{"id":402,"date":"2026-07-03T02:37:26","date_gmt":"2026-07-03T02:37:26","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=402"},"modified":"2026-08-24T13:04:16","modified_gmt":"2026-08-24T13:04:16","slug":"thermal-expansion","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/thermodynamics\/thermal-expansion\/","title":{"rendered":"Thermal Expansion Explained"},"content":{"rendered":"\n<div class=\"pf-citation\"><div class=\"eyebrow\">Definition<\/div><p>\n\nThermal expansion is the increase in a material&#8217;s length, area or volume as its temperature rises, because its atoms vibrate more vigorously and their average spacing grows. For solids, the change in length follows \u0394L = \u03b1L<sub>0<\/sub>\u0394T, where \u03b1 is the coefficient of linear expansion. Cooling reverses the effect, causing thermal contraction.\n\n<\/p><\/div>\n\n<p>On a blazing July afternoon, the Eiffel Tower stands taller than it did on New Year&#8217;s Day \u2014 by roughly the height of a coffee mug. Nobody added any iron. The Sun did all the work.<\/p>\n\n<p>That quiet growing and shrinking is happening all around you: in railway tracks, power lines, jam-jar lids and the ocean itself. This guide unpacks the physics, hands you the one-line formula engineers actually use, and lets you test it in an interactive lab.<\/p>\n\n<h2>What Is Thermal Expansion?<\/h2>\n\n<p>Picture a crowd standing still on a dance floor. Everyone fits comfortably. Now turn the music up: the same people, swaying and jumping, suddenly need far more room, and the crowd spreads outward.<\/p>\n\n<p>Atoms in a solid behave the same way. Raise the temperature and they jiggle harder about their fixed positions, nudging their neighbours a little further away. Multiply that tiny extra elbow room by trillions of atomic layers and the whole object measurably grows.<\/p>\n\n<p>More precisely, thermal expansion is the tendency of matter to change its dimensions in response to a change in temperature \u2014 expanding when heated and, for almost every material, contracting when cooled. Note that it is <em>temperature<\/em> doing the driving, not heat directly; if that distinction feels slippery, our guide to <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/thermodynamics\/heat-vs-temperature\/\">heat vs temperature<\/a> untangles it.<\/p>\n\n<p>Solids expand the least, liquids noticeably more, and gases most of all. The effect is small per degree \u2014 but it is relentless, and it ignores whatever stands in its way.<\/p>\n\n<h2>The Thermal Expansion Formula<\/h2>\n\n<p>For a rod, wire, rail or beam, the working equation is a single line:<\/p>\n\n<div class=\"pf-formula\">\u0394L = \u03b1L<sub>0<\/sub>\u0394T<\/div>\n\n<ul>\n<li><strong>\u0394L<\/strong> \u2014 change in length, in metres (m)<\/li>\n<li><strong>\u03b1<\/strong> \u2014 coefficient of linear expansion, in per degree Celsius (\u00b0C<sup>\u22121<\/sup>), identical to per kelvin (K<sup>\u22121<\/sup>) because only temperature <em>differences<\/em> enter<\/li>\n<li><strong>L<sub>0<\/sub><\/strong> \u2014 original length, in metres (m)<\/li>\n<li><strong>\u0394T<\/strong> \u2014 temperature change, in \u00b0C or K<\/li>\n<\/ul>\n\n<p>A quick taste of the numbers: warm a 2.00 m aluminium rod (\u03b1 = 23 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>) by 50 \u00b0C and it grows by \u0394L = 23 \u00d7 10<sup>\u22126<\/sup> \u00d7 2.00 \u00d7 50 = 2.3 mm. The final length is simply L = L<sub>0<\/sub> + \u0394L = L<sub>0<\/sub>(1 + \u03b1\u0394T). You can check any combination in seconds with our <a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/thermal-expansion\">Thermal Expansion Calculator<\/a>.<\/p>\n\n<h3>Area and Volume Expansion<\/h3>\n\n<p>Because every dimension grows at once, surfaces and volumes have their own versions of the law:<\/p>\n\n<div class=\"pf-formula\">\u0394A = 2\u03b1A<sub>0<\/sub>\u0394T<\/div>\n\n<div class=\"pf-formula\">\u0394V = \u03b2V<sub>0<\/sub>\u0394T,   where \u03b2 \u2248 3\u03b1 for solids<\/div>\n\n<p>Why 3\u03b1? Each of the three dimensions stretches by a factor (1 + \u03b1\u0394T), and cubing that gives 1 + 3\u03b1\u0394T to an excellent approximation when \u03b1\u0394T is tiny \u2014 which it always is in everyday conditions.