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Lami's Theorem
In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem, :\frac=\frac=\frac where ''A'', ''B'' and ''C'' are the magnitudes of the three coplanar, concurrent and non-collinear vectors, V_A, V_B, V_C, which keep the object in static equilibrium, and ''α'', ''β'' and ''γ'' are the angles directly opposite to the vectors. : Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy. Proof As the vectors must balance V_A+V_B+V_C=0, hence by making all the vectors touch its tip and tail we can get a triangle with sides A,B,C and angles 180^o -\alpha, 180^o -\beta, 180^o -\gamma. By the law of sines then \frac=\frac=\frac. Then by applying that for any angle \theta, \sin (180^o - \theta) = \sin \theta we obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Lamy
Bernard Lamy (15 June 1640 – 29 January 1715) was a French Oratorian, mathematician and theologian. Life Lamy was born in Le Mans, France. After studying there, he went to join the Maison d'Institution in Paris, and to Saumur thereafter. In 1658 he entered the congregation of the Oratory. Lamy became professor of classics at Vendôme in 1661, and at Juilly in 1663. He was ordained in 1667. After teaching a few years at Le Mans he was appointed to a chair of philosophy in the University of Angers. Here his teaching was attacked on the ground that it was too exclusively Cartesian, and Rebous the rector obtained in 1675 from the state authorities a decree forbidding him to continue his lectures. He was then sent by his superiors to Grenoble, where, thanks to the protection of Cardinal Le Camus, he again took up his courses of philosophy. In 1686 he returned to Paris, stopping at the seminary of Saint Magloire, and in 1689 he was sent to Rouen, where he spent the remain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane. Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines. Distance geometry provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them. Properties in three dimensions In three-dimensional space, two linearly independent vectors with the same initial point determine a plane through that point. Their cross product is a normal vector to that plane, and any vector orthogonal to this cross product through the initia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concurrent Lines
In geometry, lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. They are in contrast to parallel lines. Examples Triangles In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors: * A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter. * Angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter. * Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid. * Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter. Other sets of lines associated with a triangle are concurrent as well. For example: * Any median ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coplanarity
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane. Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines. Distance geometry provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them. Properties in three dimensions In three-dimensional space, two linearly independent vectors with the same initial point determine a plane through that point. Their cross product is a normal vector to that plane, and any vector orthogonal to this cross product through the initial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanical Equilibrium
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero. In addition to defining mechanical equilibrium in terms of force, there are many alternative definitions for mechanical equilibrium which are all mathematically equivalent. In terms of momentum, a system is in equilibrium if the momentum of its parts is all constant. In terms of velocity, the system is in equilibrium if velocity is constant. In a rotational mechanical equilibrium the angular momentum of the object is conserved and the net torque is zero. More generally in conservative systems, equilibrium is established at a point in configuration space where the gradient of the potential energy with respect to the generalized coordinates is zero. If a particle in equilibrium has zero velocity, that particle is in static equilibrium. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and are the lengths of the sides of a triangle, and , and are the opposite angles (see figure 2), while is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; \frac \,=\, \frac \,=\, \frac. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ''ambiguous case'') and the technique gives two possible values for the enclosed angle. The law of sines is one of two trigonometric equations commonly a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram Of Force
The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object. When more than two forces are involved, the geometry is no longer parallelogrammatic, but the same principles apply. Forces, being vectors are observed to obey the laws of vector addition, and so the overall (resultant) force due to the application of a number of forces can be found geometrically by drawing vector arrows for each force. For example, see Figure 1. This construction has the same result as moving F2 so its tail coincides with the head of F1, and taking the net force as the vector joining the tail of F1 to the head of F2. This procedure can be repeated to add F3 to the resultant F1 + F2, and so forth. Newton's proof Preliminary: the parallelogram of velocity Suppose a particle moves at a uniform rate along a line from A to B (Figure 2) in a given time (say, one second), while in the same time, the line AB moves uniformly from its position at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Embedding
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by , states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph. Example Let ''G'' be the graph of a cube, and (selecting one of its quadrilateral faces as the outer face) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with their environment. The application of Newton's second law to a system gives: : \textbf F = m \textbf a \, . Where bold font indicates a vector that has magnitude and direction. \textbf F is the total of the forces acting on the system, m is the mass of the system and \textbf a is the acceleration of the system. The summation of forces will give the direction and the magnitude of the acceleration and will be inversely proportional to the mass. The assumption of static equilibrium of \textbf a = 0 leads to: : \textbf F = 0 \, . The summation of forces, one of which might be unknown, allows that unknown to be found. So when in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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