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Calculus: Hooke's Law

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  • Type: Video Tutorial
  • Length: 5:00
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 54 MB
  • Posted: 06/27/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Applications of Integral Calculus (18 lessons, $26.73)
Calculus: Work & Hooke's Law (3 lessons, $4.95)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Applications of Integral Calculus
Work
Hooke's Law Page [1 of 2]
Okay, now when you're thinking about work, sometimes you want to think about the amount of work required to actually pull things that are hard to pull. For example, let's think about a spring. So you have a spring and you want to pull it. Well you know, gosh, it's sometimes hard to really pull. And then the further out - in fact, have you ever tried to take a spring that's really, really tightly wound - sometimes that's how I feel - and you just keep pulling it out. The more you pull it, the harder it actually is to pull, because it's stretching and it doesn't want to do it. It wants just to go back and go right back in.
Well, how would you actually figure out the force required in order to stretch out a spring? It seems intuitive that somehow, as you stretch out the spring, the force required to stretch more should increase. That is, when you have a spring stretched out, to stretch it out even more, it's even harder than if it's very, very close in. If it's close in, it's easy to move it. If it's stretched out, it's really hard to move it. So can you take that intuition and turn it into actually a physical fact? And it turns out you can, and in fact, this is exactly what Hooke did, and this is known as Hooke's law.
So Hooke's law for the force exerted in order to stretch a spring is just the following: the force to stretch a spring x units - and by the way, let me just show you the set up here. This is sort of mildly important. What I'm assuming here is that I put the spring down, and sometimes a spring has sort of a natural - you know, just how it sits naturally. I'm gong to assume that it's sitting so that this point right here is x = 0. So it's sort of sitting in the negative land and the right edge just touches that x = 0. And now what I'm going to do is I'm going to start to stretch it. And if I start to stretch it, the question is now I'm moving out x units - how much force is required in order to pull it out that far? And it turns out that Hooke's law says, that amount of force is just proportional to the length that you travel. So, in particular, it's just some constant times x. That's Hooke's law.
So let's actually look at an application of this in figuring out what work is. So suppose that I have a spring and in its natural resting position - so when it's just resting - it's 2 inches. And if you measure that, you can see that it's two inches. And I hope maybe you can't see it, which is good, because I'm faking it. So it's two inches in the resting position. From here to here is two inches. Okay, that's resting.
Now, I am told - or I actually computed this - that if I stretch the string out 10 inches from where I started - so that's right around here - the amount of force required for me to do that turns out to be 5 pounds. So a 5-pound force is required in order for me to stretch this particular spring the length 10 inches. So what that means is we're given that a 5-pound force is required in order for me to stretch the thing 10 inches.
My question now is, how much work have I done in order to stretch the spring 12 inches? So if I want to stretch it now 12 inches, how much work has really been done? Well, we know how to figure out work. We just integrate and we just integrate the force function. Unfortunately, I don't know what that constant is. But this particular piece of data will allow me to find that constant. For if I plug in, I see 5 = k x 10. That means that k actually equals . And so my force function actually equals times x.
Now that I have my force function, using Hooke's law, I can now actually figure out the work. And I'm asking now for the work that is required in order to stretch it out 12 inches. So I'm going to start by looking at an integral. I go from zero to 12 - starting position out to position 12. And then what's the force function? Well, it's given to be x - we figured that out - dx. So all I have to do is integrate that. That's not too bad. That's just going to be evaluated from zero to 12. And when I plug in 12, I have to square it, so I get , when I plugged in zero. And what does that equal? Well, that equals , which equals around 36. And what's the units? The units here - let's be very careful on the units. I was given these units in pounds. This is 5 pounds, was required, of force. So this is pounds inch. So 36 pound inches is the work that was required in order for me to stretch this spring 12 inches.
So Hooke's law actually allows us to figure out the work involved in stretching a spring. I'll see you at the next lecture.

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