<\/p>\n\n<p>Liquids have no shape of their own, so only \u03b2 is quoted: about 2.1 \u00d7 10<sup>\u22124<\/sup> \u00b0C<sup>\u22121<\/sup> for water at room temperature and roughly 9.6 \u00d7 10<sup>\u22124<\/sup> \u00b0C<sup>\u22121<\/sup> for petrol \u2014 ten to fifty times a typical solid. Gases outdo everything: at constant pressure an ideal gas expands in proportion to its absolute temperature, an effective \u03b2 of about 3.7 \u00d7 10<sup>\u22123<\/sup> K<sup>\u22121<\/sup> near 0 \u00b0C.<\/p>\n\n<p>One honest caveat. \u03b1 itself drifts slowly with temperature, so \u0394L = \u03b1L<sub>0<\/sub>\u0394T is a superb approximation for moderate swings \u2014 tens of degrees around everyday conditions \u2014 but engineers reach for tabulated data at cryogenic or furnace extremes.<\/p>\n\n<p>Reading about expansion is one thing; watching it is better. Pick a material, drag the temperature and watch the rod grow:<\/p>\n\n<div class=\"pf-sim-slot\"><div class=\"pf-sim-slot-header\"><span class=\"icon-dot\"><\/span><span class=\"label\">Thermal Expansion Lab<\/span><\/div><div class=\"pf-sim-slot-body\"><style>.pf-sim-frame{width:100%;border:none;height:600px}@media(max-width:760px){.pf-sim-frame{height:1000px}}<\/style><iframe src=\"\/labs\/thermal-expansion.html?embed=1\" class=\"pf-sim-frame\" loading=\"lazy\"><\/iframe><\/div><\/div>\n\n<h2>How Does Thermal Expansion Work?<\/h2>\n\n<p>Here is the puzzle. If heating simply made atoms vibrate harder, nothing should grow \u2014 a vibration swings as far inward as outward, and the average position would stay put.<\/p>\n\n<p>Neighbouring atoms behave like masses joined by a spring, vibrating about an equilibrium spacing in near-<a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/simple-harmonic-motion\/\">simple harmonic motion<\/a>. If the bond were a perfect spring obeying <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/hookes-law\/\">Hooke&#8217;s law<\/a>, hotter would just mean wider swings around the <em>same<\/em> average \u2014 and no expansion at all.<\/p>\n\n<p>Real bonds are lopsided. Squeeze two atoms together and they resist ferociously; pull them apart and the attraction gives way far more gently. So an energetic atom swings further into the roomy outward side than into the cramped inward side, and its <em>average<\/em> separation drifts outward. That drift, a fraction of a picometre per bond, is thermal expansion.<\/p>\n\n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/thermal-expansion-asymmetric-interatomic-potential-energy-curve.webp\" width=\"1440\" height=\"1016\" alt=\"Asymmetric interatomic potential energy curve: as temperature rises, atoms vibrate between wider limits and the midpoint of each vibration shifts to larger separations, producing thermal expansion\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:720px;display:block;margin:0 auto;\" \/><\/figure>\n<p style=\"text-align:center;font-size:14px;font-style:italic;color:#142139;\">The lopsided energy valley between neighbouring atoms. Hotter atoms (higher gold levels) swing between wider limits, and the midpoint of each swing (dashed line) drifts to the right \u2014 the material expands.<\/p>\n\n<p>The same picture explains the pecking order. Liquids hold their molecules with weaker, floppier bonds, so they expand more than solids. Gas molecules barely interact at all, so gases expand most of all.<\/p>\n\n<p>And a delicious footnote: a handful of engineered materials, such as zirconium tungstate, actually <em>shrink<\/em> as they warm across a huge temperature range, because heating twists their lattice units inward. Physics keeps exceptions on hand to keep everyone humble.<\/p>\n\n<h2>Thermal Expansion Coefficients of Common Materials<\/h2>\n\n<p>The coefficient \u03b1 is a material&#8217;s personality: how eagerly it responds to a degree of warming. Here are typical values near room temperature.