I want to demonstrate this using as\nlittle physics as possible. So this next\nexample may seem a bit contrived. You\nwill see more significant examples as our year goes on (perhaps more in E &\nM than in Mechanics). But this is a good one to begin with:<\/p>\n\n\n\n
A beam is 2 meters long. It has a density that varies along its\nlength. As a function of distance from the left end, the linear density (in\nkg\/m) is given by the function:<\/em><\/strong><\/p>\n\n\n\n Find the mass of the beam.<\/em><\/strong><\/p>\n\n\n\n Before we jump in, let me say a little\nabout what we are hoping to do.<\/p>\n\n\n\n We have a quantity that varies\ncontinuously: in this case, it\u2019s the density. \n(Also, please do not be intimidated by the fact that we are using the\nGreek letter, lambda, to represent the density function. I think we do this because it is a LINEAR\ndensity (mass\/length) and so we want a letter that begins with L.) <\/p>\n\n\n\n If we chop our beam into short pieces, we can pretend the density is constant for a given piece. Really, density does vary but if the piece is small, density doesn\u2019t change much. And we are eventually going to make the pieces reeeeally small. This is pretty much the same thing we did last time, when we pretended velocity was constant over a time interval.<\/p>\n\n\n\n We then multiply the length of each piece\nby its density to find the mass of that piece and then we add them up. <\/p>\n\n\n\n \n\nSo here\u2019s the beam, nicely chopped:\n\n<\/p>\n\n\n\n <\/p>\n\n\n\n We focus on one individual piece, say that blue one, and find its mass:<\/p>\n\n\n\n And we add up all those little masses:<\/p>\n\n\n\n As we have before, we can improve our\nestimate by chopping into more pieces \u2013 or even better, an infinite number of\npieces:<\/p>\n\n\n\n Now let\u2019s really examine that\nexpression. It sure looks a lot like a\nRiemann sum! It looks like you are\ngetting ready to find the area under the curve: \nf(x)=1+x-x2<\/sup>\/<\/em>2<\/em> <\/em>by breaking it into\nrectangles. And for that we can use the\nMagic Theorem. <\/p>\n\n\n\n We can find the anti-derivative ourselves or use Wolfram\/TI\/whatever to get: <\/p>\n\n\n\n And then see how much it changes:<\/p>\n\n\n\n We are done! And yes, that is also the area under that graph, but that\u2019s not really the point. We were not interested in finding an area. We just wanted to know the mass. But as we have just seen, the same method that lets us find areas also helps us to add up the little pieces!<\/p>\n\n\n\n Well,\nthat\u2019s all I wanted to say about the Fundamental Theorem and how it is used in\nphysics. If you have stayed with me\nthrough this entire 5-part journey, thank you so much for your persistence. I find this to be pretty challenging\nstuff. I will be happy to help you\nthrough it again when we encounter this during our year in AP Physics. I hope the time you spent working through\nthese posts makes it easier for you when you see it again. <\/em><\/p>\n\n\n\n See\nyou soon!<\/em><\/p>\n\n\n\n Mr K<\/em><\/p>\n\n\n\n Back to Part IV<\/a><\/p>\n\n\n\n Go to Summer Reading Outline<\/a><\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" So many pieces! Finally, finally we have arrived at the point all of these posts have been building toward: In physics, we only occasionally care about \u201carea\u201d. But we often care about adding up an infinite number of tiny little … Continue reading ![\"\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2021\/08\/magic21.png\")
<\/figure>\n\n\n\n![\"\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2021\/08\/magic22.png\")
<\/figure>\n\n\n\n![\"\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2021\/08\/magic23.png\")
<\/figure>\n\n\n\n![\"\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2021\/08\/magic24.png\")
<\/figure>\n\n\n\n![\"\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2021\/08\/magic25.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2022\/12\/fix1.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2022\/12\/image-1.png\")
<\/figure>\n\n\n\n