This document discusses thin section analysis of crystals and rock samples. It begins by introducing crystallographic sections using diagrams of imaginary cuts through cubes to represent likely real sections. It then discusses the appearance of sections containing parallel planes or planes perpendicular to each other. Sections through a cube contained within a larger cube are also examined. Optical properties are briefly covered, including refractive index ellipsoids and the appearance of sections through ellipsoids of revolution and non-revolution. The goal is to emphasize the geometric aspects of thin section analysis while integrating both crystallographic and optical properties.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
The document discusses key concepts related to the section properties of structural members including:
- The center of gravity is the point where the total weight of a system can be considered to be concentrated.
- The center of mass is calculated similarly using the total mass of a system rather than total weight.
- The centroid is the geometric center of an object, independent of forces or weights, and depends only on the object's shape.
- Moments of inertia measure the resistance of an area to bending and twisting forces, and are calculated based on the area properties and distance from specific axes.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
1) The document discusses concepts related to centroid and moment of inertia including: the centroid is the point where the total area of a plane figure is assumed to be concentrated; formulas are provided for finding the centroid of basic shapes; the difference between centroid and center of gravity is explained; properties and methods for finding the centroid are described such as using moments.
2) Formulas are given for moment of inertia including how it is calculated about different axes and the parallel axis theorem.
3) Example problems are provided to demonstrate calculating the centroid and moment of inertia for various shapes.
ESA Module 3 Part-B ME832. by Dr. Mohammed ImranMohammed Imran
This document discusses two-dimensional photoelasticity techniques for stress analysis. It describes various methods for separating principal stresses at interior points of a photoelastic model, including using a lateral extensometer, properties at free boundaries, Laplace's equation, shear-difference method, and oblique incidence method. It also covers scaling stresses between models and prototypes for various applications, including static, thermal, and dynamic cases. As an example, it discusses using photoelasticity to optimize the design of sheet pile sections for cofferdams.
The first moment of area of a lamina is defined as the product of the lamina's area and the perpendicular distance of its center of gravity from a given axis. It is used to determine the center of gravity of an area. To calculate the first moment of area, the area is split into segments, and the area of each segment is multiplied by its distance from the axis and summed. This gives the first moment of area, which provides information about the distribution of the area.
The document discusses moments of inertia, which are integrals related to the distribution of mass of an object and how it resists rotational changes. It provides formulas for calculating the moment of inertia for basic shapes like rectangles and circles, and explains techniques like using differential area elements and the parallel axis theorem. Sample problems demonstrate calculating moments of inertia for composite shapes.
This document provides an overview of moiré techniques. It defines moiré patterns as interference patterns produced when two similar gratings are overlaid at a slight angle. Moiré patterns can reveal differences or deformations between the two gratings. The document discusses how moiré patterns are formed and analyzed mathematically. It also outlines several applications of moiré techniques, such as displacement measurement, stress analysis, and contour mapping.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
The document discusses key concepts related to the section properties of structural members including:
- The center of gravity is the point where the total weight of a system can be considered to be concentrated.
- The center of mass is calculated similarly using the total mass of a system rather than total weight.
- The centroid is the geometric center of an object, independent of forces or weights, and depends only on the object's shape.
- Moments of inertia measure the resistance of an area to bending and twisting forces, and are calculated based on the area properties and distance from specific axes.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
1) The document discusses concepts related to centroid and moment of inertia including: the centroid is the point where the total area of a plane figure is assumed to be concentrated; formulas are provided for finding the centroid of basic shapes; the difference between centroid and center of gravity is explained; properties and methods for finding the centroid are described such as using moments.
2) Formulas are given for moment of inertia including how it is calculated about different axes and the parallel axis theorem.
3) Example problems are provided to demonstrate calculating the centroid and moment of inertia for various shapes.
ESA Module 3 Part-B ME832. by Dr. Mohammed ImranMohammed Imran
This document discusses two-dimensional photoelasticity techniques for stress analysis. It describes various methods for separating principal stresses at interior points of a photoelastic model, including using a lateral extensometer, properties at free boundaries, Laplace's equation, shear-difference method, and oblique incidence method. It also covers scaling stresses between models and prototypes for various applications, including static, thermal, and dynamic cases. As an example, it discusses using photoelasticity to optimize the design of sheet pile sections for cofferdams.
The first moment of area of a lamina is defined as the product of the lamina's area and the perpendicular distance of its center of gravity from a given axis. It is used to determine the center of gravity of an area. To calculate the first moment of area, the area is split into segments, and the area of each segment is multiplied by its distance from the axis and summed. This gives the first moment of area, which provides information about the distribution of the area.
The document discusses moments of inertia, which are integrals related to the distribution of mass of an object and how it resists rotational changes. It provides formulas for calculating the moment of inertia for basic shapes like rectangles and circles, and explains techniques like using differential area elements and the parallel axis theorem. Sample problems demonstrate calculating moments of inertia for composite shapes.
This document provides an overview of moiré techniques. It defines moiré patterns as interference patterns produced when two similar gratings are overlaid at a slight angle. Moiré patterns can reveal differences or deformations between the two gratings. The document discusses how moiré patterns are formed and analyzed mathematically. It also outlines several applications of moiré techniques, such as displacement measurement, stress analysis, and contour mapping.
This document presents information on moments of inertia based on a course titled "pre-stressed concrete". It introduces moments of inertia and how they are calculated based on the distribution of mass or area relative to a given axis. Formulas are provided for calculating the moments of inertia of basic shapes like rectangles and circles through integration. The parallel axis theorem is also described, which relates the moments of inertia about different axes passing through an object. Examples are worked through, such as finding the moment of inertia of a triangle with respect to its base.
1) Moment of inertia is the rotational analog of mass for linear motion and appears in the relationships for rotational dynamics.
2) The moment of inertia must be specified with respect to a chosen axis of rotation and is calculated by summing the products of small elements of mass and their distances from the axis.
3) The perpendicular axis theorem states that for a planar object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes in the plane.
