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Physics, science needs measurement because otherwise we're just watching

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stuff happen... We're skimming (reading the text under breath) [Bullet points about scientific measurements]

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Okay here we go so let's recall the world as it existed

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a long long time ago when we built our first experiment. Ah we remember that [Enrico Fermi pointing to two kids doing an experiment]

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pendulum so fondly, yeah and that table we made? Such a

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beautiful piece of science because without it and our handy-dandy stopwatch [Girl taking measurements of the pendulum]

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and meter stick, and scale well we'd just be watching something swing. Yeah so [Measurement tools disappear and the two kids look confused]

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whatever we do in physics we've got to take measurements and we need to take

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those measurements in the right way. So let's talk about units of measurement

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first. In the olden days units of measurement could vary from place to

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place, after all we can say something is 10 feet long but what if your foot is [Two different sized feet next to a ruler]

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bigger than ours. So scientists developed SI units, SI stands for Systeme

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International d'uintes. Yeah it's in French we showed up at school that day... [Teacher wearing a beret]

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It's basically the metric system on steroids. We're used to metric [Muscly guy wearing metric system vest]

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measurements for mass like the kilogram and for length like the meter, but SI units [Guy measuring how thick his bicep is]

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also cover electrical current, time, temperature, amount of substance and

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light. So we can measure everything in the physical world using SI units the

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measurements for the non-physical world aren't quite as precise... yet. Well now [Guy wearing a lab coat hands over an IQ score of 118 and the other guy changes the score to 198]

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even the best measurers in the world still make mistakes, no one is perfect no

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matter what our mom tells us. So we have to account for measurement errors when

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we're getting our science on. One way we can account for errors is to make

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multiple measurements or run an experiment multiple times because the [Enrico Fermi wearing a chicken costume]

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more data we have generally speaking the more confident we can be in the numbers. [Someone filling out a table of dependent variable values]

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But the old saying we just made up goes measure a bunch science once. [Book opens to show the quote]

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And sure maybe that saying doesn't make sense but nonsense has never stopped us

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before... Welcome to shmoop. Yep well even with [Clown tells a guy he can't cross the bridge]

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multiple measurements we'll still have some amount of error to deal with. Let's [Big red cross appears on the table of values]

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say we're trying to figure out just how good we are at using a stopwatch. We know [Guy picks up his stopwatch]

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we're not gonna get the time completely perfectly right we can only hit the

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button so fast no matter how much coffee we had in the morning and sometimes it's

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a lot like gallons gallons of coffee which we like.. So we do a little

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experiment we try to stop the clock as close to 10 seconds as we can, after [He stops the clock at 10.25]

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running that experiment a few times we find out that we're able to get the time

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right within three tenths of a second. But the error rate means that sometimes

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we were three tens too early and sometimes we were three tens too late. [Examples of the errors are shown in the table]

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Coffee wearing off there.. So we were plus or minus three tenths of a second in

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that trial and we record our data like this. Well our average time was 10

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seconds plus or minus 3/10 of a second. All right well another way to look at

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error rates is by percent error to figure out the percent error we have to [Boy shooting arrows at a target]

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know what our target is, we'll call that the actual number. You might see it

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referred to as the expected value or the theoretical value like in our stopwatch

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test we just did. We know that the actual number we were trying to hit was 10

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seconds and we'll use one of our measured numbers too, say 10.3

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There's a good one, we then subtract the actual number we were trying to hit from

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the measured number and we divide by the actual number. Then we multiply that [Working is shown]

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result by a hundred to find our percent error, so in this case our percent error

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would be three percent yeah not too shabby. And being good scientists we want

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our percent error to be as low as possible so let's say we did that [Scientists playing limbo]

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stopwatch experiment again. Remember just by playing around with a stopwatch you [Enrico Fermi in the chicken suit again]

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get to tell your friends, you were doing a science experiment, yeah pretty

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cool and you know we're physicists we've got to make ourselves sound cool

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whenever we can. So let's say instead of running the clock for 10 seconds we only [Guy holding his stopwatch and watching the pendulum]

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want it to run for one second, now just because we're measuring a shorter period

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of time doesn't mean our thumbs are getting any faster here people.

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So we'd still have an error rate of plus or minus three

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tens of a second, well what would our percent error be now using ten seconds [Margin of error written into the value table]

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versus one second. Well if our measured time was 1.3 seconds and our actual time

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was 1 second our equation would look like this:

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1.3 seconds minus one second gives us point three seconds divide that by our

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actual number one second and yep we still end up with 0.3 seconds don't [The equation being written out]

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multiply that by 100 and we get a percent error now of 30 percent ouch that's

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pretty high. You wouldn't want someone grading a test

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if you knew the grade they give you could be thirty percent too low. 30 [Girl looking sad holding a 70% graded test]

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percent too high you could maybe live with below.. no. [The teacher then gives the girl a test that says 130%]

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So we can see that taking a longer amount of time we can make our result

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more accurate, which is why we timed our pendulum over ten periods not 1. We [The two kids doing the pendulum experiment again]

