ShmoopTube

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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numbers we were given a linear coefficient for

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

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

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

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

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

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

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

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

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

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

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

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

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

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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]

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

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nuts... [woman hammering an aluminum block]

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