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| {{More footnotes|date=January 2009}}
| | It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner. <br><br> |
| [[File:Capstan equation diagram.svg|thumb|right|220px|]]
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| [[File:Example, no knowledge of capstan equation.png|thumb|An example of when knowledge of the capstan equation might have been useful. The bent white tube contains a cord to raise and lower a curtain. The tube is bent some 40 degrees in two places. The blue line indicates a better design.]]
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| The '''capstan equation''' or '''belt friction equation''', also known as '''Eytelwein's formula''',<ref>http://www.atp.ruhr-uni-bochum.de/rt1/currentcourse/node57.html</ref><ref>http://www.jrre.org/att_frict.pdf</ref> relates the hold-force to the load-force if a flexible line is wound around a cylinder (a [[bollard]], a [[winch]] or a [[Capstan (nautical)|capstan]])<ref name=Johnson>{{cite book | last = Johnson | first = K.L. | title = Contact Mechanics | url = http://www.knovel.com/web/portal/browse/display?_EXT_KNOVEL_DISPLAY_bookid=2685 | format = PDF | accessdate = February 14, 2011 | year = 1985}}</ref>
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| .<ref name=Attaway>{{cite conference | last = Attaway | first = Stephen W. | title = The Mechanics of Friction in Rope Rescue | url = http://www.jrre.org/att_frict.pdf | format = PDF | accessdate = February 1, 2010 | conference = International Technical Rescue Symposium | year = 1999}}</ref>
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| Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small ''holding'' force exerted on one side can carry a much larger ''loading'' force on the other side; this is the principle by which a capstan-type device operates. For instance in [[rock climbing]] with so-called [[top-roping]], a lighter person can hold (belay) a heavier person due to this effect.
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| The formula is:
| | As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. You or your web designer can customize it as per your specific needs. There are plenty of tables that are attached to this particular database. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably. However, you may not be able to find a theme that is in sync with your business. It supports backup scheduling and allows you to either download the backup file or email it to you. The Pakistani culture is in demand of a main surgical treatment. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. If you loved this report and you would like to obtain extra details relating to [http://cabkit.in/wordpressbackupplugin705551 wordpress backup plugin] kindly visit our own page. If this is not possible you still have the choice of the default theme that is Word - Press 3. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
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| :<math>T_\text{load} = T_\text{hold}\ e^{ \mu \phi} \,</math>
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| where <math>T_\text{load}</math> is the applied tension on the line, <math>T_\text{hold}</math> is the resulting force exerted at the other side of the capstan, <math>\mu</math> is the [[coefficient of friction]] between the rope and capstan materials, and <math>\phi</math> is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle <math>\phi =2\pi\,</math>).
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| Several assumptions must be true for the formula to be valid:
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| # The rope is on the verge of full sliding, i.e. <math>T_\text{load}</math> is the maximum load that one can hold. Smaller loads can be held as well, resulting in a smaller ''effective'' contact angle <math>\phi</math>.
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| # It is important that the line is not rigid, in which case significant force would be lost in the bending of the line tightly around the cylinder. (The equation must be modified for this case.) For instance a [[Bowden cable]] is to some extent rigid and doesn't obey the principles of the Capstan equation.
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| # The line is non-[[Elasticity (physics)|elastic]].
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| It can be observed that the force gain grows [[Exponential growth|exponentially]] with the coefficient of friction, the number of turns around the cylinder, and the angle of contact. Note that the radius of the cylinder has no influence on the force gain. The table below lists values of the factor <math>e^{ \mu \phi} \,</math> based on the number of turns and coefficient of friction ''μ''.
