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# elastic deformation equation

From Wikipedia

Rubber band

A rubber band (in some regions known as a binder, an elastic or elastic band, a lackey band, laggy band, lacka band or gumband) is a short length of rubber and latex formed in the shape of a loop. Rubber bands are typically used to hold multiple objects together. The rubber band was patented in England on March 17, 1845 by Stephen Perry.

## Manufacturing

Rubber bands are made by extruding the rubber into a long tube to provide its general shape, putting the tubes on mandrels and curing the rubber with heat, and then slicing it across the width of the tube into little bands.

## Material

While other rubber products may use synthetic rubber, rubber bands are primarily manufactured using natural rubber because of its superior elasticity.

Natural rubber originates from the sap of the rubber tree. Natural rubber is made from latex which is acquired by tapping into the bark layers of the rubber tree. Rubber trees belong to the spurge family (Euphorbiaceae) and live in warm, tropical areas. Once the latex has been â€œtappedâ€� and is exposed to the air it begins to harden and become elastic, or â€œrubbery.â€� Rubber trees only survive in hot, humid climates near the equator and so the majority of latex is produced in the Southeast Asian countries of Malaysia, Thailand and Indonesia.

## Rubber Band Sizes

### Measuring

A rubber band has three basic dimensions: length, width, and thickness. (See picture.)

A rubber band's length is half its circumference. Its thickness is the distance from the inner circle to the outer circle.

If one imagines a rubber band in manufacture, that is, a long tube of rubber on a mandrel, before it is sliced into rubber bands, the band's width is how far apart the slices are cut.

### Rubber Band Size Numbers

A rubber band is given a [quasi-]standard number based on its dimensions.

Generally, rubber bands are numbered from smallest to largest, width first. Thus, rubber bands numbered 8-19 are all 1/16 inch wide, with length going from 7/8&nbsp;inch to 3 1/2&nbsp;inches. Rubber band numbers 30-34 are for width of 1/8&nbsp;inch, going again from shorter to longer. For even longer bands, the numbering starts over for numbers above 100, again starting at width 1/16&nbsp;inch.

The origin of these size numbers is not clear and there appears to be some conflict in the "standard" numbers. For example, one distributor has a size 117 being 1/16&nbsp;inch wide and a size 127 being 1/8&nbsp;inch wide. However, an OfficeMax size 117 is 1/8&nbsp;inch wide. A manufacturer has a size 117A (1/16&nbsp;inch wide) and a 117B (1/8&nbsp;inch wide). Another distributor calls them 7AA (1/16&nbsp;inch wide) and 7A (1/8&nbsp;inch wide) (but labels them as specialty bands).

## Thermodynamics

Temperature affects the elasticity of a rubber band in an unusual way. Heating causes the rubber band to contract, and cooling causes expansion.

An interesting effect of rubber bands in thermodynamics is that stretching a rubber band will produce heat (press it against your lips), while stretching it and then releasing it will lead it to absorb heat, causing its surroundings to become cooler. This phenomenon can be explained with Gibb's Free Energy. Rearranging Î”G=Î”H-TÎ”S, where G is the free energy, H is the enthalpy, and S is the entropy, we get TÎ”S=Î”H-Î”G. Since stretching is nonspontaneous, as it requires an external heat, TÎ”S must be negative. Since T is always positive (it can never reach absolute zero), the Î”S must be negative, implying that the rubber in its natural state is more entangled (fewer microstates) than when it is under tension. Thus, when the tension is removed, the reaction is spontaneous, leading Î”G to be negative. Consequently, the cooling effect must result in a positive Î”G, so Î”S will be positive there.

## Red rubber bands

In 2004 in the UK, following complaints from the public about postal carriers causing litter by discarding the rubber bands which they used to keep their mail together, the Royal Mail introduced red bands for their workers to use: it was hoped that, as the bands were easier to spot than the traditional brown ones and since only the Royal Mail used them, employees would see (and feel compelled to pick up) any red bands which they had inadvertently dropped. Currently, some 342 million red bands are used every year.

## Model use

Rubber bands have long been one of the methods of powering small free-flight model aeroplanes, the rubber band being anchored at the rear of the fuselage and connected to the propeller at the front. To 'wind up' the 'engine' the propeller is repeatedly turned, twisting the rubber band. When the propeller has had enough turns, the propeller is released and the model launched, the rubber band then turning the propeller rapidly until it has unwound.

One of the earliest to use this method was pioneer aerodynamicistGeorge Cayley, who used them for powering his small experimental models. These 'rubber motors' have also been used for powering small model boats.

Geodesic

In mathematics, a geodesic (ËŒdÊ’iË�ÉµËˆdiË�zÉ¨k, ËŒdÊ’iË�ÉµËˆdÉ›sÉ¨k| , ) is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a metric, geodesics are defined to be (locally) the shortest path between points in the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it.

The term "geodesic" comes from geodesy, the science of measuring the size and shape ofEarth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.

Geodesics are of particular importance in general relativity, as they describe the motion of inertial test particles.

## Introduction

The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parametrize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parametrized "with constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional to |s&minus;t|. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimisation). Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy; the resulting shape of the band is a geodesic.

In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parametrized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map tâ†’t2 from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved space-time. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds. The article geodesic (general relativity) discusses the special case of general relativity in greater detail.