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weren't just wasting time, we were reducing our error percent trust us when we're

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just wasting time it's pretty obvious it's called YouTube. All right we also [A cat video on a monitor]

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want to be precise when we take measurements if we're doing our old

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stopwatch test while watching TV we might get a little distracted that final [A girl hands a rose to Fermi]

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rose ceremony always sucks us in you know. So we're just stopping the clock

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whenever we remember we might find that we stop it at 7 seconds and at 15

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seconds and at 13 seconds and at 5 seconds and that averages out to 10

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seconds so we could claim that our average time was accurate but we

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couldn't claim it's very precise, and precision isn't everything. Let's say we [The range of values is shown]

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were having a bad thumb day and instead of stopping the clock at 10 seconds we [Guy has a bandage on his thumb]

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stop it at 12 point four seconds and eleven point eight seconds and 12.1

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seconds and eleven point nine seconds. Well, that's pretty precise because all

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the numbers are closely bunched together, but it's also pretty what's that word oh

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yeah wrong. We want to be as close as possible to 10 seconds not 12 what [Big red cross appears]

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were you thinking not enough coffee. So in this case we would be precise but we

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wouldn't be accurate, we want both precision and accuracy in our

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measurements and our golf game. We also want some chips all this science is making

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us hungry. All right when we're looking at measurements we need to pay attention [Enrico Fermi with bowl of chips]

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to the significant figures. The term significant figures refers to the number

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of digits of a measurement that should be considered. It's not a

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Hollywood bodies thing why is that important well because it gives us

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information about how much of an error there may be with the value we're [A balloon flies into a fan and pops]

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dealing with. There are four rules to consider when determining the sigfigs of

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a measurement oh yes sigfigs is the super cool way of referring to significant [Rules of Sig Figs written on a stone]

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figures among us physics people.. sorry. Alright we busy scientists you know we

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don't have time for all the extra syllables. The first rule is that all

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nonzero numbers are significant, so a number like 472 has three sig figs and [The rule being written onto a stone]

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five point six nine seven has four the second rule zeros sandwiched between

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nonzero digits are significant so if you've got a number like five thousand

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nine then you're working with four sig figs. Now trailing zeros are only

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significant if they come after a decimal point that's the third rule. So three

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thousand only has one sig fig but three point zero zero zero has four. Keeping up

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with us here? The fourth and final rule is that zeros to the left of the first

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non-zero digit are insignificant. So a number like point zero zero three aka [Rule being written on a stone]

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three thousandth only has one sig fig, in this case the leading zeros are

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placeholders but they're considered to be insignificant which sounds harsh but

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hey that's just how science rules. Well the number of sig figs and decimal [Arrow pointing out the placeholder numbers]

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places we use when recording data is related to the amount of error in our

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measurement, looking back at our stopwatch if the amount of error is 0.3

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seconds we don't need to record the data to the nearest 10 millionth that's just

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overkill. We know we need to record our data to the nearest tenth of a second [Animal gets runover after it's already lying in the road]

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and when we're using a measuring device the error amount in our measurements

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should be 1/2 of whatever is the smallest division. So if we're using a

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scale that gives a weight to the nearest kilogram our error amount will be half a [Someone standing on the scales]

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kilogram. If we have to do math with numbers that have a different degree of

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precision sig figs in the number of decimal places have to be considered. If

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we're adding or subtracting we want to record our result using the same number

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of decimal places as the least precise measurement. So for adding 10.1

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to 0.3004 we get a result of 10.4004, but our least precise measurement [The equation is shown]

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10.1 has only one decimal place, so our result would be rounded to the nearest

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tenth. Meaning we'll be recording the result as

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10 point 4 when we're multiplying or dividing our result will be limited by

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the number with the fewest significant figures, for example if we multiply 12.13

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which has 4 sig figs by 37.0025 which has 6, we get a result of [The number of sig figs in each value is shown]

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448.840325, don't worry we used a calculator there. The lowest amount of

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sig figs in our factors is 4 so that's what we'd use when recording our data.

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Making the final result for 448.8. Now let's look at orders of magnitude, but what's

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the difference between 5 and 50 well yea 45 is one possible smart guy answer, yeah [50-5=45 appears]

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that's right. But we're not talking about subtraction 50 is 10 times larger than 5. [A fist punches the 45 away]

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Go ahead, double check our math if you're not sure now this is some high-level [Boy checking the answer on his calculator]

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stuff here.. Another way of saying this is that 50 is one order of magnitude

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greater than five. An order of magnitude is a power or exponent of 10, so 50 is

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equivalent to 5 times 10 to the first power. 500 is equal to five times ten to

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the second power. 5,000 is equal to five times 10 to the third power and so on. So

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5,000 is three orders of magnitude greater than five you get the picture

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here right and it goes in reverse too. 0.005 is equivalent to five times 10 to the [Guy in a mask steals a pictures and runs off then runs back the other way as he is chased by security]

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negative third power, why is that important well for one thing it can [Guy writing out big number]

