Stainless Steel Rods: Young’s Modulus and Shear Modulus

Stainless Steel Rods: Young’s Modulus and Shear Modulus Valentin Haemmerli Experimentally determine Young’s modulus, shear modulus and Poisson’s ratio of stainless steel rods using magnetostrictive resonance Abstract Young’s modulus (E) and the shear modulus (ÃŽ ¼) of thin stainless steel rods, as well as Poisson’s ratio (ÏÆ'), were experimentally found by determining the longitudinal and torsional resonant frequencies for different known lengths of rods using magnetostrictive resonance. Young’s modulus was found to be 140 GPa  ±17 and shear modulus 59.2 GPa  ±5.7. Poisson’s ratio was found for the rods of varying length and three of these were within right range at: 0.23 ±0.07 for the 0.417m rod, 0.13 ±0.04 for the 0.411m rod and 0.11 ±0.03 for the 0.251m rod. Introduction This experiment aimed to determine Young’s modulus (E) and the shear modulus (ÃŽ ¼) of thin stainless steel rods, as well as Poisson’s ratio (ÏÆ'), by finding the longitudinal and torsional resonant frequencies for different known lengths of rods using magnetostrictive resonance. A drive coil connected to a power amplifier was used to vary the driving frequency and excite the steel rods. The vibrations of the steel rods due to the changing magnetic field were measured using a stereo cartridge connected to an oscilloscope. Theory Magnetostriction is the effect observed when magnetic materials in an external magnetic field increase in length very slightly, due to the alignment of the microscopic domains. By rapidly reversing a magnetic field around a ferrous rod, such as the steel rods examined in this experiment, it is possible to induce vibration by the motion of the domains [1]. Young’s modulus and the shear modulus of a material determine the frequency at which it resonates in different modes. Solids can experience three main modes of vibration; longitudinal, torsional, and flexural [2]. The modes examined in this experiment are longitudinal and torsional. Longitudinal vibrations are â€Å"stretching and contracting of the beam along its own axis† [3, p. 182] of the material when a driving frequency is applied, while torsional is a twisting motion of the material. Young’s modulus determines longitudinal resonance and shear modulus determines torsional resonance. The natural frequencies for longitudinal and torsional vibration of a steel rod are given by , (1) , (2) where , L is the length of the rod, and C are the wave velocities: (3) and (4) respectively, where E and ÃŽ ¼ are Young’s and shear moduli and Ï  is density. These equations are used to relate f to 1/L and thus find the elastic moduli. Poisson’s ratio, ÏÆ', is the ratio of change in dimensions laterally and longitudinally of a material â€Å"placed under a uniform longitudinal tensile (compressive) load† and is normally around 0.3 [3, p. 4]. Davis and Opat give this as , (5) where ÃŽ ³ is given by [2] . (6) Method The method was adapted from that used by Davis and Opat in â€Å"Elastic vibrations of rods and Poisson’s ratio† [2]. Six stainless steel rods of varying lengths between 0.102 and 0.417 m were individually clamped at their centres by three pointed screws to reduce contact and thus damping. The rods were then positioned to pass through a drive coil, also close to their centre, and finally the stereo cartridge stylus was positioned at the top of the rod, off centre on the flat end, as shown in Figure 1. Figure 1: Clamp stand with rod clamped in the centre, coil clamped slightly above, and the stereo cartridge positioned above the rod to pick up vibrations. Foam used under clamp stand to attempt to reduce back ground vibrations. This positioning allowed for the detection of and distinction between longitudinal and torsional resonances. The two outputs of the stereo cartridge each respond to different component of motion of the stylus at 45ËÅ ¡ to the horizontal. Figure 2 is a diagram of the stereo cartridge stylus and placement on the end of the rod from Davis and Opat [2] which shows how it was possible to differentiate between longitudinal and torsional modes. Whenever resonance occurred and the two channels were in phase it was longitudinal as both directions of motion moved up and down at the same time. When resonance occurred out of phase it was torsional as the rotation meant the two directions of motion were outputting opposite signals. Figure 2: From Davis and Opat’s â€Å"Elastic vibrations of rods and Poisson’s ratio† [2]. Stereo stylus design (a) and placement on the rod (b) The drive coil was connected to a power amplifier and the output frequency was varied. The two outputs