R05010105 A P P L I E D M E C H A N I C Sguestd436758
This document contains an examination for Applied Mechanics from the I B.Tech Supplimentary Examinations held in August/September 2008. It consists of 8 questions related to topics in applied mechanics, including forces, friction, belts, moments of inertia, and kinematics. Students were instructed to answer any 5 of the 8 questions, with all questions carrying equal marks. The questions involve calculations related to mechanical systems, structures, and motion.
Properties of surfaces-Centre of gravity and Moment of InertiaJISHNU V
The document discusses properties of surfaces, including centre of gravity and moment of inertia. It defines key terms like centre of gravity, centroid, area moment of inertia, radius of gyration, and mass moment of inertia. Methods for calculating these properties are presented for basic shapes like rectangles, triangles, circles, and composite shapes. Theorems like the perpendicular axis theorem and parallel axis theorem are also covered. Examples are provided for determining the moment of inertia of various plane figures and structures.
1. The document defines basic concepts related to solid geometry including direction ratios of lines, equations of straight lines and planes, different forms of the equation of a sphere, properties of spheres like touching spheres and tangent planes, right circular cones and cylinders.
2. It provides examples to find the equations of spheres, cones and cylinders given certain conditions like vertices, radii, axes etc.
3. Several exercises are given to find equations of spheres, cones and cylinders based on different scenarios like passing through points, intersecting planes or circles orthogonally.
This document discusses finding the centroid of solids of revolution. It explains that the centroid of a solid generated by revolving a plane area about an axis will lie on that axis. To find the coordinates of the centroid, one takes the moment of an elementary disc about the coordinate axes and sums these moments by treating the discs as an integral. Two examples are worked through to demonstrate finding the coordinates of the centroid. Five exercises are then posed asking to find the centroid coordinates for solids generated by revolving given bounded regions.
This document discusses the concept of centroid and provides formulas to calculate the centroid of different geometric shapes. It defines centroid as the point within an object where the downward force of gravity appears to act. The centroid allows an object to remain balanced when placed on a pivot at the centroid point. Formulas are given for finding the centroid of triangles, rectangles, circles, semicircles, right circular cones, and composite figures. Real-life applications of centroid calculation in construction and engineering are also mentioned.
Chapter 6: Pure Bending and Bending with Axial ForcesMonark Sutariya
This summary provides the key points about pure bending and bending with axial forces from the document:
1. Pure bending occurs when a beam segment is in equilibrium under bending moments alone, with examples being a cantilever loaded at the end or a beam segment between concentrated forces.
2. For beams with symmetric cross-sections, plane sections remain plane after bending according to the fundamental flexure theory. The elastic flexure formula gives the normal stress as proportional to the bending moment and the distance from the neutral axis.
3. The second moment of area, or moment of inertia, represents the beam's resistance to bending and is used to calculate maximum bending stresses. The elastic section modulus is a ratio of the moment
This document discusses methods for determining areas, volumes, centroids, and moments of inertia of basic geometric shapes. It begins by introducing the method of integration for calculating areas and volumes. Standard formulas are provided for areas of rectangles, triangles, circles, sectors, and parabolic spandrels. Formulas are also provided for volumes of parallelepipeds, cones, spheres, and solids of revolution. The concepts of center of gravity, centroid, and center of mass are defined. Equations are given for calculating the centroids of uniform bodies, plates, wires, and line segments. Methods for finding centroids of straight lines, arcs, semicircles, and quarter circles are illustrated.
This document discusses the concept of moment of inertia. It defines moment of inertia and provides formulas for calculating it for different objects like thin rods, rings, spheres, and rectangular shapes. It also discusses related concepts like torque, angular acceleration, angular momentum, angular impulse, work done by torque, and angular kinetic energy. Examples are provided to demonstrate calculations for these concepts. The key objectives are to understand moment of inertia and be able to calculate it for basic shapes, as well as understand how it relates to other rotational motion concepts.
This document discusses centroids and centers of mass. It defines the centroid of a set of points as the point where the sum of first moments is equal to zero. The position vector of the centroid is given by the weighted average of the position vectors of the individual points. Similarly, the center of mass of a system of particles is the weighted average of their position vectors, where the weights are the particle masses. Methods for finding the centroid of curves, surfaces, solids and continuous bodies are presented using integrals. The first moment of an area is introduced and related to the area centroid. Properties of centroids and centers of mass, such as their dependence on reference frames and behavior under decomposition, are covered.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
This document discusses moment of inertia calculations for non-symmetric structural shapes. It provides examples of calculating the neutral axis location, transformed moment of inertia about the strong axis, and moment of inertia about the weak axis for "T-shaped" beams. The process involves determining the centroid of the overall shape, then using the parallel axis theorem to calculate the transformed moment of inertia by summing the moments of inertia of individual pieces after accounting for the distance from each piece's centroid to the neutral axis.
ESA Module 5 Part-B ME832. by Dr. Mohammed ImranMohammed Imran
1. The document describes the moire fringes technique for experimental stress analysis. Moire fringes occur when two similar patterns are overlaid, allowing strains to be measured.
2. There are two approaches to analyzing moire fringe patterns - the geometrical approach regards fringes as intersections of grids, while the displacement approach uses fringes to determine displacements.
3. The distance between bright or dark fringes equals the master grid pitch divided by the strain. Fringes indicate loci of equal displacement and allow calculating strains from measured displacements.
Mass moment of inertia measures an object's resistance to angular acceleration and is defined as the integral of the second moment of mass elements about an axis. It depends on the axis chosen and the distribution of mass. For composite objects, the total mass moment of inertia is found by summing the contributions of each component. The parallel axis theorem allows calculating mass moments of inertia about parallel axes.
This document discusses calculating the moment of inertia for composite cross-sections made up of multiple simple geometric shapes. It introduces the parallel axis theorem, which allows calculating the moment of inertia of each individual shape about a common reference axis so that the individual values can be added to determine the total moment of inertia of the composite cross-section. Several examples are provided to demonstrate calculating moments of inertia for composite areas using this approach.