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save us from carpal tunnel when we're dealing with really big or really tiny

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numbers. It's a lot easier to write 5 times 10 to the 12th power than it is to [Fermi writing the numbers on a blackboard]

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write out five trillion, and it can help us deal with those sigfigs we were just

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talking about. If we have a number with a lot of digits but not a lot of sig figs

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or sig figs we can make it easier on our eyes by

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expressing it using orders of magnitude. 5,300,000 has seven digits but only two [A huge number is written out and zeros keep being added]

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sig figs so we can write it as five point three times ten to the sixth and

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not worry about the insignificant figures. But we can also use orders of [The example is written out]

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magnitude to do some pretty cool estimations, see there are these things

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called Fermi questions. Fermi questions are problems where you don't have a lot

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of data to work with and you're just trying to make a good guesstimation of

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the right answer. So what's a Fermi anyway well this guy's a Fermi yep [Guy on stage with a microphone, Fermi puts his hand up]

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Enrico Fermi right here. I was a physicist who was really good at this

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kind of thing I was even able to make a pretty good estimate of how much energy [Explosion goes off]

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an atomic bomb would release, but that's kind of a downer so why don't we tackle

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something a little tastier, like how many jellybeans would fit inside the [Guy playing American football is tackled]

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Washington Monument that's a good one. You know Washington Monument right big [The monument opens in the the middle and a jelly bean is put inside]

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pointy thing in Washington DC well the monument has a volume of

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31,1776 cubic meters, and a jelly bean has a volume of about one cubic centimeter. And

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that's all the info we need to make a good guess at this problem so 31,1776

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cubic meters has four orders of magnitude and one cubic centimeter has

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negative six orders of magnitude, which may not seem obvious at first so let's

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break that down. A centimeter is 1/100 of a meter right, so that's negative two [A giant wrecking smashes into skyscrapers and knocks them over]

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orders of magnitude but we're not just dealing with length here our unit of the

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measurement is cubed adding another exponent into the mix. So we have to [Fermi talking]

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multiply our exponents here, which is how we get to the negative sixth order of

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magnitude. Another way of putting it is that one cubic centimeter equals ten to

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the negative six cubic meters, don't worry we'll go more into this kind of

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stuff later when we tackle unit analysis. Yes we can't wait to get there either. [Girl flicking through a calendar and unit analysis appears]

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Okay so how far apart are our two starting numbers in terms of orders of [Parents look annoyed at kids messing about in the back of a car because they're bored]

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magnitude, well if we put them on a number line we see that they're ten

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orders of magnitude apart we can find this mathematically too using dimensional [The area on the number line is highlighted]

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analysis. Now how does that work, well we thought you'd never ask..

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So we want to figure out how many jellybeans it takes to fill a monument

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or to say it a little differently how many jellybeans do we need per monument.

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Sounds like a ratio to us and we can express it as a fraction. Of course every [Monuments filled with jelly beans]

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fraction needs a numerator and a denominator. Well we'll put in the unit [Mom with her kids, she is the fraction and the kids are the denominator and numerator]

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we want for the numerator which is jellybeans and the unit we want for the

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denominator AKA the monument in the denominator, fit there [The kids are replaced by the monument and jelly bean]

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never thought you'd be using jellybeans and monuments as, in a fraction did you...

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Alright so we know we want our final result to be expressed as jellybeans per

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monument, and since we know our orders of magnitude we can set up an equation the [Fermi points to the monument filling up with jelly beans]

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volume of the monument has four orders of magnitude, meaning there are 10 to the

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fourth cubic meters per monument and there was only one jelly bean per every

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ten to the negative six cubic meters. So when we cancel out the meters and do the

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math we find that there are 10 to the 10th jellybeans per monument and now we [The equation is shown]

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can plug in our real numbers to this equation, remember the area of the

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monument is 31,776 cubic meters and using the right order of magnitude that

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would be three point one seven seven six times 10 to the fourth. We're only

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working with one jelly bean here so we don't need to change that, it turns out

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that three point one seven seven six times one equals three point one seven

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seven six yeah we really like that kind of math. So we find that three point one

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seven seven six to the tenth order of magnitude jellybeans will fit inside the [The monument fills up with beans]

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Washington Monument if it was hollow. But wait, remember what we said about sig [Record player stops]

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figs and multiplication, yeah we don't either...

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Wait it's coming back to us, we said that when we're multiplying or dividing our

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result will be limited by the number with the fewest significant figures. The

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volume of the monument has five sig figs but we're only dealing with one

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jellybean which has one sig fig. So our answer should be expressed using one sig

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fig. Meaning that about three times 10 to the

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10th jellybeans will fit inside the Washington Monument that's ten billion [Final answer is shown]

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jellybeans, which makes our teeth hurt just thinking about it. Okay so there may [Guy opens a door and thousands of jelly beans pour out]

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not have been much actual sciencing done just now, it was a lot of numbers [Fermi holding a ball on a string]

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measury stuff but it's not like you can separate physics from math it's a

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pretty much package deal there. Yeah that's Sheldon...

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