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id="toc-Strings_in_modern_physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Strings_in_modern_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Strings in modern physics</span> </div> </a> <ul id="toc-Strings_in_modern_physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" 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href="https://bh.wikipedia.org/wiki/%E0%A4%A4%E0%A4%BE%E0%A4%A8_(%E0%A4%AD%E0%A4%93%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="तान (भओतिकी) – Bhojpuri" lang="bh" hreflang="bh" data-title="तान (भओतिकी)" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%86%CE%B5%CE%BB%CE%BA%CF%85%CF%83%CE%BC%CF%8C%CF%82" title="Εφελκυσμός – Greek" lang="el" hreflang="el" data-title="Εφελκυσμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tensi%C3%B3n_(mec%C3%A1nica)" title="Tensión (mecánica) – Spanish" lang="es" hreflang="es" data-title="Tensión (mecánica)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B4%D8%B4_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کشش (فیزیک) – Persian" lang="fa" hreflang="fa" data-title="کشش (فیزیک)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tension_m%C3%A9canique" title="Tension mécanique – French" lang="fr" hreflang="fr" data-title="Tension mécanique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%A5%EB%A0%A5" title="장력 – Korean" lang="ko" hreflang="ko" data-title="장력" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A4%A8%E0%A4%BE%E0%A4%B5_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="तनाव (भौतिकी) – Hindi" lang="hi" hreflang="hi" data-title="तनाव (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Mekanikala_tenso" title="Mekanikala tenso – Ido" lang="io" hreflang="io" data-title="Mekanikala tenso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tarikan_(fisika)" title="Tarikan (fisika) – Indonesian" lang="id" hreflang="id" data-title="Tarikan (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tensione_(meccanica)" title="Tensione (meccanica) – Italian" lang="it" hreflang="it" data-title="Tensione (meccanica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%99%D7%97%D7%95%D7%AA" title="מתיחות – Hebrew" lang="he" hreflang="he" data-title="מתיחות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%8E%E0%B2%B3%E0%B3%86%E0%B2%A4" title="ಎಳೆತ – Kannada" lang="kn" hreflang="kn" data-title="ಎಳೆತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%B0%D0%BB%D2%93%D0%B0%D1%8E_%D0%BC%D0%B5%D0%BD_%D1%81%D1%8B%D2%93%D1%8B%D0%BB%D1%83" title="Ұлғаю мен сығылу – Kazakh" lang="kk" hreflang="kk" data-title="Ұлғаю мен сығылу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Tansyon" title="Tansyon – Haitian Creole" lang="ht" hreflang="ht" data-title="Tansyon" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B4%A8%E0%B4%A8_%E0%B4%AC%E0%B4%B2%E0%B4%82" title="തനന ബലം – Malayalam" lang="ml" hreflang="ml" data-title="തനന ബലം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BC%B5%E5%8A%9B" title="張力 – Japanese" lang="ja" hreflang="ja" data-title="張力" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Si%C5%82a_naci%C4%85gu" title="Siła naciągu – Polish" lang="pl" hreflang="pl" data-title="Siła naciągu" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tens%C3%A3o_(f%C3%ADsica)" title="Tensão (física) – Portuguese" lang="pt" hreflang="pt" data-title="Tensão (física)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/For%C8%9B%C4%83_de_%C3%AEntindere" title="Forță de întindere – Romanian" lang="ro" hreflang="ro" data-title="Forță de întindere" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%B0_%D0%BD%D0%B0%D1%82%D1%8F%D0%B6%D0%B5%D0%BD%D0%B8%D1%8F" title="Сила натяжения – Russian" lang="ru" hreflang="ru" data-title="Сила натяжения" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Forca_e_tensioni_(fizik%C3%AB)" title="Forca e tensioni (fizikë) – Albanian" lang="sq" hreflang="sq" data-title="Forca e tensioni (fizikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tension_(mechanics)" title="Tension (mechanics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tension (mechanics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Mekanisk_sp%C3%A4nning" title="Mekanisk spänning – Swedish" lang="sv" hreflang="sv" data-title="Mekanisk spänning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D1%82%D1%8F%D0%B3" title="Розтяг – Ukrainian" lang="uk" hreflang="uk" data-title="Розтяг" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BC%B5%E5%8A%9B" title="張力 – Cantonese" lang="yue" hreflang="yue" data-title="張力" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BC%B5%E5%8A%9B" title="張力 – Chinese" lang="zh" hreflang="zh" data-title="張力" 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Tension_(mechanics)&redirect=no" class="mw-redirect" title="Tension (mechanics)">Tension (mechanics)</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Pulling force transmitted axially – opposite of compression</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress (mechanics)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tug_Of_War_Tension.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tug_Of_War_Tension.png/300px-Tug_Of_War_Tension.png" decoding="async" width="300" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tug_Of_War_Tension.png/450px-Tug_Of_War_Tension.