<\/p>\n\n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr>\n<th style=\"text-align:left;padding:10px 12px;border-bottom:2px solid #C8932A;\">Material<\/th>\n<th style=\"text-align:left;padding:10px 12px;border-bottom:2px solid #C8932A;\">\u03b1 (\u00d710<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>)<\/th>\n<th style=\"text-align:left;padding:10px 12px;border-bottom:2px solid #C8932A;\">Worth knowing<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Fused quartz<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">0.5<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Lab glassware that shrugs off thermal shock<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Invar (Fe\u2013Ni alloy)<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">\u22481.2<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Engineered <em>not<\/em> to move \u2014 precision instruments, clock pendulums<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Pyrex (borosilicate glass)<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">3.3<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Why oven dishes survive temperature jumps ordinary glass cannot<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Ordinary glass<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">\u22489<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Boiling water in a cold tumbler can crack it via uneven expansion<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Steel \/ iron<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">11\u201313<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">The workhorse value behind rails, bridges and rebar<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Concrete<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">\u224812<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Matches steel almost exactly \u2014 the quiet reason reinforced concrete works<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Copper<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">17<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Hot-water pipework needs room to creak and move<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Brass<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">19<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Paired with steel in bimetallic strips<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Aluminium<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">23<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Roughly double steel \u2014 allow for it in window frames and cladding<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Lead<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">29<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">One of the most expansive common metals<\/td><\/tr>\n<tr><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Ice (at 0 \u00b0C)<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">51<\/td><td style=\"padding:9px 12px;border-bottom:1px solid #D9CFB8;\">Even ice expands as it warms \u2014 right up until it melts<\/td><\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n<p><em>Values are typical near 20 \u00b0C and shift slightly with alloy, composition and source. For engineering work, always use the datasheet for your exact material.<\/em><\/p>\n\n<p>For liquids the quoted figure is \u03b2: roughly 210 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup> for water at room temperature, 182 \u00d7 10<sup>\u22126<\/sup> for mercury and about 960 \u00d7 10<sup>\u22126<\/sup> for petrol. That last number is why a brim-full fuel tank weeps on a hot afternoon \u2014 worked problem 5 below puts a litre figure on it.<\/p>\n\n<h2>Real-World Examples of Thermal Expansion<\/h2>\n\n<h3>The Eiffel Tower \u2014 a giant thermometer<\/h3>\n\n<p>Run the formula on 300 m of iron with \u03b1 = 12 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup> and a 40 \u00b0C seasonal swing in metal temperature: \u0394L = 12 \u00d7 10<sup>\u22126<\/sup> \u00d7 300 \u00d7 40 \u2248 0.14 m. That is roughly 14 cm of vertical growth every summer \u2014 consistent with <a href=\"https:\/\/theconversation.com\/the-eiffel-tower-gets-bigger-every-summer-heres-why-261904\" target=\"_blank\" rel=\"noopener\">engineering analyses that put the figure at 12\u201315 cm<\/a>. The Tower is, in effect, a 330-metre thermometer.