of the stereo cartridge were connected to the two channels of an oscilloscope. In this way, it was possible to vary the frequency until the amplitude shown on the oscilloscope was a maximum and record the frequency. This was repeated for rods of different length. Also recorded were the mass and diameter of each rod analysed in order to find the density since each steel rod had slightly different composition. Results Figure 3 shows the resonant frequency plotted against the reciprocals of the lengths of steel rods. Also plotted is a line of best fit by least squares method with intercept 0 as a result of equation (1), if 1/L =0, f=0. The error bars on the frequency are the standard errors found by regression. Error in the equipment for frequency was 2Hz and insignificant compared to the large random error. Error bars in the reciprocal length comes from the percentage error of the measurements due to an equipment error of  ±0.003m. As can be seen, the line of best fit is outside of the error boxes created by these errors and this suggests that the data is not very reliable and that there are not enough points for the line of best fit to be very accurate. Figure 3: Resonant frequencies (kHz) of longitudinal vibrations for n=1 (fundamental) plotted against the reciprocals of the lengths of the rods (m-1). The gradient of the fitted line in Figure 3 is 2.095 kHz  ±0.129. Using equations (1) and (3) with n=1, this gives E=140 GPa  ±17 using Ï steel =7970 kg m-3 [3, p. 435], or using the average of the densities of steel recorded (Ï  =8020 kg m-3  ±700) E=141 GPa  ±20. Similar to Figure 3, Figure 4 shows the fundamental resonant frequencies for torsional vibrations of the same rods. Figure 4: Resonant frequencies (kHz) of torsional vibrations for n=1 (fundamental) plotted against the reciprocals of the lengths of the rods (m-1). The gradient of the fitted line in Figure 3 is 1.363 kHz  ±0.066. Using equations (2) and (4) with n=1, this gives ÃŽ ¼= 59.2 GPa  ±5.7 using Ï steel =7970 kg m-3 [3, p. 435], or using the average of the densities of steel recorded (Ï  =8020 kg m-3  ±701) ÃŽ ¼=59.6 GPa  ±7.8. Poisson’s Ratio (ÏÆ') is found from the longitudinal and torsional resonant frequencies of the same rod and the same mode (n=1) using equations (5) and (6). This quantity varies for each rod, again evidence of a large random error in the resonant frequencies. Table 1 shows the different values of ÏÆ'. The errors for Poisson’s ratio are calculated based on the random error in each of the resonant frequencies. Table 1: Poisson’s Ratio for different lengths of rods for mode n=1 from equations (5) and (6) Discussion Young’s modulus and shear modulus are in the same order of magnitude as literature values, with experimentally determined E=140GPa compared to a literature value of around 180 GPa for stainless steel [4] or 194 according to Blevins [3]. Experimentally determined shear modulus was found to be ÃŽ ¼=59.2GPa compared to 77.2Gpa [5]. The result for the shear modulus is more accurate, and this is confirmed by the smaller random error. The errors due to the equipment for these measurements are very small, since the frequency could be varied to within 0.1 Hz and differences in amplitudes on the oscilloscope could be observed within 2Hz. However, with only 5 points, and no modes higher than n=1 to confirm the resonant frequencies, as well as a limited few lengths, there are not enough points of data to obtain a truly accurate result. For Poisson’s Ratio, there is a large variation between the values for each rod, which is in part linked to the large random error in the frequency values themselves, but which may also be due in part to the differences in the type of steel used in each rod. They are almost all of the right order of magnitude, and some are very close to the literature value of 0.265 [3, p. 435]. One major problem encountered was the fact that no resonant frequencies above 15kHz were observed. Whether this is a limitation of the stereo cartridge or due to the extremely small width of the resonances at these high frequencies, or a combination of both, is unclear. However, it may be possible to detect resonances at higher frequencies with a more sensitive stereo cartridge or a more accurate power amplifier. While the power amplifier used was adjustable to 0.1Hz at low frequencies, above 10kHz this was reduced to 1Hz. Another improvement to the method is to use more and longer rods. This is similar to the problem addressed above of high frequency resonances being difficult to detect. No resonances were found for the shortest rod available because all, including the n=1 mode, were