### Examples

The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points (like the North pole and the South pole), then there are infinitely many shortest paths between them.

## Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curveÎ³: Iâ†’ M from an interval I of the reals to the metric spaceM is a geodesic if there is a constantvâ‰¥ 0 such that for any tâˆˆ I there is a neighborhood J of t in I such that for any t1, t2âˆˆ J we have

d(\gamma(t_1),\gamma(t_2))=v|t_1-t_2|.\,

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parametrization, i.e. in the above identity v = 1 and

d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|.\,

Question:I already know the equation for a non-deforming CV. Could you give me the equation for the deformable control volume?

Answers:There isn't one. A control volume, by definition, is a region of space which remains constant in volume and is used to track energy flow in to and out of the region. You will need to define your own version of the first law of thermodynamics if you wish for it to be for a deformable region of space. Remember: the first law of thermodynamics is nothing more than conservation of energy. How do forms of energy exist in you system of interest?

Question:I'm completing a comprehension question and I'm a bit confused. We're relating the elasticity of a bouncy ball polymer (assuming it follows hooks law) to it's spring constant/force constant. Short question: Does a higher k mean more or less elastic than a small k? Long version of the question: F = -k*x and v (frequency) = 1/2pi(k/m)^1/2 I'm awfully confused about what exactly "elasticity" means. Is it the ease with which an object stretches for a given force (and still returns to its original shape) or the ease with which is returns to its original shape for a given force? A higher k value means that it will return to its shape faster, and in my mind that seems like it's more elastic. Higher force constants in my elastomer bouncy ball result in a more elastic collision between the floor and the bouncy ball and a higher bounce (energy is more efficiently converted from elastic potential to kinetic energy, instead of propagating as heat throughout a distorting ball) Higher force constant also means that there will be a higher frequency of oscillation. Yet one line of my text read " Higher spring constant means the material is less elastic...higher frequency of oscillation." This doesn't make sense to me, doesn't a higher frequency mean its capable of returning to its shape more readily for a given amount of force applied?

Answers:I do not like the adjective "elastic"...except when just stating that it relates to the force types of elasitcity. It is very unclear whether "elastic" as a quantifying adjective means hard to deform or easy to deform. Often times, science uses it to mean hard to deform, but common usage uses it to mean easy to deform. SO CONFUSING. From now on, elasticity means nothing more than a classification of forces and a subject of study. It is like electricity, electricity is not "what you fill a battery with" or what you get from the socket...electricity is just a subject in physics. Same with elasticity...just a name for subject in physics. Elastic is a good adjective for force...as in the force in the cord is an elastic force, or as in elastic forces hold a body's shape together when gravity is insignificant. ------------------------- So for that reason, use the words stiff and flexible to indicate what you really mean. Stiff means hard to deform. More force or stress required per unit deformation distance or strain. Flexible means easy to deform. More deformation distance or strain results per unit force or stress. The spring constant (the k-value) indicates how stiff a spring is. You can call the k-value the spring stiffness if you want. The plural of stiffness in this context is "measures of stiffness". There are numerous examples of measures of stiffness...both at the structural member level (spring constant, torsional constant), and at the individual material fiber level (Young's modulus, Shear modulus, Bulk modulus). It IS TRUE that higher k-value means a higher frequency. That you can always count upon. Mass (inertia) will make the vibration slower.

Question:A car moving at speed v undergoes a one-dimensional collision with an identical car initially at rest. The collision is neither elastic nor fully inelastic; 2/17 of the initial kinetic energy is lost. Find the velocities of the two cars after the collision. Express your answer in units of v Cannot figure out how to get started. Any help is appreciated.

Answers:Let equal mass of both cars = m Initial velocity of 2nd car, u = 0 Let v' and u' be the final velocities of the 1st and 2nd car respectively. Momentum is always conserved. So, mv + 0 = mv' + mu' => v = v' + u' ... ( 1 ) 2/17 of initial K.E. is lost => 15/17 of initial K.E. = Final K.E. => (15/17)(1/2)mv^2 = (1/2)mv'^2 + (1/2)mu'^2 => (15/17) v^2 = v'^2 + u'^2 ... ( 2 ) From equations ( 1 ) and ( 2 ), (v' + u')^2 - (v'^2 + u'^2) = v^2 - (15/17)v^2 => 2v'u' = (2/17) v^2 => (v' - u')^2 = (v' + u')^2 - 4v'u' = v^2 - (4/17)v^2 = (13/17)v^2 => v' - u' = v (13/17) = 0.874 v ... ( 3 ) Solving equations ( 1 ) and ( 3 ), v' = 0.987 v and u' = 0.063v You can refer to my free educational website www.schoolnotes4u.com and download study materials without requiring any form of registration.

Question:what are the dimensions of Xnth in the standard equation Xnth = u + a/2 (2n-1) where u is intial velocity ,a is uniform accleration and n is time? what are the 2 essential conditions in each isothermal and adiabatic process to take place? thank you

Answers:For a perfectly rigid body, the strain produced is zero, no matter how much the stress, so the Young's modulus of such a body is infinite. THe dimensions should be that of distance as this is the equation for the distance covered in the nth second. In isothermal process, the temperature should remain constant. In adiabatic proces, there should be no exchange of temperature between the system and the surroundings. The temperature of the system might increase but it should not be distributed to the surroundings. SO the enclosure of such a system should be perfect insulators of heat.