How to find moment of inertia of rigid bodiesAnaya Zafar
The document provides expressions for calculating the moment of inertia of various regularly shaped rigid bodies about different axes of rotation. It discusses:
1) Calculating moment of inertia using integral methods, considering small elements of the rigid body.
2) Examples of calculating moment of inertia for a rod, rectangular plate, circular ring, thin circular plate, hollow cylinder, solid cylinder, hollow sphere, and solid sphere.
3) Key steps involve identifying the small element, elemental mass, and integrating the expression for elemental moment of inertia over the body.
- A triangle is a three-sided polygon with three angles that sum to 180 degrees. Triangles can be classified based on side length (scalene, isosceles, equilateral) or angle type (acute, right, obtuse).
- The triangle inequality theorem states that any side of a triangle must be less than the sum of the other two sides and greater than the difference of the other two sides.
- A quadrilateral is a four-sided polygon. Quadrilaterals can be simple or complex, and simple ones can be convex or concave. The interior angles of any simple quadrilateral sum to 360 degrees.
- A circle is the set of all points in a plane equid
Here are the key points made in the response:
- Men commit battering against current or former female partners much more often than women battering men. This is consistent with national crime survey data.
- Women's use of force is often in self-defense, retaliation, or to express emotions like anger, stress or frustration. Men's use of violence is often to control and exercise power over their partners. So the motivations differ between men and women.
- The damage caused by women using force does not equal the damage caused by men.
- Therefore, gender symmetry does not exist in domestic violence. Rates of domestic violence are not equivalent between men and women as the gender symmetry argument suggests. Women are not
This document presents information on moments of inertia based on a course titled "pre-stressed concrete". It introduces moments of inertia and how they are calculated based on the distribution of mass or area relative to a given axis. Formulas are provided for calculating the moments of inertia of basic shapes like rectangles and circles through integration. The parallel axis theorem is also described, which relates the moments of inertia about different axes passing through an object. Examples are worked through, such as finding the moment of inertia of a triangle with respect to its base.
1) Moment of inertia is the rotational analog of mass for linear motion and appears in the relationships for rotational dynamics.
2) The moment of inertia must be specified with respect to a chosen axis of rotation and is calculated by summing the products of small elements of mass and their distances from the axis.
3) The perpendicular axis theorem states that for a planar object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes in the plane.
R05010105 A P P L I E D M E C H A N I C Sguestd436758
This document contains an examination for Applied Mechanics from the I B.Tech Supplimentary Examinations held in August/September 2008. It consists of 8 questions related to topics in applied mechanics, including forces, friction, belts, moments of inertia, and kinematics. Students were instructed to answer any 5 of the 8 questions, with all questions carrying equal marks. The questions involve calculations related to mechanical systems, structures, and motion.
Properties of surfaces-Centre of gravity and Moment of InertiaJISHNU V
The document discusses properties of surfaces, including centre of gravity and moment of inertia. It defines key terms like centre of gravity, centroid, area moment of inertia, radius of gyration, and mass moment of inertia. Methods for calculating these properties are presented for basic shapes like rectangles, triangles, circles, and composite shapes. Theorems like the perpendicular axis theorem and parallel axis theorem are also covered. Examples are provided for determining the moment of inertia of various plane figures and structures.
1. The document defines basic concepts related to solid geometry including direction ratios of lines, equations of straight lines and planes, different forms of the equation of a sphere, properties of spheres like touching spheres and tangent planes, right circular cones and cylinders.
2. It provides examples to find the equations of spheres, cones and cylinders given certain conditions like vertices, radii, axes etc.
3. Several exercises are given to find equations of spheres, cones and cylinders based on different scenarios like passing through points, intersecting planes or circles orthogonally.
This document discusses finding the centroid of solids of revolution. It explains that the centroid of a solid generated by revolving a plane area about an axis will lie on that axis. To find the coordinates of the centroid, one takes the moment of an elementary disc about the coordinate axes and sums these moments by treating the discs as an integral. Two examples are worked through to demonstrate finding the coordinates of the centroid. Five exercises are then posed asking to find the centroid coordinates for solids generated by revolving given bounded regions.
This document discusses the concept of centroid and provides formulas to calculate the centroid of different geometric shapes. It defines centroid as the point within an object where the downward force of gravity appears to act. The centroid allows an object to remain balanced when placed on a pivot at the centroid point. Formulas are given for finding the centroid of triangles, rectangles, circles, semicircles, right circular cones, and composite figures. Real-life applications of centroid calculation in construction and engineering are also mentioned.
Chapter 6: Pure Bending and Bending with Axial ForcesMonark Sutariya
This summary provides the key points about pure bending and bending with axial forces from the document:
1. Pure bending occurs when a beam segment is in equilibrium under bending moments alone, with examples being a cantilever loaded at the end or a beam segment between concentrated forces.
2. For beams with symmetric cross-sections, plane sections remain plane after bending according to the fundamental flexure theory. The elastic flexure formula gives the normal stress as proportional to the bending moment and the distance from the neutral axis.
3. The second moment of area, or moment of inertia, represents the beam's resistance to bending and is used to calculate maximum bending stresses. The elastic section modulus is a ratio of the moment
This document discusses methods for determining areas, volumes, centroids, and moments of inertia of basic geometric shapes. It begins by introducing the method of integration for calculating areas and volumes. Standard formulas are provided for areas of rectangles, triangles, circles, sectors, and parabolic spandrels. Formulas are also provided for volumes of parallelepipeds, cones, spheres, and solids of revolution. The concepts of center of gravity, centroid, and center of mass are defined. Equations are given for calculating the centroids of uniform bodies, plates, wires, and line segments. Methods for finding centroids of straight lines, arcs, semicircles, and quarter circles are illustrated.
This document discusses the concept of moment of inertia. It defines moment of inertia and provides formulas for calculating it for different objects like thin rods, rings, spheres, and rectangular shapes. It also discusses related concepts like torque, angular acceleration, angular momentum, angular impulse, work done by torque, and angular kinetic energy. Examples are provided to demonstrate calculations for these concepts. The key objectives are to understand moment of inertia and be able to calculate it for basic shapes, as well as understand how it relates to other rotational motion concepts.