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| {| class="wikitable" style="text-align:right;"
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| ! rowspan="2" | Number<br/>of turns !! colspan="7" | Coefficient of friction ''μ''
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| ! scope="col" | 0.1
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| ! scope="col" | 0.2
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| ! scope="col" | 0.3
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| ! scope="col" | 0.4
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| ! scope="col" | 0.5
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| ! scope="col" | 0.6
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| ! scope="col" | 0.7
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| |-
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| ! scope="row" | 1
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| | 1.9 || 3.5 || 6.6 || 12 || 23 || 43 || 81
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| |-
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| ! scope="row" | 2
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| | 3.5 || 12 || 43 || 152 || 535 || {{gaps|1|881}} || {{gaps|6|661}}
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| |-
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| ! scope="row" | 3
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| | 6.6 || 43 || 286 || {{gaps|1|881}} || {{gaps|12|392}}|| {{gaps|81|612}}|| {{gaps|437|503}}
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| |-
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| ! scope="row" | 4
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| | 12 || 152 || {{gaps|1|881}} || {{gaps|23|228}} || {{gaps|286|751}} || {{gaps|3|540|026}} || {{gaps|43|702|631}}
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| |-
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| ! scope="row" | 5
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| | 23 || 535 || {{gaps|12|392}} || {{gaps|286|751}} || {{gaps|6|635|624}} || {{gaps|153|552|935}} || {{gaps|3|553|321|281}}
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| |}
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| From the table it is evident why one seldom sees a [[sheet (sailing)|sheet]] (a rope to the loose side of a sail) wound more than three turns around a winch. The force gain would be extreme besides being counter-productive since there is risk of a [[riding turn]], result being that the sheet will foul, form a knot and not run out when [[eased]] (by slacking grip on the [[tail]] (free end), or in land talk, one lets go of the hold end. It is both ancient and modern practice for anchor capstans and jib winches to be slightly flared out at the base, rather than cylindrical, to prevent the rope (anchor [[warp]] or sail sheet) from sliding down. The rope wound several times around the winch can slip upwards gradually, with little risk of a riding turn, provided it is [[tailed]] (loose end is pulled clear), by hand or a self-tailer.
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| For instance, the factor 153552935 means, in theory, that a newborn baby would be capable of holding the weight of two {{USS|Nimitz|CVN-68|6}} supercarriers (97 000 ton each, but for the baby it would be only a little more than 1 kg).
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| == Proof of the capstan equation ==
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| '''1. Circular coordinates'''
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| : <math>\varphi=\frac{s}{r}\qquad{d}\varphi=\frac{ds}{r}\qquad\frac{d\varphi}{ds}=\frac{1}{r}</math> (1), (2), (3)
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| <br /> | |
| [[File:Capstan eqn1.svg|right|thumb]]
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| Let <math>\mathbf{s}_{u}</math> and <math>\mathbf{n}_{u}</math> denote [[unit vectors]];
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| : <math>\mathbf{s}_{u}=-\sin\varphi\cdot{\mathbf{x}}_{u}+\cos\varphi\cdot{\mathbf{y}}_{u}</math> (4)
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| : <math>\mathbf{n}_{u}=\cos\varphi\cdot{\mathbf{x}}_{u}+\sin\varphi\cdot{\mathbf{y}}_{u}</math> (5)
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| Then from (5)
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| : <math>\frac{d}{d\varphi}{\mathbf{s}}_u = -\cos\varphi\cdot{\mathbf{x}}_u -\sin\varphi\cdot{\mathbf{y}}_u=-{\mathbf{n}}_u</math> (6)
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| : <math>\frac{d}{ds}=\frac{d}{d\varphi}\cdot\frac{d\varphi}{ds}=\frac{d}{d\varphi}\cdot\frac{1}{r}=\frac{1}{r}\cdot\frac{d}{d\varphi}</math> (7)
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| From (6) and (7), it follows that
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| : <math>\frac{d}{ds}{\mathbf{s}}_u=\frac{1}{r}\cdot\frac{d}{d\varphi}{\mathbf{s}}_u=-\frac{1}{r}\cdot{\mathbf{n}}_u.</math> '''(8)'''
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| '''2. Forces on cordage in general'''
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| Now, let's study a piece of cord in general, subject to an arbitrary force. Let <math>s\,</math> denote the length of the cord and let the force ''per unit length'' be <math>\mathbf{q}(s)</math>. Consider a short piece <math>\Delta{s}\,</math> of the cord and introduce the cross-sectional force <math>\mathbf{T}(s)\,</math>.
