 2 V, and charge,Q. kg m2 current, I and Volt V= Cs 2 resistance, R current, I and Watt P= resistance, R work, w and Watt P= time, t current, I and Watt P= potential, V kg m force, F and Joule s2 length, mass, m and Joule E= velocity, v mass, m, Joule E= acceleration, a length, Plank’s const, h Plank’s kg m velocity, v E= Constant, h s length, Mr. C's Dimensional Analysis Page 5 of 5 Answers: 1. 750 runts 2. 29250 seifs 3. 214,500 boleighs 4. 4719 boleighs 5. 2.08 runts 6. 54.1 yaunts 7. 82290 yaunts 8. 585 seifs 9. 48 cups 10. 24 pints 11. 7 gal 12. 56 pints 13. 28 qt 14. 896 oz 15. 4184g x 20OC x 4.184J/g OC = 350,117J 16. You need to first realize that it is heated 30OC. To end up with g on top, you need to invert the heat capacity 1gOC/4.184J it can now be seen that to cancel the J and the OC that you will need a J on top and a OC on the bottom: 1gOC/4.184J x 3000J/30 OC = 23.9g 17. 2.33 g 18. 1062g = 1.062 kg 19. 157.6 g 20. .228 mol