This document discusses centroids and centers of mass. It defines the centroid of a set of points as the point where the sum of first moments is equal to zero. The position vector of the centroid is given by the weighted average of the position vectors of the individual points. Similarly, the center of mass of a system of particles is the weighted average of their position vectors, where the weights are the particle masses. Methods for finding the centroid of curves, surfaces, solids and continuous bodies are presented using integrals. The first moment of an area is introduced and related to the area centroid. Properties of centroids and centers of mass, such as their dependence on reference frames and behavior under decomposition, are covered.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
This document discusses moment of inertia calculations for non-symmetric structural shapes. It provides examples of calculating the neutral axis location, transformed moment of inertia about the strong axis, and moment of inertia about the weak axis for "T-shaped" beams. The process involves determining the centroid of the overall shape, then using the parallel axis theorem to calculate the transformed moment of inertia by summing the moments of inertia of individual pieces after accounting for the distance from each piece's centroid to the neutral axis.
ESA Module 5 Part-B ME832. by Dr. Mohammed ImranMohammed Imran
1. The document describes the moire fringes technique for experimental stress analysis. Moire fringes occur when two similar patterns are overlaid, allowing strains to be measured.
2. There are two approaches to analyzing moire fringe patterns - the geometrical approach regards fringes as intersections of grids, while the displacement approach uses fringes to determine displacements.
3. The distance between bright or dark fringes equals the master grid pitch divided by the strain. Fringes indicate loci of equal displacement and allow calculating strains from measured displacements.
Mass moment of inertia measures an object's resistance to angular acceleration and is defined as the integral of the second moment of mass elements about an axis. It depends on the axis chosen and the distribution of mass. For composite objects, the total mass moment of inertia is found by summing the contributions of each component. The parallel axis theorem allows calculating mass moments of inertia about parallel axes.
This document discusses calculating the moment of inertia for composite cross-sections made up of multiple simple geometric shapes. It introduces the parallel axis theorem, which allows calculating the moment of inertia of each individual shape about a common reference axis so that the individual values can be added to determine the total moment of inertia of the composite cross-section. Several examples are provided to demonstrate calculating moments of inertia for composite areas using this approach.
How to find moment of inertia of rigid bodiesAnaya Zafar
The document provides expressions for calculating the moment of inertia of various regularly shaped rigid bodies about different axes of rotation. It discusses:
1) Calculating moment of inertia using integral methods, considering small elements of the rigid body.
2) Examples of calculating moment of inertia for a rod, rectangular plate, circular ring, thin circular plate, hollow cylinder, solid cylinder, hollow sphere, and solid sphere.
3) Key steps involve identifying the small element, elemental mass, and integrating the expression for elemental moment of inertia over the body.
- A triangle is a three-sided polygon with three angles that sum to 180 degrees. Triangles can be classified based on side length (scalene, isosceles, equilateral) or angle type (acute, right, obtuse).
- The triangle inequality theorem states that any side of a triangle must be less than the sum of the other two sides and greater than the difference of the other two sides.
- A quadrilateral is a four-sided polygon. Quadrilaterals can be simple or complex, and simple ones can be convex or concave. The interior angles of any simple quadrilateral sum to 360 degrees.
- A circle is the set of all points in a plane equid
Here are the key points made in the response:
- Men commit battering against current or former female partners much more often than women battering men. This is consistent with national crime survey data.
- Women's use of force is often in self-defense, retaliation, or to express emotions like anger, stress or frustration. Men's use of violence is often to control and exercise power over their partners. So the motivations differ between men and women.
- The damage caused by women using force does not equal the damage caused by men.
- Therefore, gender symmetry does not exist in domestic violence. Rates of domestic violence are not equivalent between men and women as the gender symmetry argument suggests. Women are not
The document discusses the difference between bulk/macrotexture and microtexture. Bulk texture provides information about the overall orientation distribution in a material but not the spatial location of orientations. Microtexture provides information about individual grain orientations and neighbors from EBSD mapping. Bulk texture is represented by pole figures and ODFs from XRD, while microtexture uses orientation image maps from EBSD. The document also explains how pole figures are obtained from XRD by rotating the sample to satisfy Bragg's law for different crystal planes and orientations.
1. The document discusses various engineering drawing concepts including scales, conic sections, engineering curves, and units of measurement.
2. Scales include plain, diagonal, and comparative scales. Conic sections include ellipses, parabolas, and hyperbolas. Engineering curves include cycloids, epicycloids, hypocycloids, and involutes.
3. The document provides examples of scale construction and contains questions related to engineering drawing concepts.
An isosceles triangle is a triangle with at least two equal sides. This means two angles of the triangle are also equal. An isosceles triangle can be a special case of an equilateral triangle, which has all three sides equal. It can also be an isosceles right triangle, which has one 90 degree angle. Formulas are provided for calculating the height, area, inradius, and centroid of an isosceles triangle.
This powerpoint includes:
Triangles and Quadrangles
Definition, Types, Properties, Secondary part, Congruency and Area
Definitions of Triangles and Quadrangles
Desarguesian Plane
Mathematician Desargues and His Background
Harmonic Sequence of Points/Lines
Illustrations and Animated Lines.
Crystallography and X-ray diffraction (XRD) Likhith KLIKHITHK1
Atoms in materials are arranged into crystal structures and microstructures.
Periodic arrangement of atoms depends strongly on external factors such as temperature, pressure, and cooling rate during solidification. Solid elements and their compounds are classified into amorphous, polycrystalline, and single crystalline materials. The amorphous solid materials are isotropic in nature because their atomic arrangements are not regular and possess the same properties in all directions. In contrast, the crystalline materials are anisotropic because their atoms are arranged in regular and repeated pattern, and their properties vary with direction. The polycrystalline materials are combinations of several crystals of varying shapes and sizes. The properties of polycrystalline materials are strongly dependent on distribution of crystals sizes, shapes, and orientations within the individual crystal. Diffraction pattern or intensities of X-ray diffraction techniques are used for characterizing and probing arrangement of atoms in each unit cell, position of atoms, and atomic spacing angles because of comparative wavelength of X-ray to atomic size.The X-ray diffraction, which is a non-destructive technique, has wide range of material analysis including minerals, metals, polymers, ceramics, plastics, semiconductors, and solar cells. The technique also has wide industry application including aerospace, power generation, microelectronics, and several others. The X-ray crystallography remained a complex field of study despite wide industrial applications.
