1.
Torus
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In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a shape and is called a torus of revolution. Real-world examples of objects include inner tubes, swim rings, and the surface of a doughnut. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, a solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, in topology, a ring torus is homeomorphic to the Cartesian product of two circles, S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1, the ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces an object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any space that is topologically equivalent to a torus. R is known as the radius and r is known as the minor radius. The ratio R divided by r is known as the aspect ratio, a doughnut has an aspect ratio of about 2 to 3. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is 2 + z 2 = r 2, or the solution of f =0, algebraically eliminating the square root gives a quartic equation,2 =4 R2. The three different classes of standard tori correspond to the three aspect ratios between R and r, When R > r, the surface will be the familiar ring torus. R = r corresponds to the torus, which in effect is a torus with no hole. R < r describes the self-intersecting spindle torus, when R =0, the torus degenerates to the sphere. When R ≥ r, the interior 2 + z 2 < r 2 of this torus is diffeomorphic to a product of an Euclidean open disc, the losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. In traditional spherical coordinates there are three measures, R, the distance from the center of the system, and θ and φ. As a torus has, effectively, two points, the centerpoints of the angles are moved, φ measures the same angle as it does in the spherical system. The center point of θ is moved to the center of r and these terms were first used in a discussion of the Earths magnetic field, where poloidal was used to denote the direction toward the poles
2.
Doughnut
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A doughnut or donut is a type of fried dough confectionery or dessert food. The doughnut is popular in countries and prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls. Doughnuts are usually deep-fried from a dough, and typically either ring-shaped or without a hole. Other types of batters can also be used, and various toppings and flavorings are used for different types, such as sugar, chocolate, or maple glazing. In addition to flour, doughnuts may also include such ingredients as water, leavening, eggs, milk, sugar, oil/shortening, the two most common types are the ring doughnut and the filled doughnut—which is injected with fruit preserves, cream, custard, or other sweet fillings. A small spherical piece of dough may be cooked as a doughnut hole, once doughnuts have been fried, they may be glazed with a sugar icing, spread with icing or chocolate, or topped with powdered sugar or sprinkles or fruit. Other shapes include rings, balls, and flattened spheres, as well as ear shapes, twists, Doughnut varieties are also divided into cake and yeast-risen type doughnuts. Donuts are often accompanied by coffee when they are purchased at shops or fast food restaurants. This smaller piece of dough can be cooked and served as a hole or added back to the batch to make more doughnuts. A disk-shaped doughnut can also be stretched and pinched into a torus until the center breaks to form a hole, alternatively, a doughnut depositor can be used to place a circle of liquid dough directly into the fryer. There are two types of ring doughnuts, those made from a dough for raised doughnuts, or those made from a special type of cake batter. Yeast-raised doughnuts contain about 25% oil by weight, whereas cake doughnuts oil content is around 20%, cake doughnuts are fried for about 90 seconds at approximately 190 °C to 198 °C, turning once. Yeast-raised doughnuts absorb more oil because they take longer to fry, about 150 seconds, cake doughnuts typically weigh between 24 g and 28 g, whereas yeast-raised doughnuts average 38 g and are generally larger, and taller when finished. After frying, ring doughnuts are often topped, raised doughnuts are generally covered with a glaze. Cake doughnuts can also be glazed, or powdered with confectioners sugar, or covered with cinnamon and they are also often topped with cake frosting and sometimes sprinkled with coconut, chopped peanuts, or sprinkles. Doughnut holes are small, bite-sized doughnuts that were made from the dough taken from the center of ring doughnuts. Traditionally, doughnut holes are made by frying the dough removed from the portion of the doughnut. Consequently, they are smaller than a standard doughnut and tend to be spherical
3.