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/83/Tug_Of_War_Tension.png 2x" data-file-width="582" data-file-height="276" /></a><figcaption>Nine men pull on a rope. The rope in the photo extends into a drawn illustration showing adjacent segments of the rope. <br /><br /> One segment is duplicated in a free body diagram showing a pair of action-reaction forces of magnitude T pulling the segment in opposite directions, where <b>T</b> is transmitted axially and is called the tension force. This end of the rope is pulling the <a href="/wiki/Tug_of_war" title="Tug of war">tug of war</a> team to the right. <br /><br /> Each segment of the rope is pulled by the two neighboring segments, stressing the segment in what is also called tension.</figcaption></figure> <p><b>Tension</b> is the pulling or stretching <a href="/wiki/Force" title="Force">force</a> transmitted axially along an object such as a string, rope, chain, rod, <a href="/wiki/Truss" title="Truss">truss</a> member, or other object, so as to stretch or pull apart the object. In terms of force, it is the opposite of <a href="/wiki/Compression_(physics)" title="Compression (physics)"><i>compression</i></a>. Tension might also be described as the action-reaction pair of forces acting at each end of an object. </p><p>At the atomic level, when atoms or molecules are pulled apart from each other and gain <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> with a <a href="/wiki/Restoring_force" title="Restoring force">restoring force</a> still existing, the restoring force might create what is also called tension. Each end of a string or rod under such tension could pull on the object it is attached to, in order to restore the string/rod to its relaxed length. </p><p>Tension (as a transmitted force, as an action-reaction pair of forces, or as a restoring force) is measured in <a href="/wiki/Newton_(unit)" title="Newton (unit)">newtons</a> in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (or <a href="/wiki/Pounds-force" class="mw-redirect" title="Pounds-force">pounds-force</a> in <a href="/wiki/Imperial_units" title="Imperial units">Imperial units</a>). The ends of a string or other object transmitting tension will exert forces on the objects to which the string or rod is connected, in the direction of the string at the point of attachment. These forces due to tension are also called "passive forces". There are two basic possibilities for systems of objects held by strings:<sup id="cite_ref-Physics_1-0" class="reference"><a href="#cite_note-Physics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> either <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> is zero and the system is therefore in equilibrium, or there is acceleration, and therefore a <a href="/wiki/Net_force" title="Net force">net force</a> is present in the system. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tension_in_one_dimension">Tension in one dimension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tension_(physics)&action=edit&section=1" title="Edit section: Tension in one dimension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tension_figure.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Tension_figure.svg/300px-Tension_figure.svg.png" decoding="async" width="300" height="353" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Tension_figure.svg/450px-Tension_figure.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Tension_figure.svg/600px-Tension_figure.svg.png 2x" data-file-width="331" data-file-height="389" /></a><figcaption>The tension in a tetherball rope</figcaption></figure> <p>Tension in a string is a non-negative <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">vector quantity</a>. Zero tension is slack. A string or rope is often idealized as one dimension, having fixed length but being massless with zero <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross section</a>. If there are no bends in the string, as occur with <a href="/wiki/Vibration" title="Vibration">vibrations</a> or <a href="/wiki/Pulley" title="Pulley">pulleys</a>, then tension is a constant along the string, equal to the magnitude of the forces applied by the ends of the string. By <a href="/wiki/Newton%27s_third_law" class="mw-redirect" title="Newton's third law">Newton's third law</a>, 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 will still have constant tension along its length in the idealized situation that the pulleys are <a href="/wiki/Mass" title="Mass">massless</a> and <a href="/wiki/Friction" title="Friction">frictionless</a>. A <a href="/wiki/Vibrating_string" class="mw-redirect" title="Vibrating string">vibrating string</a> vibrates with a set of <a href="/wiki/Frequencies" class="mw-redirect" title="Frequencies">frequencies</a> that depend on the string's tension. These frequencies can be derived from <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>. Each microscopic segment of the string pulls on and is pulled upon by its neighboring segments, with a force equal to the tension at that position along the string. </p><p>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 <a href="/wiki/Net_force" title="Net force">net force</a> on that segment of the string, causing an acceleration. This net force is a <a href="/wiki/Restoring_force" title="Restoring force">restoring force</a>, and the motion of the string can include <a href="/wiki/Transverse_wave" title="Transverse wave">transverse waves</a> that solve the equation central to <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg [}\tau (x){\frac {\mathrm {d} \rho (x)}{\mathrm {d} x}}{\bigg ]}+v(x)\rho (x)=\omega ^{2}\sigma (x)\rho (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>+</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg [}\tau (x){\frac {\mathrm {d} \rho (x)}{\mathrm {d} x}}{\bigg ]}+v(x)\rho (x)=\omega ^{2}\sigma (x)\rho (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792dab0d9855f0f76d4bfa08aa1c7feafe21672a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.388ex; height:6.343ex;" alt="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg [}\tau (x){\frac {\mathrm {d} \rho (x)}{\mathrm {d} x}}{\bigg ]}+v(x)\rho (x)=\omega ^{2}\sigma (x)\rho (x)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b371a381e15c71d8fc4ec43cf14b156f02a0d35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.267ex; height:2.843ex;" alt="{\displaystyle v(x)}"></span> is the force constant per unit length [units force per area], <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae09ff47b50183fbfd1ea5697c63963ec9eefa20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.469ex; height:2.843ex;" alt="{\displaystyle \sigma (x)}"></span> is the ...., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27315ea6ad300a4cc07c8b03397ce6aac0cd944c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.341ex; height:2.843ex;" alt="{\displaystyle \tau (x)}"></span> is the ...., and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc60ab391d9835017f0778767fb25a54402d20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle \omega ^{2}}"></span> are the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> for resonances of transverse displacement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32a26d8795457b2f5c2bdc078758dcbbc71b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.341ex; height:2.843ex;" alt="{\displaystyle \rho (x)}"></span> on the string,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> with solutions that include the various <a href="/wiki/Scale_of_harmonics" title="Scale of harmonics">harmonics</a> on a <a href="/wiki/Stringed_instrument" class="mw-redirect" title="Stringed instrument">stringed instrument</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Tension_of_three_dimensions">Tension of three dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tension_(physics)&action=edit&section=2" title="Edit section: Tension of three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tension is also used to describe the force exerted by the ends of a three-dimensional, continuous material such as a rod or <a href="/wiki/Truss" title="Truss">truss</a> member. In this context, tension is analogous to <a href="/wiki/Pressure#Negative_pressures" title="Pressure">negative pressure</a>. A rod under tension <a href="/wiki/Elongation_(mechanics)" class="mw-redirect" title="Elongation (mechanics)">elongates</a>. The amount of elongation and the <a href="/wiki/Structural_load" title="Structural load">load</a> that will cause failure both depend on the force per cross-sectional area rather than the force alone, so <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> = axial force / cross sectional area is more useful for engineering purposes than tension. Stress is a 3x3 matrix called a <a href="/wiki/Tensor" title="Tensor">tensor</a>, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{11}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9010ce404b2240a94a8b55e1f8f90f5f6c6080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.204ex; height:2.009ex;" alt="{\displaystyle \sigma _{11}}"></span> element of the stress tensor is tensile force per area, or compression force per area, denoted as a negative number for this element, if the rod is being compressed rather than elongated. </p><p>Thus, one can obtain a scalar analogous to tension by taking the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of the stress tensor.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="System_in_equilibrium">System in equilibrium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tension_(physics)&action=edit&section=3" title="Edit section: System in equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A system is in equilibrium when the sum of all forces is zero.<sup id="cite_ref-Physics_1-1" class="reference"><a href="#cite_note-Physics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {\vec {F}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\vec {F}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed5bd908393dee70d8a0ef6aa1b482d663522f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.774ex; height:3.843ex;" alt="{\displaystyle \sum {\vec {F}}=0}"></span> </p><p>For example, consider a system consisting of an object that is being lowered vertically by a string with tension, <i>T</i>, at a constant <a href="/wiki/Velocity" title="Velocity">velocity</a>. The system has a constant velocity and is therefore in equilibrium because the tension in the string, which is pulling up on the object, is equal to the <a href="/wiki/Weight" title="Weight">weight</a> <a href="/wiki/Force" title="Force">force</a>, mg ("m" is mass, "g" is the acceleration caused by the <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">gravity of Earth</a>), which is pulling down on the object.<sup id="cite_ref-Physics_1-2" class="reference"><a href="#cite_note-Physics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {\vec {F}}={\vec {T}}+m{\vec {g}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\vec {F}}={\vec {T}}+m{\vec {g}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721b37779525c203642ba26555a98304838ca2ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.667ex; height:3.843ex;" alt="{\displaystyle \sum {\vec {F}}={\vec {T}}+m{\vec {g}}=0}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="System_under_net_force">System under net force</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tension_(physics)&action=edit&section=4" title="Edit section: System under net force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A system has a net force when an unbalanced force is exerted on it, in other words the sum of all forces is not zero. Acceleration and net force always exist together.