This document provides information on spherical trigonometry. It defines key terms like great circles, small circles, spherical angles, and spherical triangles. It describes properties of these concepts, such as every great circle passing through the center of a sphere. Formulas for solving spherical triangles are presented, including the Sine Formula, Cosine Formula, and Haversine Formula. Examples show how to use these formulas to calculate unknown sides and angles of spherical triangles given other information.
This document discusses different types of graphical representations used in geology. It begins by defining graphs and graphical representation. It then discusses why engineering geologists use graphs, including to visualize and predict geologic events. It provides examples of log-normal graphs, log-log graphs, triangular diagrams, and polar graphs. It also discusses equal interval, equal angle, and equal area projections of a sphere and how they are used to plot surface data onto a flat surface.
This report summarizes research on the motion of particles on curves. It was found that:
1) The center of mass of 3 points on an ellipse that divide its perimeter evenly traces out a smaller ellipse of the same shape.
2) The maximum product of distances between 4 particles on a rectangle occurs when particles are at the corners for small rectangles, but 2 particles move off the corners for larger rectangles.
3) The center of mass of n points on a square that divide its perimeter evenly traces out a smaller square n times for odd n, and remains fixed at the center for even n.
Kepler's laws describe the motion of planets orbiting the Sun. Kepler's first law states that planets follow elliptical orbits with the Sun at one focus. The more distant the foci, the more elongated the ellipse. Kepler's second law says that a line connecting a planet to the Sun sweeps out equal areas in equal times. A planet moves fastest when closest to the Sun. Kepler's third law relates the square of a planet's orbital period to the cube of its average distance from the Sun.
Kepler's laws describe the motion of planets orbiting the Sun. Kepler's First Law states that planets follow elliptical orbits with the Sun at one focus. The more distant the foci, the more elongated the ellipse. Kepler's Second Law says that a line connecting a planet to the Sun sweeps out equal areas in equal times. A planet moves fastest when closest to the Sun. Kepler's Third Law relates the square of a planet's orbital period to the cube of its average distance from the Sun.
A hexagonal tiling with reflections of angle π/6 is recognized as the dihedral group D6. After identifying 3 different kaleidoscopic points, the tiling is found to be from the wallpaper group p6m. An equilateral triangle tiling with a 3-fold rotational symmetry of 2π/3 is also identified, belonging to the wallpaper group p31m after determining an independent 3-fold gyration point. The document further discusses extensions to Coxeter polygons, orbifolds, frieze groups, and classifications of different brick wall patterns.
- Crystallographic points, directions and planes are specified using indexing schemes like Miller indices.
- Materials can be single crystals or polycrystalline aggregates of randomly oriented grains, leading to anisotropic or isotropic properties respectively.
- A crystal's diffraction pattern in reciprocal space is determined by its real space lattice and atomic structure. The reciprocal lattice is constructed geometrically from the real lattice and maps planes in real space to points in reciprocal space.
This document discusses crystal structures and lattice planes. It begins by defining a crystal as having atoms or molecules arranged in a periodic three-dimensional pattern. The smallest repeating unit of a crystal structure is called the unit cell, which is defined by its axial lengths and interaxial angles. There are 14 possible lattice arrangements known as Bravais lattices. Miller indices are used to describe lattice planes, which are determined by taking the reciprocals of the intercepts of a plane with the crystal axes and multiplying by the least common denominator. Bragg's law of diffraction is mentioned as the reason X-rays are useful for crystallography due to their wavelengths being on the order of atomic distances.
The document summarizes the theory and operation of the Michelson interferometer. It describes how the interferometer uses a beam splitter and two mirrors to split an incoming light beam into two paths that later recombine, creating an interference pattern. Key points covered include the factors that influence the interference patterns observed, such as the coherence and wavelength of the light source. Applications like measuring refractive indices and displacements are also mentioned. Instructions for setting up and using a basic Michelson interferometer in the laboratory are provided.
This document discusses congruent triangles and their properties. It begins by introducing triangles and their historical uses. It then defines different types of triangles based on their sides and angles. The document explains the triangle congruence postulates of SSS, SAS, ASA, and uses examples to prove triangles congruent. It also discusses using congruent triangles to solve real-world problems involving structures. The document warns that there is no SSA or AAA postulate for triangle congruence. It concludes by discussing properties of isosceles and equilateral triangles.
The document discusses the logo of the Mathematical Association of America (MAA), which is an icosahedron. It begins by describing how to construct an icosahedron out of three rectangles placed along perpendicular axes. This construction leads to two questions: 1) Can the three rectangles be rearranged into the desired configuration without taking one apart? 2) How symmetrical is an icosahedron? Additional constructions are used to answer these questions by showing that the rectangles are linked together, preventing their rearrangement, and that the icosahedron has 60 orientation-preserving symmetries forming the alternating group on 5 symbols.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
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Petrographic notes
1. INTRODUCTION
(Translated from Spanish)
Thin Sections are two dimensional cuts of bodies with crystallographic and optical properties belonging to three
dimensions, so a good knowledge of Solid Geometry, can be of great help in studies of these sections with the
petrographic microscope. These notes are intended to emphasize geometric aspects, taking into account, first
crystallography, then optics, and finally an integration of both. We use the Miller indices only for simple planes:
CRYSTALLOGRAPHIC SECTIONS.
The idea is to do many imaginary cuts to a cube, in order to obtain an approximation to the most likely real random
cuts. Note that a good example would be the volcanic rocks, where the majority of crystals can take their own forms.