<\/p>\n\n<h3>Bridges that breathe<\/h3>\n\n<p>The same arithmetic scales up fast for bridges. A 1 km steel span facing a 40 \u00b0C swing needs almost half a metre of breathing room \u2014 enough to crumple the deck if it has nowhere to go.<\/p>\n\n<p>That is what expansion joints are for: the comb-toothed metal strips your car thuds over at each end of a bridge. They let the deck grow and shrink freely instead of grinding against its abutments.<\/p>\n\n<figure style=\"margin:32px auto;max-width:640px;text-align:center;\">\n  <img decoding=\"async\" src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/07\/BridgeExpansionJoint-scaled.jpg\" alt=\"Bridge expansion joint allowing thermal expansion of the road deck\" loading=\"lazy\" style=\"width:100%;height:auto;border-radius:4px;\" width=\"1286\" height=\"1920\">\n  <figcaption style=\"font-size:13px;color:#1F2E47;font-style:italic;margin-top:8px;\">An expansion joint lets a bridge deck grow and shrink with temperature instead of cracking.<\/figcaption>\n<\/figure>\n\n<h3>Railway tracks and sun kinks<\/h3>\n\n<p>Old jointed track left small gaps between rails \u2014 the source of the classic clickety-clack \u2014 precisely so summer heat had somewhere to go. Modern continuous-welded rail takes the opposite approach: it is stretched and clamped at a chosen neutral temperature, so hot weather builds compression in the steel instead of movement.<\/p>\n\n<p>Push past the design limit in a heatwave and the track can buckle sideways into a &#8220;sun kink&#8221;, which is why rail operators impose speed restrictions on extreme days.<\/p>\n\n<h3>Bimetallic strips<\/h3>\n\n<p>Bond a strip of brass to a strip of steel and warm them: the brass expands about one and a half times as much, so the strip has no choice but to curl. That reliable curl opened and closed the contacts in classic thermostats, kettle cut-outs and car indicator flashers for decades.<\/p>\n\n<h3>The stuck jar lid<\/h3>\n\n<p>Run a stubborn jar lid under hot water and it twists free. The thin metal lid heats faster and expands more than the glass beneath it, loosening its grip \u2014 the same trick, scaled up, lets engineers heat a seized bearing to slide it off a shaft.<\/p>\n\n<h3>The rising sea<\/h3>\n\n<p>The oceans absorb more than 90% of the extra heat trapped by greenhouse gases, and warm water simply takes up more room than cold. According to <a href=\"https:\/\/climate.nasa.gov\/vital-signs\/ocean-warming\/\" target=\"_blank\" rel=\"noopener\">NASA&#8217;s sea-level monitoring<\/a>, thermal expansion alone accounts for roughly a third of the rise measured by satellites since 2004 \u2014 no added water required.<\/p>\n\n<h2>Why Water Breaks the Rules<\/h2>\n\n<p>Between 0 \u00b0C and 4 \u00b0C, water does the opposite of almost everything else: warm it and it <em>contracts<\/em>; cool it and it expands. Fresh water is densest at about 4 \u00b0C, as the <a href=\"https:\/\/www.usgs.gov\/water-science-school\/science\/water-density\" target=\"_blank\" rel=\"noopener\">USGS water-science data<\/a> confirms \u2014 this is water&#8217;s famous anomalous expansion.<\/p>\n\n<p>Freezing is stranger still. Hydrogen bonds lock the molecules into an open hexagonal lattice with more empty space than the liquid had, so ice occupies about 9% more volume \u2014 which is why it floats, and why an unlagged pipe bursts in a hard frost.<\/p>\n\n<p>The consequences are life-sized. In winter, the densest 4 \u00b0C water sinks to the bottom of a lake while ice forms only at the top, insulating everything below; fish overwinter in liquid water beneath a frozen lid. Were water ordinary, lakes would freeze from the bottom up.<\/p>\n\n<h2>Common Misconceptions About Thermal Expansion<\/h2>\n\n<h3>&#8220;Heating a ring makes the hole shrink&#8221;<\/h3>\n\n<p>It feels intuitive \u2014 surely the metal expands inward and squeezes the gap? In fact every dimension of an object scales up together, exactly like enlarging a photograph, so the hole grows along with everything else.<\/p>\n\n<p>Mechanics rely on this daily: heat a seized nut or a bearing and its bore widens enough to break free.