Rope
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A rope is a group of yarns, plies, or strands that are twisted or braided together into a larger and stronger form. Ropes have tensile strength and so can be used for dragging and lifting, as a result, they cannot be used for pushing or similar compressive applications. Rope is thicker and stronger than similarly constructed cord, line, string, fiber rope is made from fiber, whereas wire rope is made from wire. Rope may be constructed of any long, stringy, fibrous material, synthetic fibre ropes are significantly stronger than their natural fibre counterparts, but also possess certain disadvantages, including slipperiness. Common natural fibres for rope are manila hemp, hemp, feathers, linen, cotton, coir, jute, straw, synthetic fibres in use for rope-making include polypropylene, nylon, polyesters, polyethylene, Aramids and acrylics. Some ropes are constructed of mixtures of several fibres or use co-polymer fibres, wire rope is made of steel or other metal alloys. Ropes have been constructed of fibrous materials such as silk, wool, and hair. Rayon is a fibre used to make decorative rope. The twist of the strands in a twisted or braided rope serves not only to keep a rope together, without any twist in the rope, the shortest strand would always be supporting a much higher proportion of the total load. Rope is of paramount importance in fields as diverse as construction, seafaring, exploration, sports, theatre, and communications, to fasten rope, many types of knots have been invented for countless uses. Pulleys redirect the force to another direction, and can create mechanical advantage so that multiple strands of rope share a load. Winches and capstans are machines designed to pull ropes, such ropes normally use a kernmantle construction, as described below. Static ropes, used for example in caving, rappelling, and rescue applications, are designed for minimal stretch, the UIAA, in concert with the CEN, sets climbing-rope standards and oversees testing. Any rope bearing a GUIANA or CE certification tag is suitable for climbing, climbing ropes, however, do cut easily when under load. Keeping them away from sharp rock edges is imperative, rock climbing ropes come with either a designation for single, double or twin use. A single rope is the most common and it is intended to be used by itself, single ropes range in thickness from roughly 9mm to 11mm. Smaller ropes are lighter, but wear out faster, double ropes are thinner ropes, usually 9mm and under, and are intended for use as a pair. These ropes offer a margin or security against cutting, since it is unlikely that both ropes will be cut, but they complicate belaying and leading
4.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
5.
Compression (physics)
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It is contrasted with tension or traction, the application of balanced outward forces, and with shearing forces, directed so as to displace layers of the material parallel to each other. The compressive strength of materials and structures is an important engineering consideration, in uniaxial compression the forces are directed along one direction only, so that they act towards decreasing the objects length along that direction. If the stress vector itself is opposite to x, the material is said to be under compression or pure compressive stress along x. In a solid, the amount of compression depends on the direction x. If the stress vector is purely compressive and has the same magnitude for all directions and this is the only type of static compression that liquids and gases can bear. In a mechanical longitudinal wave, or compression wave, the medium is displaced in the waves direction, when put under compression, every material will suffer some deformation, even if imperceptible, that causes the average relative positions of its atoms and molecules to change. The deformation may be permanent, or may be reversed when the compression forces disappear, in the latter case, the deformation gives rise to reaction forces that oppose the compression forces, and may eventually balance them. Liquids and gases cannot bear steady uniaxial or biaxial compression, they will deform promptly and permanently, however they can bear isotropic compression, and may be compressed in other ways momentarily, for instance in a sound wave. The deformation may not be uniform and may not be aligned with the compression forces, what happens in the directions where there is no compression depends on the material. Most materials will expand in those directions, but some special materials will remain unchanged or even contract, by inducing compression, mechanical properties such as compressive strength or modulus of elasticity, can be measured. Compression machines range from small table top systems to ones with over 53 MN capacity. Gases are often stored and shipped in highly compressed form, to save space, slightly compressed air or other gases are also used to fill balloons, rubber boats, and other inflatable structures. Compressed liquids are used in equipment and in fracking. In internal combustion engines the explosive mixture gets compressed before it is ignited, in the Otto cycle, for instance, the second stroke of the piston effects the compression of the charge which has been drawn into the cylinder by the first forward stroke. This compression, moreover, obviates the shock which would otherwise be caused by the admission of the steam for the return stroke. Buckling Container compression test Compression member Compressive strength Longitudinal wave P-wave Rarefaction Strength of materials
6.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
7.
