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Re: Enrichment
Posted:
Aug 21, 2012 4:56 PM
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Dave, thanks for your feedback. I will bring up the max and min problem for a parabola with one of my algebra students, as we have discussed this issue before. Thank you for taking the time to hash it all out.
I'm a great believer in simplifying math, particularly in physics problems. My favorite is the calculation of the velocity of molecules. Rudolph Clausius applies calculus to the simple problem of finding the pressure exerted on the wall of a container by a gas. After looking at his presentation, I thought it could be much simplified, and I used a cube and divided the molecules into three groups, each group bouncing against parallel sides. From this and very simple algebra, one can arrive at an expression for the mean molecular speed in terms of the density and pressure of a gas. As it turns out, James P. Joule used this same method (1848) to determine the theoretical velocity of molecules several years before Clausius.
When I took physics in high school, I was totally baffled as to why temperature was associated with the speed of molecules. No explanation was given. It was stated as a fact. I think teaching this part of physics has not changed much. But the simplified formula, mean molecular velocity = sqrt(3 x Pressure/Density) would have explained it all, and it is easily derivable from first principles.
Another example is the pendulum. Solving the motion of a pendulum is a complex problem, and it is difficult to avoid using calculus. At worst, it involves elliptical functions, although some say that can be avoided. Furthermore, it would appear to require the use of D'Alembert's principle. Newton says the central forces cancel out, but they in fact do not. There is a net acceleration toward the point of suspension, which is why the bob travels in an arc. However, if one uses a conical pendulum instead of a back and forth pendulum -- in other words, a pendulum that swirls around in a circle instead of back and forth -- the problem is solved rather simply using the most elementary algebra and the equation for circular motion. It is a good way to teach circular motion, and the equation for the period of a conical pendulum is the same as that of a regular one.
In my undergraduate days, I stared at the pendulum problem for hours on end, attempting, without success, to make sense of it a la Newton. One must use D'Alembert's principle to solve it.
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