<\/p>\n\n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/thermal-expansion-metal-ring-before-after-heating.webp\" width=\"1440\" height=\"720\" alt=\"Thermal expansion - A metal ring before and after heating: after heating, both the outer edge and the inner hole are larger than the dashed original outlines \u2014 the hole grows, it does not shrink\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:720px;display:block;margin:0 auto;\" \/><\/figure>\n<p style=\"text-align:center;font-size:14px;font-style:italic;color:#142139;\">Heat a ring and every dimension scales up \u2014 outer edge and inner hole alike. Expansion enlarges the whole picture; it never squeezes the gaps.<\/p>\n\n<h3>&#8220;Everything expands when heated&#8221;<\/h3>\n\n<p>Most things, not everything. Water between 0 \u00b0C and 4 \u00b0C contracts as it warms; a stretched rubber band pulls <em>shorter<\/em> when heated, thanks to its writhing polymer chains; and negative-thermal-expansion ceramics such as zirconium tungstate shrink over enormous temperature ranges.<\/p>\n\n<h3>&#8220;The atoms themselves get bigger&#8221;<\/h3>\n\n<p>They don&#8217;t. Each atom stays the same size; it is the average <em>spacing<\/em> between atoms that grows, because their vibrations are lopsided. Expansion is a story about gaps, not about swelling particles.<\/p>\n\n<h3>&#8220;The effect is so tiny it can&#8217;t matter&#8221;<\/h3>\n\n<p>Per degree it is tiny \u2014 parts per million. But block the movement and the forces are brutal: a rigidly clamped material develops thermal stress \u03c3 = E\u03b1\u0394T, so structural steel denied just 30 \u00b0C of expansion carries about 72 MPa of compression, roughly a third of the way to yielding mild steel. Small displacement, enormous force.<\/p>\n\n<h2>How Thermal Expansion Connects to Other Thermodynamics Ideas<\/h2>\n\n<p>Temperature is, at heart, a measure of the average <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/kinetic-energy-formula\/\">kinetic energy<\/a> of a material&#8217;s particles. Thermal expansion is simply the geometry of matter responding to that energy \u2014 the shape of the bond deciding what jiggling atoms do to the space between them.<\/p>\n\n<p>Heating a material therefore does two jobs at once: it raises the temperature by an amount set by the material&#8217;s <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/thermodynamics\/specific-heat-capacity\/\">specific heat capacity<\/a>, and it nudges the dimensions outward. Liquid-in-glass thermometers exploit the second job to measure the first.<\/p>\n\n<p>Zoom out further and expansion is where thermodynamics starts doing work: a gas expanding against a piston trades internal energy for mechanical output, the bookkeeping governed by the <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/thermodynamics\/laws-of-thermodynamics\/\">laws of thermodynamics<\/a>.<\/p>\n\n<h2>Worked Problems<\/h2>\n\n<p>Work through these in order \u2014 each one adds a new twist. Carry the units and keep \u03b1 in its full \u00d710<sup>\u22126<\/sup> form until the final line.<\/p>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 1<\/div><div class=\"pf-problem-question\">A 25.0 m steel footbridge girder (\u03b1 = 12 \u00d7 10^-6 \u00b0C^-1) warms from 5 \u00b0C at dawn to 35 \u00b0C in the afternoon. How much longer does it get?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Linear expansion applies: \u0394L = \u03b1L<sub>0<\/sub>\u0394T, with \u0394T = 35 \u2212 5 = 30 \u00b0C.<\/p>\n<p>Step 2: Substitute: \u0394L = (12 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>)(25.0 m)(30 \u00b0C).<\/p>\n<p>Step 3: \u0394L = 9.0 \u00d7 10<sup>\u22123<\/sup> m.<\/p>\n<p><strong>Answer: \u0394L = 9.0 mm \u2014 about the width of a pencil, from sunshine alone.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 2<\/div><div class=\"pf-problem-question\">An aluminium rod (\u03b1 = 23 \u00d7 10^-6 \u00b0C^-1) is exactly 2.000 m long at 20 \u00b0C. Find its length at 170 \u00b0C.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: \u0394T = 170 \u2212 20 = 150 \u00b0C, and L = L<sub>0<\/sub>(1 + \u03b1\u0394T).<\/p>\n<p>Step 2: \u0394L = (23 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>)(2.000 m)(150 \u00b0C) = 6.9 \u00d7 10<sup>\u22123<\/sup> m = 6.9 mm.<\/p>\n<p>Step 3: L = 2.000 m + 0.0069 m.<\/p>\n<p><strong>Answer: L = 2.0069 m (an increase of 6.9 mm).