1. SIMPLE CUBE
In figure 2, we begin with a section coincident with the frontal face of the cube (100). If we turn the section 90° with
the axis of rotation indicated by the arrows, we will obtain several rectangles and two squares. Two of the sides of
rectangles are the same than the cube’s side (a) and the maximum length that can reach the other two is the diagonal
of a face of the cube (section 3).
Now (figure 3) we begin with section 3 of the previous case: a plane containing the diagonals of two opposite faces of
the cube, one of which is the axis of rotation. Initially we have a rectangle (cut 1) then isosceles trapezoids (cut 2) and
finally an equilateral triangle (cut 3). If we continue turning the cross section in the same direction, we get isosceles
triangles up to a square.
In figure 4, we begin with the same type of section that the previous case, but the axis of rotation is now bisector of
the section and parallel to the diagonal of upper face (001). After the initial rectangle, we get a regular hexagon (cut
2), then a rhombus (cut 3) whose major axis is the main diagonal of the cube. If we continue turning in the same
direction 90°, we get rhombus, whose major axis reduces until we get a square.
(100) (010) (001) (101) (011) (110) (111) (210)
1
2
3
1
2 3
a
2
a
1 2
3
1 2 3
3
2. Now (Figure 5) we have the same initial section than before, but now we move the section in a parallel way. We get
several rectangles and a square section. It should be noted that the length of two sides of the rectangles, is the same
than the side of the cube.
The initial section in figure 6, is an equilateral triangle (the section of a tip of the cube), then we get isosceles
triangles, trapezoids (not shown) and finally we obtain a rectangle.
So far, we can see that most likely sections that we can obtain of random cuts of a cube are rectangles, triangles
and trapezoids. Less likely are hexagons, squares and rhombuses. The rectangles have two sides equal to the
side of the cube. The size of the sides of the triangles may vary from minuscule to the diagonal of one face of
the cube.
2. CUBE WITH INNER PLANES PARALLEL TO A FACE.
A set of equally spaced parallel planes, will show the smaller thickness and higher density (amount of traces per unit
area) in a perpendicular section. Cut 1 of figure 7 is perpendicular to x, y and z planes. Cut 2 is tilted and the traces of
the planes in this section are thicker and more spaced (less dense).
4
a 3
a
1 2 3
1
2
a
a
5
6
a
3. With this short introduction, let us consider a cube
than contains a set of planes parallel to one face
(figure 8).
Figure 9 shows the same type of cuts that figure 2. Section 1 is the frontal side of the cube (001) and does not cut
the inner planes. Section 2 cut only one plane with a weak slope. The trace of the plane in this section is then thick.
Amphibole of Sotará volcano, Colombia. Note the presence
of two cleavages, one thinner and denser and the other
thicker and less dense. If the objective (x40) is moved
slightly, the thinner cleavage does not seem to move, while
the other shows a neat movement. The thickness and dense
difference is due to section, almost perpendicular for the
thinner and tilted for the thicker one. One section
perpendicular to both cleavages, show them with the same
thickness and density. Plane polarized light. x10.
Thin cleavage
Thick cleavage
8
x y z
2
x y z
1
2
x y
z7
1
4. The other sections cut all the planes with increasing inclination, therefore the thickness of the traces decreases and
density increases.
Figure 10 shows the same cuts than figure 3, but the axis of rotation is now the diagonal of the top face of the cube
(001). In order to obtain the different sections it should be noted that in all cases, one of the edges of the sections is
contained in the frontal face of the cube (100) and then parallel to the inner planes. Therefore, the traces of these
planes in all sections will be parallel to this border line. Note also that the direction of rotation is toward the upper
face (001) which is perpendicular to the inner planes. Therefore, the thickness of traces decrease and their density
rise gradually.
2
3
4
5
1
9
1
2
3
4
5
3
10
1
2
1
2
3
5. 3. CUBE WITH TWO INNER PLANES MUTUALLY PERPENDICULAR.
(Figure 11)
Figure 12 shows the same sections than figures 2 and 9. Note that cuts are perpendicular to X planes therefore their
traces have the minimum thickness and the maximum density in all them. For traces of Y planes is the same case than
figure 9. It should be noted that traces of both planes are perpendicular to each other, but for one family, their
thickness and density will vary.
11
y
3
x
1
x
2
3
y
4
1
2
3
4x y
12
6. Figure 13 shows exactly the same cut that section 3 of figure 10 or section 3 of figure 3. The cut is an equilateral
triangle, where one side is the diagonal of the front face (100) and then parallel to Y planes. Another side is the
diagonal of (010) face and then parallel to X planes. Therefore the traces of both planes in the section will have the
same thickness and density since the slope of cut is the same for both planes. Note that the angle between both traces
is 60º because is an equilateral triangle.
Note in figure 14 that vertical sides of cuts are parallel to both inner planes, therefore their traces will be parallel
each other. The thickness and density will depend of the slope of cut and will be the same only for section 3, but
note that the direction of inclination is opposite.
13
x
y
x
y
1
2 3
4
x y
1
y xy
2
3
y x
4
x
14
7. 4. CUBE INSIDE ANOTHER CUBE.
Figure 15 shows a cube included in the center of another, twice its size.
Figure 16 shows three sections, the second one in
perspective for more clarity. It is clear, that the
probability that a random section cuts the inner cube,
will be low if the size is small, but if its size approach
the size of the external cube, more random sections
will contain both cubes.
We could see an analogy with all these figures and Thin Sections. Inner planes could be cleavage or twin planes. The
cube inside another one is similar to zoning of minerals or the external portion of a crystal altered by some chemical
reaction with his environment. It is important to note that although the thin sections are essentially two dimensional
bodies, their thickness (30 microns) is of great help in finding particular sections of a mineral. When the objective is
displaced slightly (40x would be appropriate), a cleavage or twin plane perpendicular to the thin section, will present
a fine trace and remain static in the field of view, but seems thicker and to move more or less insofar as the slop is
farther away of perpendicular to the thin section.
15
16
2
1 3
1 2
3
2
Plagioclases. Nevado del Ruiz. Colombia. Cross
polarized light. The left crystal shows the traces
of twin planes very thin and their density is high
suggesting that they are perpendicular to the
section. The right crystal instead, shows the twin
traces thicker and less dense. x4.
