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An Introduction to Materials
Published in Paul J. Hazell, Armour, 2023
where E is defined as the modulus of elasticity or Young’s modulus of the material. This relationship was originally discovered by Robert Hooke in 1678 and is sometimes referred to as Hooke’s law. At an atomic level, ε is a measure of the increase (or decrease) in atomic spacing due to the applied load. As the load is increased,the inter-atomic spacing increases,and when the load is removed,the atoms return to their equilibrium position. The greater the attraction between atoms—that is,the stronger the bonding—the greater the load required to increase the inter-atomic spacing. Certain ceramic armour materials are good examples of materials with high values of E due to their strong atomic bonding. Materials with relatively weak ionic bonding tend to possess relatively low values of E.
Petroleum Geophysical Survey
Published in Muhammad Abdul Quddus, Petroleum Science and Technology, 2021
The downward pulling gravitational force is measured by a spring. A spring is made by applying a twisting force to a thin, soft, flexible and long metallic bar. The mechanical force needed to twist the metallic bar is resisted by the opposing resistance of the bar. The resistance force is equal to the applied mechanical force but in the opposite direction. Thus a spring is a store of mechanical energy. A vertically suspended spring will resist the downward gravitational force, at the cost of the elongation of the spring. Hooke’s law states that ‘the force needed to elongate the spring by some distance, is proportional to the change in length (∆r) of the spring’. The elongation force is provided by the earth’s gravitational downward force (F) acting on the mass (m) of the spring. Therefore
Physical Properties of Infrared Optical Materials
Published in Paul Klocek, Handbook of Infrared Optical Materials, 2017
James Steve Browder, Stanley S. Ballard, Paul Klocek
This discussion of elastic properties is applicable only when the elastic deformation of the crystal is small, that is, less than 1% in research studies and less than 2% for practical applications. When a crystal is deformed to such an extent that Hooke’s law is no longer obeyed, the crystal is said to be plastically deformed. The elastic limit is used to indicate the stress above which plastic deformation occurs and below which Hooke’s law is obeyed. For metals, the stress–strain curve shows an abrupt change at the elastic limit; for most optical materials the stress–strain curve gradually changes from a straight line to a curve with a decreasing slope. In the apparent absence of a unique departure from Hooke’s law, an apparent elastic limit must be defined. This limit is taken as that stress on the stress–strain curve where the slope is half the slope at the origin of the stress–strain curve.
Students’ responses to the question: how does a computer do curve fitting?
Published in International Journal of Mathematical Education in Science and Technology, 2022
During the physics laboratory work, the students worked in 11 groups of 2–4 persons each and collected data from a proportional relation (between voltage and current). Each group typed their collected measurement data into a computer program and got the output as a scatterplot of data and regression line displayed together with the equation for the regression line. During this work, the researcher interviewed them with the following initial question. What mathematical ideas do you think a computer program uses when fitting a line to data? Give some spontaneous suggestions and conjectures!In the observed mathematics classroom, the students worked in nine groups with two to four students in each group. Here, the teacher opened the lesson by presenting the task and closed the lesson with a teacher-led whole-class discussion in which the students presented their work. Between the teacher’s opening and closing of the lesson, the students did about 20 min group work (which was about half of the lesson) on the following task. There are computer programs for fitting a line to data in a diagram. What mathematical ideas could be behind such computer programs? Suggest at least two methods for determining a line that fits well to measured data!The students were also told that they are expected to give principal ideas for how their suggested methods for curve fitting work, but that they do not need to do any computer programming or perform any lengthy calculations, but in that case, only highlight principal ideas in their suggestion. As tools for their mathematical work, they had paper, pencil, ruler and calculators. The historical context for mathematicians exploring over-determined systems of linear equations (1) as curve fitting was astronomy (Katz, 1998). This study instead uses Hooke’s law from physics as data context since it is a proportionality, and the students in this study are familiar with it. Hooke’s law describes how a spring extends when applying a force to it. In the task formulation, the students got the data in Table 1 and a scatter plot graph of these data. Hence, the task and its context explicitly restricted the students to the case of fitting a proportionality to data.