<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 3<\/div><div class=\"pf-problem-question\">A 12.000 m copper busbar (\u03b1 = 17 \u00d7 10^-6 \u00b0C^-1) must not lengthen by more than 10.0 mm. What is the maximum temperature rise it can tolerate?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Rearrange \u0394L = \u03b1L<sub>0<\/sub>\u0394T for the unknown: \u0394T = \u0394L \/ (\u03b1L<sub>0<\/sub>).<\/p>\n<p>Step 2: Substitute: \u0394T = (0.0100 m) \/ [(17 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>)(12.000 m)].<\/p>\n<p>Step 3: \u0394T = 0.0100 \/ (2.04 \u00d7 10<sup>\u22124<\/sup>) \u00b0C = 49.0 \u00b0C.<\/p>\n<p><strong>Answer: \u0394T(max) \u2248 49 \u00b0C.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 4<\/div><div class=\"pf-problem-question\">A steel washer (\u03b1 = 12 \u00d7 10^-6 \u00b0C^-1) has a hole of diameter 12.00 mm at 20 \u00b0C. What is the hole&#039;s diameter at 220 \u00b0C?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: A hole expands exactly as if it were made of the surrounding material, so \u0394d = \u03b1d<sub>0<\/sub>\u0394T with \u0394T = 200 \u00b0C.<\/p>\n<p>Step 2: \u0394d = (12 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>)(12.00 mm)(200 \u00b0C) = 0.029 mm.<\/p>\n<p>Step 3: d = 12.00 mm + 0.029 mm.<\/p>\n<p><strong>Answer: d \u2248 12.03 mm \u2014 the hole gets BIGGER, not smaller.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 5<\/div><div class=\"pf-problem-question\">A 60.0 L steel fuel tank is filled to the brim with petrol at 15 \u00b0C and left in the sun until both reach 35 \u00b0C. Taking \u03b2(petrol) = 9.6 \u00d7 10^-4 \u00b0C^-1 and \u03b1(steel) = 12 \u00d7 10^-6 \u00b0C^-1, how much petrol overflows?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Both expand; the spill is the difference. \u0394T = 20 \u00b0C, and for the steel tank \u03b2(tank) = 3\u03b1 = 36 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>.<\/p>\n<p>Step 2: Petrol: \u0394V = (9.6 \u00d7 10<sup>\u22124<\/sup>)(60.0 L)(20) = 1.152 L. Tank: \u0394V = (36 \u00d7 10<sup>\u22126<\/sup>)(60.0 L)(20) = 0.043 L.<\/p>\n<p>Step 3: Overflow = 1.152 \u2212 0.043 = 1.109 L.<\/p>\n<p><strong>Answer: about 1.1 litres spills \u2014 the liquid wins by a factor of nearly thirty.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 6<\/div><div class=\"pf-problem-question\">A steel rail (\u03b1 = 12 \u00d7 10^-6 \u00b0C^-1, Young&#039;s modulus E = 200 GPa) is clamped so it cannot expand at all. What compressive stress builds up when its temperature rises by 30 \u00b0C?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Blocked expansion is equivalent to compressing the rail by strain \u03b5 = \u03b1\u0394T, so \u03c3 = E\u03b1\u0394T.<\/p>\n<p>Step 2: Substitute: \u03c3 = (200 \u00d7 10<sup>9<\/sup> Pa)(12 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>)(30 \u00b0C).<\/p>\n<p>Step 3: \u03c3 = 7.2 \u00d7 10<sup>7<\/sup> Pa.<\/p>\n<p><strong>Answer: \u03c3 = 72 MPa of compression \u2014 roughly a third of mild steel&#8217;s yield stress, from a warm day.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 7<\/div><div class=\"pf-problem-question\">A clock keeps perfect time at 20 \u00b0C using a brass pendulum (\u03b1 = 19 \u00d7 10^-6 \u00b0C^-1). How many seconds does it lose per day at 35 \u00b0C?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: The period is T = 2\u03c0\u221a(L\/g), so T \u221d \u221aL and a small length change gives \u0394T\/T = \u00bd(\u0394L\/L) = \u00bd\u03b1\u0394\u03b8, with \u0394\u03b8 = 15 \u00b0C.<\/p>\n<p>Step 2: Fractional slowing = \u00bd(19 \u00d7 10<sup>\u22126<\/sup>)(15) = 1.43 \u00d7 10<sup>\u22124<\/sup>. A longer pendulum swings slower, so the clock runs slow.<\/p>\n<p>Step 3: Time lost per day = (1.43 \u00d7 10<sup>\u22124<\/sup>)(86 400 s) = 12.3 s.<\/p>\n<p><strong>Answer: about 12.3 seconds lost per day \u2014 why precision clocks used low-\u03b1 invar pendulums.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<h2>Frequently Asked Questions<\/h2>\n\n<details class=\"pf-faq-item\"><summary>What causes thermal expansion?