<sup id="cite_ref-Physics_1-3" class="reference"><a href="#cite_note-Physics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {\vec {F}}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\vec {F}}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8cff2d59e2efc9b5165d94c3f09494ff998b18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.774ex; height:3.843ex;" alt="{\displaystyle \sum {\vec {F}}\neq 0}"></span> </p><p>For example, consider the same system as above but suppose the object is now being lowered with an increasing velocity downwards (positive acceleration) therefore there exists a net force somewhere in the system. In this case, negative acceleration would indicate that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |mg|>|T|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |mg|>|T|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df82e0a7f4547b67be6545f851161f92a001bda1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.478ex; height:2.843ex;" alt="{\displaystyle |mg|>|T|}"></span>.<sup id="cite_ref-Physics_1-4" class="reference"><a href="#cite_note-Physics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {\vec {F}}={\vec {T}}-m{\vec {g}}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\vec {F}}={\vec {T}}-m{\vec {g}}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d623c743a1618aeb5fe57d58ebef14239793f45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.667ex; height:3.843ex;" alt="{\displaystyle \sum {\vec {F}}={\vec {T}}-m{\vec {g}}\neq 0}"></span> </p><p>In another example, suppose that two bodies A and B having masses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecebe334d5cadc3ffcf245eb02919034d7a2ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}"></span>, respectively, are connected with each other by an inextensible string over a frictionless pulley. There are two forces acting on the body A: its weight (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}=m_{1}g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1}=m_{1}g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279e9b41c452c07848d8e5e04151c680a6528660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.027ex; height:2.009ex;" alt="{\displaystyle w_{1}=m_{1}g}"></span>) pulling down, and the tension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> in the string pulling up. Therefore, the net force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c7fbf174fe8b06eacc2a6b0bb2e1badd1c7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{1}}"></span> on body A is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1}-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85414bf80b9308627aaa2305ab207babbb3626b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.195ex; height:2.509ex;" alt="{\displaystyle w_{1}-T}"></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}a=m_{1}g-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>a</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>g</mi> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}a=m_{1}g-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99def74cf24e2450549d2700f320e75fd677a883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.11ex; height:2.509ex;" alt="{\displaystyle m_{1}a=m_{1}g-T}"></span>. In an extensible string, <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> applies. </p> <div class="mw-heading mw-heading2"><h2 id="Strings_in_modern_physics">Strings in modern physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tension_(physics)&action=edit&section=5" title="Edit section: Strings in modern physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>String-like objects in <a href="/wiki/Special_relativity" title="Special relativity">relativistic</a> theories, such as the <a href="/wiki/QCD_string" class="mw-redirect" title="QCD string">strings</a> used in some models of interactions between <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a>, or those used in the modern <a href="/wiki/String_theory" title="String theory">string theory</a>, also possess tension. These strings are analyzed in terms of their <a href="/wiki/World_sheet" class="mw-redirect" title="World sheet">world sheet</a>, and the <a href="/wiki/Energy" title="Energy">energy</a> is then typically proportional to the length of the string. 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Seventh Edition, Brooks/Cole Cengage Learning, 2008.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">A. Fetter and J. Walecka. (1980). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=n54oAwAAQBAJ&q=Tension">Theoretical Mechanics of Particles and Continua</a>. New York: McGraw-Hill.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Jayachandran, Arul. <i><a rel="nofollow" class="external text" href="https://www.docsity.com/en/nxnmsxmnzxcmnhzcnhzcxnhczxjhzcxjhzmsnnmnmsznmz/5331958/">Design of Tension Members: Mechanical Properties and Block Shear Failure, Exercises of Civil Engineering</a></i> April 9, 2014. <a href="/wiki/Illinois_Institute_of_Technology" title="Illinois Institute of Technology">Illinois Institute of Technology</a></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline 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