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AP Physics 2: 1.3 Changes and Conservation Laws. To what temperature does the aluminum sheet need to be heated in order for the square peg to fit through the hole?

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Transcript

00:04

And here's your shmoop du jour brought to you by square pegs and

00:06

round holes some days we feel like an octagonal peg in a world of triangular [Man trying to fit octagonal shape in triangular hole]

00:10

holes, well those are usually days when we've gotten a lot of geometry homework

00:14

The following may be useful information here we go the linear coefficient of aluminum

00:19

is 23 times 10 to the negative 6 centimeters per degree celsius the

00:24

linear coefficient of iron is 11.8 times 10 to the negative 6 centimeters per [Aluminum and iron linear coefficiencies]

00:29

degree celsius according to a common expression square pegs can't fit into

00:33

round holes however it doesn't take thermal expansion coefficients into [Man points to thermal expansion coefficient]

00:38

account. Well a square iron peg has sides of two times square root of two

00:42

centimeters and needs to pass through an aluminum sheet with a round hole of [Square block of aluminum and a metal sheet]

00:47

diameter 3.99 centimeters both objects are initially 20 degrees Celsius...To what

00:54

temperature does the aluminum sheet need to be heated in order for the square peg

00:58

to fit through the hole and here are the potential answers... All right well this

01:04

question really speaks to our attitude whenever taking on a DIY project if [Woman doing DIY project]

01:09

something doesn't fit beat it into shape or in this case, heat it into shape

01:16

alright which may be why everything we build falls apart after about five [Color block tower falls down]

01:20

minutes first of all we need to know how much bigger this hole needs to be the

01:24

dimensions of the peg will remain the same and the widest part of the peg will

01:27

be the diagonal the minimum diameter of a whole. To find the length of the

01:31

diagonal we multiply the length of one side times the square root of two which [Formula for length of diagonal]

01:36

gives us a diagonal of four centimeters that means the diameter the hole has to

01:40

increase by point 01 centimeters an equation for linear thermal expansion is

01:45

the change in length equals the product of the linear coefficient the original

01:50

length and the change in temperature so we need to find the change in [Thermometer increases]

01:54

temperature that equals the point 0 1 centimeter change in length we can

01:58

rearrange the equation to solve for the change in temperature and plug in our [A plug and a light bulb switches on]

02:02

numbers we were given a linear coefficient for

02:05

aluminum at the start which is 23 times 10 to the negative 6 centimeters per

02:08

degree celsius we'll round that down to 20 to make it a little easier and we'll

02:12

round the diameter of the hole up to 4 centimeters now let's solve this thing [Person solving a rubiks cube]

02:16

the change in temperature equals the change in length over the coefficient

02:20

times the starting length that means point 01 centimeters over 20 times 10

02:25

to the negative 6 degrees Celsius times 4 centimeters when we do the math we

02:30

find that the change in temperature equals 125 degrees Celsius since the [Aluminum block and thermometer rising in temperature]

02:34

starting temp was 20 degrees the final temp needed for this peg to fit is 145

02:40

degrees Celsius which means our answer is none of the options match 145 degrees

02:46

but remember we ballparked the number on the linear coefficient since we made the [linear coefficients on a ball park]

02:50

denominator a little smaller our result was a little bigger than it should have

02:54

been the only answer close to 145 degrees 129 degrees and we can be

02:58

confident that B is the correct choice and remember if you're in a situation

03:02

like this trust physics to make it work don't just brute force it that's [girl wearing a face mask using a blow torch]

03:06

assuming you have a blowtorch handy if not, just grab a hammer and well go

03:11

nuts... [woman hammering an aluminum block]

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