too high. With longer rods, and more data points, a more accurate result could be In some cases, it was difficult to record data accurately or to detect resonances due to background vibrations. For example, the movement of a chair 5m away was enough to create a very unstable oscilloscope trace due to the sensitivity of the stereo cartridge to low frequencies. This was the case despite efforts to reduce the background vibrations by placing the clamp stand set up on foam. An extension to non-ferrous materials was attempted by using a small piece of steel with two longer pieces of aluminium attached with screws on either side. However, only one strong resonance was detected, which was not close to the predicted resonance of aluminium, and since the issues mentioned above meant that it was difficult to obtain enough data even for steel rods it was decided not to pursue this. As Davis and Opat put it, â€Å"Inhomogeneities in the structure of the rod can lead to coupling of the different vibrational modes and the description of the oscillating rod rapidly becomes more complex.† [2]. A more appropriate method for generating vibrations in rods of non-ferrous materials is outlined by Meiners and may be found in â€Å"Physics Demonstration Experiments† on page 439 [6]. Conclusion The longitudinal and torsional resonance frequencies for stainless steel rods of varying known length were measured and used to determine Young’s modulus of 140 GPa  ±17 and shear modulus of 59.2 GPa  ±5.7 using literature values for density of steel. Poisson’s ratio was found for the rods of varying length and three of these were within right range at: 0.23 ±0.07 for the 0.417m rod, 0.13 ±0.04 for the 0.411m rod and 0.11 ±0.03 for the 0.251m rod. The random error in the resonance frequencies was large, which meant that none of the results are very accurate. The accuracy could be improved with more data form more rods. References Acknowledgements Thanks to collaborator in Data Collection: Bivu Nepaune 1

We Must Educate the Children of Illegal Immigrants Essay example -- Cr

Immigration laws have been a subject of debate throughout American history, especially in states such as California and Texas, where immigrant populations are high. Recently, some citizens have been questioning whether we should continue to educate the children of illegal immigrants. While this issue is steeped in emotional controversy, we must not allow divisive "us against them" rhetoric to cloud our thinking. Yes, educating undocumented immigrants costs us, but not educating them would cost us much more. Those who propose barring the children of illegal immigrants from our schools have understandable worries. They worry that their state taxes will rise as undocumented children crowd their school systems. They worry about the crowding itself, given the loss of quality education that comes with large class sizes. They worry that school resources will be deflected from their children because of the linguistic and social problems that many of the newcomers face. And finally, they worry that even more illegal immigrants will cross our borders because of the lure of free education. This last worry is probably unfounded. It is unlikely that many parents are crossing the borders solely to educate their children. More likely, they are in desperate need of work, economic opportunity, and possibly political asylum. As Charles Wheeler of the National Immigration Law Center asserts, "There is no evidence that access to federal programs acts as a magnet to foreigners or that further restrictions would discourage illegal immigrants" (qtd. in "Exploiting"). The other concerns are more legitimate, but they can be addressed by less drastic measures than barring children from schools. Currently the responsibility o... ...re in school, we can make sure they are inoculated properly, and we can teach them the facts about health and disease. Do we really want thousands of uneducated children growing up on the streets, where we have little control over them? Surely not. The lure of the streets is powerful enough already. Only by inviting all children into safe and nurturing and intellectually engaging schools can we combat that power. Our efforts will be well worth the cost. Works Cited Edmondson, Brad. "Life without Illegal Immigrants." American Demographics May 1996: 1. "Exploiting Fears." Admissions Decisions: Should Immigration Be Restricted? 7 Oct. 1996. Public Agenda. 10 Feb. 1999. http://www.vote-smart.org/issues/Immigration/chap2/imm2itx.html. McCarthy, Martha M. "Immigrants in Public Schools: Legal Issues." Educational Horizons 71 (1993): 128-30.