Plagioclase. Nevado del Ruiz volcano. Colombia. Left image plane
polarized light. Right image cross polarized light. Note both cleavages
almost mutually perpendicular and the trace of albite twin very thin.
These characteristics belong to a section very close to perpendicular to
a axis o [100]. The probability to find this section is very low but very
interesting, because it allows the better determination for composition
in routine methods of Thin Sections. x10.
8. OPTICS
Although these notes are not intended as a manual of Optical Mineralogy, always is useful to recall some basic
concepts that are handled in this discipline.
In anisotropic crystals, the speed of light can vary according to its direction of vibration. The refractive index is the
ratio between the speed of light in vacuum with respect to its velocity in the medium considered. The light used in
Petrography typically is orthoscopic and then is possible to associate directly a direction vibration of light with a
refractive index and in that way simplify the reasoning used in the determination of anisotropic sections.
If the refractive indices of a crystal are put all together in a point with the same direction in space that have the
vibration of light associated with each of them, the resulting envelope is an ellipsoid called the indicatrix. Depending
on the crystal symmetry this will be an ellipsoid of revolution (Tetragonal and Hexagonal systems) or not
(Orthorhombic, Monoclinic and Triclinic systems). It is useful to remember that the indicatrix is an artifact and by its
construction, its sections must necessarily pass through its center.
SECTIONS OF A REVOLUTION ELLIPSOID (17 and 18)
We take as the axis of rotation, the major axis of the ellipsoid but
rationing is essentially the same if we take the minor axis. A
perpendicular section to the major axis will be a circumference since
all points on the ellipse to rotate, describe a circle perpendicular to the
axis of rotation. A parallel section to major axis will be an ellipse with
the largest eccentricity can be obtained, that will decrease with the
angle of cut. Note that the minor axis is contained in all sections.
SECTIONS OF A ELLIPSOID NOT OF REVOLUTION
In this case the ellipsoid has three axes: large, medium
and small that are mutually perpendicular. Sections
perpendicular to one of these axes, contain the other
two. If we take the section that contains the major and
minor axis of the ellipsoid (figure 20), somewhere in
the ellipse, will give a distance to the center, equal to
the intermediate axis of the ellipsoid. If we continue
with the same procedure for cuts parallel to the axis of
the ellipsoid (Figure 21), we obtain a circular area
whose radius is the length of the intermediate axis of
the ellipsoid. These sections will be isotropic and its
perpendicular is called the optical axis. There will be
two circular sections is an ellipsoid not of revolution.
17
18
20
Np
Ng
Nm
19 great
petty
middle
9. There are several ways to symbolize the refractive indices. In
an effort to emphasize its size, is used here Ng (g great) for the
major axis of the ellipsoid (Figure 22), Np (p petty) for the
smaller one and Nm for the middle axis (these are equivalent
to the vibration directions Z, X and Y). In the case of an
anisotropic section, we will use n'p and n'g when we only
know the relative size between the two indices.
The birefringence of a mineral, is the difference between its major and minor refractive indices, that is, between
the major and minor axes of the indicatrix. The birefringence of a section is the difference between the major and
minor indices of the section. It is clear then that for a given mineral, the birefringence of the sections will range
from zero (the refractive indices of the section are equal, which corresponds to circular sections of the indicatrix) to
a maximum value that coincides with the nominal value given for the mineral (the section contains then the major
and minor axis of the indicatrix).
Usually the light used in the petrographic microscope is white and normal to the thin section (orthoscopic) being
polarized according to the direction of the Polarizer that is often taken NS; over the thin section, is the Analyzer
whose polarization direction is perpendicular or EW. Polarizer and Analyzer arranged in this way (crossed
polarizers) do not allow the passage of light. If we interpose between the two, another polarizer with its polarization
direction at 45 degrees of both, fwe see that there will be light transmission (Figure 23). This can be seen by vector
decomposition (Figure 24).
Clearly, if the direction of the intermediate polarizer coincides with either the Polarizer or Analyzer, no light is
transmitted.
Ng
Nm
1
1
Np
Nm
Ng
21
Circular
section
Ng
Np
Nm
22
Polarized
direction
23
Polarizer
Analyzer
10. The sections of the indicatrix, other than circular
sections, can be seen as a polarizer, with two
polarization directions mutually perpendicular. If any
of these directions coincides with the Polarizer of the
microscope, the beam of light coming from it, will
break down in the section into two beams with
mutually perpendicular vibration with different speeds
(different refractive indices).
If the analyzer is crossed, the two rays from the section are decomposed vectorially, interfering with each other,
resulting in a characteristic color (interference colours), which will depend on the birefringence of the section and
its thickness. These two factors together, constitute what is called the retardation. It is therefore important to
remember that the observed color (with crossed polarizers), depends not only on the birefringence of the section, but
also its thickness. Thus using the same thickness for the sections (typically 30 microns), the interference colours
varies only with the birefringence of the section.
In the event that one of the indices of the anisotropic section, match with the direction of the Polarizer (Figure 25) it
will exists only one direction of vibration in the section parallel to the Polarizer (the other index is 90 degrees to the
Polarizer and can not provide components). With crossed nicols, there is no light transmission (extinction position).
In order to observe the optical characteristics of one index of a section in natural light (without the Analizer) it must
be parallel to Polarizer. This is accomplished by taking the section to extinction (crossed nicols), and remove the
Analyzer. The relief and color observed, belong to the index parallel to Polarizer. It is important to note that for
different positions, relief and color will be intermediate between both indices.
REAL CASE
HYPERSTHENE
Orthorhombic system. Two good cleavages {210} that cut a axis at
half distance and b axis at unity, but a is almost twice b in length,
that means that they are almost perpendicular (88°). Np (X)= 1.712
light rose-brown. Nm(Y)=1.724 pale yellow-green. Ng(Z)=1.727
pale grey-green. Biaxial Negative (the minor index is the bisector
of optical axes). 2V between 50° - 60° (Tröger, 1971). With these
values we can see that the birefringence of Hypersthene (Ng-Np) is
0.015 that is, an interference color of orange first order for 30
microns.