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| Balancing the forces, we get
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| : <math>\mathbf{T}(s+\Delta{s})-\mathbf{T}(s)+\mathbf{q}(s)\cdot\Delta{s}=0, </math> (9)
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| [[File:Capstan eqn2.png|right|thumb]]
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| : <math>\frac{\mathbf{T}(s+\Delta{s})-\mathbf{T}(s)}{\Delta{s}}=-\mathbf{q}(s). </math> (10) | |
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| Letting <math>\Delta{s}\rightarrow0</math>, we conclude that
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| : <math>\frac{d}{ds}\mathbf{T}(s)=-\mathbf{q}(s).</math> '''(11)'''
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| '''3. A line around a capstan'''
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| A line is wound around a cylinder(a bollard or a capstan). In this case the curvature of the line is circular which makes the problem easier. Let <math>s\,</math> be the length of the line from a point A where the line makes contact with the cylinder. At the point <math>s\,</math> on the short piece <math>\Delta{s}\,</math> of the line acts a force from the cylinder that can be subdivided into a ''tangential'' component <math>t\Delta{s}\,</math> (friction) and a ''normal'' component <math>n\Delta{s}\,</math>. That is to say that
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| : <math>\mathbf{q}(s)=t\cdot\mathbf{s}_u + n\cdot\mathbf{n}_u</math> (12)
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| [[File:Capstan eqn3.png|right|thumb]]
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| With the cross-sectional force <math>\mathbf{T}(s)\,</math> (which is tangential) we get
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| : <math>\mathbf{T}(s)=T(s)\cdot\mathbf{s}_u</math> (13)
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| From (11), (12) and (13), it follows that
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| : <math>\frac{d}{ds}T(s)\cdot\mathbf{s}_u = -t\cdot\mathbf{s}_u - n\cdot\mathbf{n}_u</math> '''(14)'''
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| Derivative of a product and (8) imply that
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| : <math>\frac{d}{ds}T(s)\cdot\mathbf{s}_{u}=\frac{dT(s)}{ds}\cdot\mathbf{s}_{u}+T(s)\cdot\frac{d}{ds}\mathbf{s}_u</math>
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| : <math>= \frac{dT(s)}{ds}\cdot\mathbf{s}_u-\frac{T(s)}{r}\cdot\mathbf{n}_u=-t\cdot\mathbf{s}_u-n\cdot\mathbf{n}_u</math> '''(15)'''
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| Identifying components in '''(15)''', we get
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| : <math>\frac{dT(s)}{ds}=-t</math> (16)
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| and
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| : <math>\frac{T(s)}{r}=n.</math> (17)
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| Dividing (16) by (17), we get
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| : <math>
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| \frac{dT(s)}{ds}/\frac{T(s)}{r}=-\frac{t}{n}</math> (18)
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| From (18) and reciprocal of (2), we get
| |
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| : <math>\mathrm{LHS}=\frac{dT(s)}{T(s)}\cdot\frac{r}{ds}=\frac{dT(s)}{T(s)}\cdot\frac{1}{d\varphi}=\frac{1}{T(S)}\cdot\frac{dT(s)}{d\varphi}.</math> (19)
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| From (18) and (19) it follows that
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| : <math>\frac{1}{T(s)}\cdot\frac{dT(s)}{d\varphi}=-\frac{t}{n}.</math> (20)
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| Let <math>\mu=\frac{t}{n}</math> (21) be the ''coefficient of friction'' (no slip). Then
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| : <math>\frac{1}{T}\cdot\frac{dT}{d\varphi}=-\mu</math> (22) : <math>\Rightarrow</math>
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| : <math>\frac{1}{T}\cdot{dT}=-\mu\cdot{d}\varphi</math> '''(23)'''
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| Integration of '''(23)''' yields
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| : <math>\int_{T_\text{load}}^{T_\text{hold}} \frac{1}{T} \cdot{dT} = \int_0^\phi -\mu \cdot {d}\varphi</math> (24)
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| : <math>\ln T_\text{hold} - \ln T_\text{load} = \ln\frac{T_\text{hold}}{T_\text{load}} = -\mu\cdot\phi</math> (25) <math>\Rightarrow</math>
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| : <math>\frac{T_\text{hold}}{T_\text{load}}={e}^{-\mu\cdot\phi}</math> (26)
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| Finally,
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| : <math>T_\text{hold}=T_\text{load}\cdot{e}^{-\mu\cdot\phi}\quad\text{ or } \quad T_\text{load} = T_\text{hold}\cdot{e}^{\mu\cdot\phi} </math>
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| ==See also==
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| *[[Belt friction]]
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| *[[Frictional contact mechanics]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * Arne Kihlberg, Kompendium i Mekanik för E1, del II, Göteborg 1980, 60–62.
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| {{DEFAULTSORT:Capstan Equation}}
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| [[Category:Concepts in physics]]
| |
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