An advice leaflet for teenagers on how to lead a healthier lifestyle

To start with, if you want to look good and stay fit, you should exercise and eat a healthy and balanced diet. Today, 25% of the world’s population are dying from heart disease which is caused by fat blocking the artery which carries blood that goes to the heart and that’s how people die. You can stop this problem by exercising. What good will this do to you? Well, when you run or do any sort of exercise, your body will burn the fat and make you feel lighter. You can argue with me that fat is needed for keeping you warm in the winter. True, but it’s needed in limited amounts. If you exercise, it exercises not only your body but also your heart and will also control your breathing rate. Secondly, you should eat a balanced diet. That is eat carbohydrates: bread, cornflakes, wheat etc. more every day to give you energy to start your day; vitamins and minerals- fruit and veg to boost up your immune system: defence for your body to protect you against harmful diseases and fats and oils for storage and warmth in your body- e.g. chocolate. You should also eat proteins e.g. egg for growth and repair of cells. Thirdly, you should get out, cycle or play football with your friends and feel the fresh, clean and pure breeze rushing on to your face and making you feel refreshed. It also gives you a break from study and calms your mind from stress. You should at least go out for an hour every day. So overall what I am trying to do is to get you all to agree with me that you should exercise, get outside and feel the fresh air and eat a balanced diet for your healthy lifestyle, healthy body and healthy mind. Pictures to fit the above points: An unfit body: A fit body: Let’s see what you have learned. Fill in the quick quiz below: 1. What do you need in order to have a healthy diet? 2. What do you need to do in order to remain fit? 3. What does going outside to play etc. do to you? Submit your answers no later than Tuesday 7th February 2011 on this address: 12, Fitchburg province, Tennessee, US 00345 Website- www.healthyadvicemagazine.com

Financial and Analysis Project - 7698 Words

OXFORD BROOKES UNIVERSITY [pic] The Identification Of Key Factors That Influence The Levels Of Motivation Of Employees In An Organization (Topic 6) A Research and Analysis Project for the BSc. [Hons] in Applied Accounting Name of Student: Andre Adimoolah ACCA Registration # : 1119620 Date: 13th March 2009 Number of Words: 5,630 Acknowledgement and Thanks: This project would not have existed today if it had not been for the following persons that I would like to thank: ↠ My family, teachers, friends, and anyone who have contributed to the success of this project. ↠ The management and staff of Methanol Holdings (Trinidad) Limited (MHTL) for their valuable contribution by providing the necessary data†¦show more content†¦In 2003, following a group restructuring, MHTL became the parent company of the operating methanol companies, which was continued in 2004 by way of a group amalgamation and dissolution of the subsidiaries. By virtue of this reorganization and amalgamation, MHTL has acquired over 25 years experience in the methanol business. Trinidad and Tobago Methanol Company (TTMC) is headquartered in the Point Lisas Industrial Estate in Trinidad, West Indies at the site of its methanol complex which has five methanol plants with a total production capacity of over 4 million metric tonnes annually. The company’s most recent plant, the M5000 plant, is rated as the largest methanol plant in the world with a designed capacity of 5,400 metric tonnes per day. This company was a state owned company which began operations in 1 984 with the TTMC I plant, the first world scale methanol plant to be constructed and operated in Latin America with a rated annual capacity of 460,000 metric tonnes. In 1996 the company commenced another production train with a second methanol plant (TTMC II) which had a rated annual capacity of 550,000 metric tonnes. |[pic] |Show MoreRelatedFinancial Analysis Of A New Project1338 Words   |  6 PagesFinancial analysis of a new project Introduction The following paper analyzes a project from financial perspectives using the capital budgeting techniques like Net Present Value (NPV) and Internal Rate of Return (IRR). Background My dad has a textile business, involved in embroidery and painting of the fabric. I have been visiting my dad’s office complex and observing the whole process of clothes manufacture. The most important asset for the business is a large machine required for whole paintingRead MoreFinancial Forecasting and Project Analysis14473 Words   |  58 PagesFINANCIAL FORECASTING AND CAPITAL BUDGETING ANALYSIS Ronald W. Spahr Professor and Chair, Department of Finance, Insurance and Real Estate Fogelman College of Business and Economics University of Memphis, Memphis, TN 