Tension (physics)
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Tension is the opposite of compression. At the atomic level, when atoms or molecules are pulled apart each other and gain potential energy with a restoring force still existing. Each end of a string or rod under such tension will pull on the object it is attached to, to restore the string/rod to its relaxed length. In physics, tension, as a force, as an action-reaction pair of forces. The ends of a string or other object transmitting tension will exert forces on the objects to which the string or rod is connected and these forces due to tension are also called passive forces. Tension in a string is a scalar quantity. A string or rope is often idealized as one dimension, having length but being massless with zero cross section. If there are no bends in the string, as occur with vibrations or pulleys, then tension is a constant along the string, by Newtons Third Law, these are the same forces exerted on the ends of the string by the objects to which the ends are attached. If the string curves around one or more pulleys, it still have constant tension along its length in the idealized situation that the pulleys are massless and frictionless. A vibrating string vibrates with a set of frequencies that depend on the strings tension and these frequencies can be derived from Newtons laws of motion. Each microscopic segment of the string pulls on and is pulled upon by its neighboring segments, tension = τ where x is the position along the string. If the string has curvature, then the two pulls on a segment by its two neighbors will not add to zero, and there will be a net force on that segment of the string, causing an acceleration. With solutions that include the various harmonics on a stringed instrument, tension is also used to describe the force exerted by the ends of a three-dimensional, continuous material such as a rod or truss member. Such a rod elongates under tension, a system is in equilibrium when the sum of all forces is zero. ∑ F → =0 For example, consider a system consisting of an object that is being lowered vertically by a string with tension, T, at a constant velocity. ∑ F → = T → + m g → =0 A system has a net force when a force is exerted on it. Acceleration and net force always exist together, ∑ F → ≠0 For example, consider the same system as above but suppose the object is now being lowered with an increasing velocity downwards therefore there exists a net force somewhere in the system. In this case, negative acceleration would indicate that | m g | > | T |
8.
Hoist (device)
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A hoist is a device used for lifting or lowering a load by means of a drum or lift-wheel around which rope or chain wraps. It may be operated, electrically or pneumatically driven and may use chain. The load is attached to the hoist by means of a lifting hook, the basic hoist has two important characteristics to define it, Lifting medium and power type. The lifting medium is either wire rope, wrapped around a drum, or load-chain, the power can be provided by different means. Common means are hydraulics, electrical and air driven motors, both the wire rope hoist and chain hoist have been in common use since the 1800s, however mass production of an electric hoist did not start until the early 1900s and was first adapted by Germany. A hoist can be built as one unit, designed for cost-effective purchasing and moderate use, or it can be built as a built-up custom unit, designed for durability. The built-up hoist will be more expensive, but will also be easier to repair. Package units were regarded as being designed for light to moderate usage. Built-up units are designed for heavy to severe service, but over the years that market has decreased in size since the advent of the more durable packaged hoist. A machine shop or fabricating shop will use an integral-package hoist, while a Steel Mill or NASA would use a unit to meet durability, performance. NASA has also seen a change in the use of package hoists, the NASA Astronaut training pool, for example, utilizes cranes with packaged hoists. The more commonly used hoist in todays worldwide market is an electrically powered hoist and these are either the chain type or the wire rope type. See the Hoists Manufacturers Institute site for true life calculation which is based on load, in todays modern world for the North American market there are a few governing bodies for the industry. The Overhead Alliance is a group that represents Crane Manufacturers Association of America, Hoist Manufacturers Institute and these product counsels of the Material Handling Industry of America have joined forces to create promotional materials to raise the awareness of the benefits to overhead lifting. The members of group are marketing representatives of the member companies. Common small portable hoists are of two types, the chain hoist or chain block and the wire rope or cable type. Chain hoists may have a lever to actuate the hoist or have a loop of operating chain that one pulls through the block which then activates the block to take up the main lifting chain. For a given rated load wire rope is lighter in weight per unit length, the lift chain of a chain hoist is far larger than the liftwheel over which chain may function
"Chain". Encyclopædia Britannica. 5 (11th ed.). 1911.
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