Figure 26 is a schematic perspective view of hypersthene. The refractive indices coincide with crystallographic axes.
The section (001), that is perpendicular to c axis (Orthorhombic system) contains a and b axis and then Np and Nm
indices. The birefringence of this section (Nm-Np) is 0.012 yellow first order for 30 microns of thickness. The
Polarizer
Analyzer
Middle
PolarizerVibration
of light
de la luz
24
Polarizer
Anisotropic
Section25
10
0
(210)
(010)
Ng c
Np
b
Nm
a
Optical axis26
11. cleavages are perpendicular to section then their traces are very thin, perpendicular each other and stay static if the
objective (x40) is moved slightly. In plane polarized light, if the major index of the section (Nm) is parallel to
Polarizer, the color of section will be yellowish hue. If we rotate the section 90º (Np will be parallel to Polarizer), we
will see a pinkish hue. In crossed polarized light, the extinction will be symmetric with respect to traces of cleavages.
Section (100) is perpendicular to a axis and then contains b and c axis and therefore Np and Ng indices. The
birefringence is the same than the mineral (Ng – Np) 0.015 orange first order. The cleavages are at 45º to the
section and then is difficult to observe them. The section is pleochroic between a greenish hue (Ng parallel to the
polarizer) and pink (Np aligned with the polarizer).
The section (010) contains Ng and Nm indices. The birefringence will be (Ng – Nm) 0.003 dark gray. For the same
reason as above, it will be difficult to observe the cleavages. The colors of the indices are greenish, therefore the
pleochroism is not obvious. The section is perpendicular to bisector of optical axis and can therefore be seen the
interference figure well centered.
The probability to obtain strictly these three sections is very small. However the sections close to them show
similar characteristics.
A section (210) that is parallel to one of the cleavages (figure 28)
contains Ng index while the minor axis of the section, will be
between Np and Nm therefore the birefringence is between 0.003
and 0.015. We may consider a white hue of interference more or
less. The color in plane polarized light will be between a greenish
(Ng) and pinkish hue (n’p). We can see only one cleavage with a
trace very thin because is almost perpendicular to section.
In figure 29 we start with a (001) section. In the first case, the cuts are directed toward the face (100) and they
remain parallel to the b axis, therefore, all sections contain the Np index. The major index of the initial section is
Nm and Ng for the final one, then the major index of the section (n’g) will have a value between these two indices.
The interference colors will be between yellow and orange. The traces of cleavages will become thicker and the
angle between them gradually decreases. If the objective is moved slightly the traces will move to opposite sides,
because although the angle is the same, the direction of inclination is opposite. Extinctions remain symmetrical
with respect to cleavages.
In the second case, the cuts are directed toward the face (010), remaining parallel to the a axis. Note that the plane
formed by the optical axes and indices Ng and Np is perpendicular to all sections and then the cuts will contain the
index Nm of the mineral. The other index in the initial section is Np and in the final one is Ng, so it will be a section
where this index is Nm and the section is isothropic or cyclic and perpendicular to one of optical axes. This section is
not pleochroic. The birefringence decrease then progressively from the initial section, to zero (cyclic section), before
rising slightly to a final birefringence of 0.003 (face (010)). As in the previous case the traces of cleavages thicken
progressively, the angle between them diminishes and extinctions remain symmetrical.
Starting from the same previous cut (001) towards (210), it can be seen that in this case, the sections will be almost
perpendicular to one of the cleavages. The situation is similar to the cuts in Figure 12. The traces of cleavage
perpendicular to the sections, will be fine, keep the same density and remain static when slightly displace the
(001) section (100) section (010) section
Nm
Ng
Np
Ng
Nm
27
Np
Ng
n‘p
28
12. microscope objective. The traces of the other cleavage will thicken, its density will decrease and appear to move
more strongly as the tilt angle decreases, when slightly move the objective. Note that only the initial section contains
two of the indices of the mineral, while the final contains only one (Ng). Both indices of the other cuts will have
intermediate values. The sections of the indicatrix are not evident in this case, however, the major index of initial
section bisects the traces of cleavages, while in the final one, this index is parallel to the only visible cleavage. It
could be seen that the major index of sections will be progressively close to the fine cleavage and the extinctions are
not parallel which means that extinctions for these sections are neither straight nor symmetric.
Toward (100) Face
n‘g
n‘g
Np
Np
Toward (010) Face
Cyclic section
Np
(001) Section
(001)Nm
Nm
n‘g
Nm
Nm
29
Toward (210) Face
n‘g
n‘p
n‘g
n‘p
Hypersthene. Section close to (001) face.
The traces of two cleavages are thin and
almost perpendicular each other. Yellow first
order in crossed polarized light (30 microns).
Left, plane polarized light. Right, crossed
nicols. Andesite of Nevado del Ruiz
volcano.Colombia. x10.
13. EPILOGUE
From all these examples, it can be seen the importance of the Solid Geometry. A good knowledge of the ellipsoids,
together with the crystalline forms, will allow more reliable identification of the crystals and a better understanding of
their textures. In short, a three-dimensional ‘vision’, can go beyond the simple identification of minerals and is a
necessary starting point in structural studies.
BIBLIOGRAPHY
SHELLEY, David. Manual of Optical Mineralogy. 1975. Elsevier.
STOIBER, Richard; MORSE, Stearns. Crystal identification with the polarizing microscope. 1994. Chapman & Hall.
TRÖGER, W.E.. Optische Bestimmung der gesteinsbildenden Minerale. 1971.
Hypersthene. Section close to (010) face.
Cleavages are no visible. Pleochroism is clear
and interference colors are grayish. The
section shows the interference figure well
centered.
Hypersthene. Section close to (100) face. This
section does not show cleavages, as they are far
from perpendicular to section. Pleochroism very
clear between greenish and pinkish colors. In
crossed polarized light, the interference color is
orange of first order (30 microns) and is the same
that the mineral (contains the major and minor
indices of mineral).