38152-3120 Office phone: (901) 678-1747 or 5930, Fax: (901) 678-0839 spahr@memphis.edu January 10, 2011 FINANCIAL FORECASTING AND CAPITAL BUDGETING ANALYSIS Course Description This course covers fundamental concepts and techniques of financial forecasting and financialRead MoreFinancial Analysis Project : Starbucks2056 Words   |  9 PagesDavid Pulido Mr. Norcio Financial Management 22 April 2015 Financial Analysis Project Starbucks is a privileged company connected to millions of customers every day with great products in the market and more than 20,000 retail stores in 65 countries. Starbucks day to day mission is to serve the best coffee, which is why they have all their coffee grown under the best standards of quality. Starbuck’s coffee employees or the people in charge to get the coffee, travel to coffee farms in differentRead MoreA Financial Statement Analysis Project for Introductory Financial Accounting5953 Words   |  24 PagesEducation Volume 6, 2009, 83-96 A FINANCIAL STATEMENT ANALYSIS PROJECT FOR INTRODUCTORY FINANCIAL ACCOUNTING Marilyn B. Misch Business Administration Division Pepperdine University Malibu, California USA Carolyn A. Galantine Business Administration Division Pepperdine University Malibu, California USA ABSTRACT This paper describes a financial statement analysis project useful in both preparerbased and user-based introductory courses in financial accounting. The project requires students to analyze trendsRead MoreA Financial Statement Analysis Project for Introductory Financial Accounting5944 Words   |  24 PagesVolume 6, 2009, 83-96 A FINANCIAL STATEMENT ANALYSIS PROJECT FOR INTRODUCTORY FINANCIAL ACCOUNTING Marilyn B. Misch Business Administration Division Pepperdine University Malibu, California USA Carolyn A. Galantine Business Administration Division Pepperdine University Malibu, California USA ABSTRACT This paper describes a financial statement analysis project useful in both preparerbased and user-based introductory courses in financial accounting. The project requires students to analyzeRead MoreFinancial Analysis Xacc280 Final Project1784 Words   |  8 PagesFinancial Analysis/ Final Project XACC/280 Eden Lord Financial Analysis, Final Project Based upon my knowledge learned on financial reporting, I had compared to companies reporting statistics. The two companies in comparison are PepsiCo Incorperated and The Coca-Cola Company in which both have reported annual statistics for 2004 and 2005. During my comparison of net incomes, gross expenses, stock statistics, and assets accumulations, I have suggested some strategies for each business to take intoRead MoreFinancial Statement Analysis Project Apple vs Dell5273 Words   |  22 PagesFINANCIAL STATEMENT ANALYSIS PROJECT Comparison between Apple Inc. and Dell Inc. Accounting and Financial Analysis Spring 2013 Done by: Table of Contents: * Introduction * Horizontal Analysis * Significant Percentages * Ratios * Inventory and Depreciation * Liabilities * Cash Flow Analysis * Audit Report Analysis * Investor Analysis * Conclusion Introduction Apple Inc. (Apple), incorporated on January 3, 1977, Apple s headquarter isRead MoreSports Direct Financial Situation Analysis and Their New Project Investment in Morocco5386 Words   |  22 PagesPART I: COMPANY ANALYSIS 5 1- Introduction of the company 5 1.1 History: 5 1.2. The industry: 5 1.3. Sports Direct today: 6 2. Analysis on the financing structure of the company 6 2.1. Sports Direct financial statements: 7 2.2. JJB Sports financial statements: 9 2.3. Ratios of Sports Direct and JJB Sports 12 3. The weighted average cost of capital 15 3.1. Cost of equity: 15 3.2. Cost of debt: 15 3.3. WACC: 16 PART II: PROJECT INVESTMENT ANALYSIS 17 1- PreliminaryRead MoreFinals Project - Financial Analysis of Pepsi Co. and Coca-Cola1940 Words   |  8 PagesCo. and Coca-Cola’s Vertical, Horizontal, and Ratio Analysis XACC/280 Submitted by: Lataikeii Evans Date: April 8, 2011 Daneene Barton Arbitrating a company’s monetary physical stability inculpate their financial statements (income statements, balance sheets, and statement of cash flow). These statements must follow the GAAP standards. Sidewise from the statements that are involved in balancing a company or in my case comparing the financial stability of the two successful companies’ three componentsRead MoreFinancial Statement Analysis Project- Liz Clairborne Inc and Jones Apparel Group4723 Words   |  19 PagesFINANCIAL STATEMENT ANALYSIS PROJECT LIZ CLAIRBORNE INC AND JONES APPAREL GROUP TABLE OF CONTENTS I. INTRODUCTION - 1 - II. HORIZONTAL ANALYSIS OF JONES APPAREL GROUP - 2 - III. SIGNIFICANT PERCENTAGE INCREASES - 3 - IV. FINANCIAL RATIO ANALYSIS - 5 - 1. Current Ratio - 5 - 2. Gross Margin Percentage - 6 - 3. Profit Margin Percentage - 6 - 4. Accounts Receivable Turnover and Days to Collect - 7 - 5. Allowance for Doubtful Accounts to Accounts Receivable - 8 - 6. Inventory