MatS_3011_S12_HW_5

MatS_3011_S12_HW_5 - You may do it manually or use a...

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University of Minnesota Department of Chemical Engineering and Materials Science MatS 3011: Introduction to Materials Science and Engineering Spring Semester 2012 Homework #5 (Due March 7, 2012) 1) (15 pts) Callister 6.29. 2) (15 pts) Callister 6.41. 3) (10 pts) Callister 6.49. 4) (5 pts) Callister 6.54. 5) (5 pts) Callister 7.20. 6) (5 pts) Callister 7.21. 7) (10 pts) Callister 7.30. 8) (10 pts) Callister 7.31. 9) (10 pts) Callister 7.38. 10) (15 pts) 4 specimens of the same alloy prepared under different conditions were tested for their mechanical properties and the grain sizes were examined using an optical microscope. The results of the experiments are as follows. Specimen # Yield Strength (MPa) Average grain size (mm) 1 60 0.1 2 90 0.03 3 130 0.01 4 150 0.007 Make an appropriate plot of the data to determine the constants, σ o and k y in the Hall-Petch equation. Use least squares method to fit a line to your plot (also known as linear regression).
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Unformatted text preview: You may do it manually or use a software or your calculator. You may learn about least squares method by reading about it in one of your math texts or on the web. If you have never done this before in a class you can read about it under Least Squares on Wikipedia.org. Here is how it works. If you have a set of N x and y values and want to fit a line y=mx+b you do the following. Calculate the sum of all x, S x =x 1 +x 2 +x 3 +x N Calculate the sum of all y, S y =y 1 + 2 +y 3 +y N Calculate the sum of all x 2 , S xx =x 1 2 + x 2 2 + x 3 2 + x N 2 Calculate the sum of all xy, S xy =x 1 y 1 + x 2 y 2 + x 3 y 3 + x N y N Slope is xy x y xx x x NS S S m NS S S = and the intercept is y x S mS b N = . Most of your calculators and programs like excel use these formulas to do linear regression....
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MatS_3011_S12_HW_5 - You may do it manually or use a...

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