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Static 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 eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Derivative Test
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. First-derivative test The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch" and remains increasing or remains decreasing, then no highest or least value is achieved. One can examine a funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Integrity And Failure
Structural integrity and failure is an aspect of engineering that deals with the ability of a structure to support a designed structural load (weight, force, etc.) without breaking and includes the study of past structural failures in order to prevent failures in future designs. Structural integrity is the ability of an item—either a structural component or a structure consisting of many components—to hold together under a load, including its own weight, without breaking or deforming excessively. It assures that the construction will perform its designed function during reasonable use, for as long as its intended life span. Items are constructed with structural integrity to prevent catastrophic failure, which can result in injuries, severe damage, death, and/or monetary losses. ''Structural failure'' refers to the loss of structural integrity, or the loss of load-carrying structural capacity in either a structural component or the structure itself. Structural failure is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jenga
''Jenga'' is a Game of skill, game of physical skill created by British board game designer and author Leslie Scott (game designer), Leslie Scott and marketed by Hasbro. The name comes from the Swahili language, Swahili word "" which means 'to build or construct'. Players take turns removing one block at a time from a tower constructed of 54 blocks. Each block removed is then placed on top of the tower, creating a progressively more unstable structure. The game ends when the tower falls over. Rules ''Jenga'' is played with 54 wooden blocks. Each block is three times as long as it is wide, and one fifth as thick as its length – . Blocks have small, random variations from these dimensions so as to create imperfections in the stacking process and make the game more challenging. To begin the game, the blocks are stacked into a solid rectangular tower of 18 layers, with three blocks per layer. The blocks within each layer are oriented in the same direction, with their long sides tou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rock Balancing
Rock balancing (also stone balancing, or stacking) is a form of recreation or artistic expression in which Rock (geology), rocks are piled in balanced stacks, often in a precarious manner. Conservationists and park services have expressed concerns that the arrangements of rocks can disrupt animal habitats, accelerate soil erosion, and misdirect hikers in areas that use Cairn, cairns as navigation waypoints. Process During the 2010s, rock balancing became popular around the world, popularised through images of the rocks being shared on social media. Balanced rocks vary from simple stacks of two or three stones, to arrangements of round or sharp stones balancing in precarious and seemingly improbable ways. Professional rock-balancing artist Michael Grab, who can spend hours or minutes on a piece of rock balancing, says that his aim when stacking the stones is "to make it look as impossible as possible", and that the larger the size of the top rock, the more improbable the str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ship Stability
Ship stability is an area of naval architecture and ship design that deals with how a ship behaves at sea, both in still water and in waves, whether intact or damaged. Stability calculations focus on center of mass#center of gravity, centers of gravity, buoyancy, centers of buoyancy, the metacenters of vessels, and on how these interact. History Ship stability, as it pertains to naval architecture, has been taken into account for hundreds of years. Historically, ship stability calculations relied on rule of thumb calculations, often tied to a specific system of measurement. Some of these very old equations continue to be used in naval architecture books today. However, the advent of calculus-based methods of determining stability, particularly Pierre Bouguer's introduction of the concept of the metacenter in the 1740s ship model basin, allow much more complex analysis. Master shipbuilders of the past used a system of adaptive and variant design. Ships were often copied from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reaction (physics)
As described by the third of Newton's laws of motion of classical mechanics, all forces occur in pairs such that if one object exerts a force on another object, then the second object exerts an equal and opposite reaction force on the first. The third law is also more generally stated as: "To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."This translation of the third law and the commentary following it can be found in the "Mathematical Principles of Natural Philosophy, Principia" opage 20 of volume 1 of the 1729 translation The attribution of which of the two forces is the action and which is the reaction is arbitrary. Either of the two can be considered the action, while the other is its associated reaction. Examples Interaction with ground When something is exerting force on the ground, the ground will push back with equal force in the opposite direction. In certain f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point
In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and a relative maxima and minima, maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface (mathematics), surface that ''curves up'' in one direction, and ''curves down'' in a different dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagram Of A Ball Placed In A Neutral Equilibrium
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram. Overview The term "diagram" in its commonly used sense can have a general or specific meaning: * ''visual information device'' : Like the term "illustration", "diagram" is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables. * ''specific kind of visual display'' : This is the genre that shows qualitative data with shapes that are connected by lines, arrows, or other visual links. In science the term is used in both ways. For example, Anderson (1997) stated more generally: "diagrams are pictorial, yet abstract, represent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign Function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as \sgn x or \sgn (x). Definition The signum function of a real number x is a piecewise function which is defined as follows: \sgn x :=\begin -1 & \text x 0. \end The law of trichotomy states that every real number must be positive, negative or zero. The signum function denotes which unique category a number falls into by mapping it to one of the values , or which can then be used in mathematical expressions or further calculations. For example: \begin \sgn(2) &=& +1\,, \\ \sgn(\pi) &=& +1\,, \\ \sgn(-8) &=& -1\,, \\ \sgn(-\frac) &=& -1\,, \\ \sgn(0) &=& 0\,. \end Basic properties Any real number can be expressed as the product of its absolute value and its sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagram Of A Ball Placed In A Stable Equilibrium
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram. Overview The term "diagram" in its commonly used sense can have a general or specific meaning: * ''visual information device'' : Like the term "illustration", "diagram" is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables. * ''specific kind of visual display'' : This is the genre that shows qualitative data with shapes that are connected by lines, arrows, or other visual links. In science the term is used in both ways. For example, Anderson (1997) stated more generally: "diagrams are pictorial, yet abstract, represent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