<\/summary><div class=\"pf-faq-item-answer\">\n\nThermal expansion is caused by atoms vibrating more energetically as temperature rises. Because the bond between neighbouring atoms resists compression far more fiercely than stretching, hotter atoms swing further outward than inward, so their average separation increases. Summed over trillions of atomic layers, that microscopic drift becomes a visible change in size.\n\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>What is the formula for thermal expansion?<\/summary><div class=\"pf-faq-item-answer\">\n\nThe linear thermal expansion formula is \u0394L = \u03b1L<sub>0<\/sub>\u0394T, where \u03b1 is the coefficient of linear expansion, L<sub>0<\/sub> the original length and \u0394T the temperature change. For areas, \u0394A = 2\u03b1A<sub>0<\/sub>\u0394T; for volumes, \u0394V = \u03b2V<sub>0<\/sub>\u0394T, where \u03b2 \u2248 3\u03b1 for solids. Liquids and gases are described by \u03b2 alone, since they have no fixed shape.\n\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>Do holes get bigger or smaller when a material is heated?<\/summary><div class=\"pf-faq-item-answer\">\n\nHoles get bigger. A heated object scales up in every dimension, like a photograph being enlarged, so a hole expands exactly as if it were filled with the surrounding material. That is why heating a seized nut or bearing loosens it, and why a heated ring slips over a ball it previously could not pass.\n\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>Why does water expand when it freezes?<\/summary><div class=\"pf-faq-item-answer\">\n\nWater expands on freezing because hydrogen bonds lock its molecules into an open hexagonal ice lattice containing more empty space than the liquid. The result is roughly a 9% jump in volume, which is why ice floats, why ponds freeze from the top down, and why unprotected pipes burst in a hard frost.\n\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>Which materials expand the most when heated?<\/summary><div class=\"pf-faq-item-answer\">\n\nGases expand the most, liquids next, and solids least. Among everyday solids, lead (\u03b1 \u2248 29 \u00d7 10<sup>\u22126<\/sup> \u00b0C<sup>\u22121<\/sup>) and aluminium (\u224823 \u00d7 10<sup>\u22126<\/sup>) are near the top, while invar and fused quartz barely move at all \u2014 which is exactly why they are chosen for precision instruments that must hold their dimensions.\n\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>Why do bridges need expansion joints?<\/summary><div class=\"pf-faq-item-answer\">\n\nBridges need expansion joints because a long deck changes length by centimetres \u2014 up to about half a metre for a kilometre of steel over a 40 \u00b0C swing. Without room to move, the blocked expansion would generate stresses of tens of megapascals (\u03c3 = E\u03b1\u0394T), enough to buckle members or crack the structure over repeated seasons.\n\n<\/div><\/details>\n","protected":false},"excerpt":{"rendered":"<p>Why do bridges have gaps, and why is the Eiffel Tower taller in summer? Thermal expansion explained with the \u0394L = \u03b1L\u2080\u0394T formula, real coefficients, worked examples and an interactive lab.<\/p>\n","protected":false},"author":1,"featured_media":403,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[235,237,236,233,234,28],"class_list":["post-402","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-thermodynamics","tag-anomalous-expansion-of-water","tag-coefficient-of-thermal-expansion","tag-linear-expansion","tag-thermal-expansion","tag-thermal-stress","tag-thermodynamics"],"_links":{"self":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/402","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/comments?post=402"}],"version-history":[{"count":10,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/402\/revisions"}],"predecessor-version":[{"id":1650,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/402\/revisions\/1650"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media\/403"}],"wp:attachment":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media?parent=402"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/categories?post=402"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/tags?post=402"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}