{"id":336,"date":"2026-06-25T15:53:04","date_gmt":"2026-06-25T15:53:04","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=336"},"modified":"2026-08-24T13:04:23","modified_gmt":"2026-08-24T13:04:23","slug":"elastic-potential-energy","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/elastic-potential-energy\/","title":{"rendered":"What Is Elastic Potential Energy?"},"content":{"rendered":"\n<div class=\"pf-citation\"><div class=\"eyebrow\">Definition<\/div><p>\nElastic potential energy is the energy stored in an elastic object \u2014 such as a spring, a rubber band, or a trampoline \u2014 when it is stretched or compressed from its natural length. For an ideal spring that obeys Hooke&#8217;s law, it equals one-half of the spring constant times the displacement squared (EPE = \u00bdkx<sup>2<\/sup>), measured in joules (J).\n<\/p><\/div>\n<p>Pull back a bowstring and hold it. Nothing is moving, yet you can feel that the bow is loaded \u2014 straining to snap forward the instant you let go. That stored, ready-to-release energy is <strong>elastic potential energy<\/strong>, and it is hiding in far more of your day than you might think.<\/p>\n<p>It is in the trampoline that throws a child skyward, the squashed suspension spring soaking up a pothole, and the wound mainspring driving a mechanical watch. In every case, something elastic has been deformed, energy has been tucked away inside it, and that energy is waiting to come back out as motion.<\/p>\n<h2>What Is Elastic Potential Energy?<\/h2>\n<p>Elastic potential energy is the energy an object stores when it is deformed \u2014 stretched, compressed, bent, or twisted \u2014 and which it gives back when it returns to its original shape. The key word is <em>elastic<\/em>: the object must spring back. Squash a lump of clay and it stays squashed, so it stores almost nothing; squash a spring and it pushes right back.<\/p>\n<p>Think of it as energy you can put in and get out again. The work you do stretching a spring does not vanish. It is held in the spring as potential energy, ready to be released as kinetic energy the moment you release your grip.<\/p>\n<p>More precisely, elastic potential energy is the energy stored in any object that obeys <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/hookes-law\/\">Hooke&#8217;s law<\/a> when it is displaced by a distance <em>x<\/em> from its equilibrium position. It is one member of a wider family of stored energy \u2014 the broader idea of <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/what-is-energy-in-physics\/\">energy in physics<\/a> as the capacity to do work.<\/p>\n<h2>The Elastic Potential Energy Formula<\/h2>\n<p>For an ideal spring, the stored energy is given by one compact expression:<\/p>\n<div class=\"pf-formula\">EPE = \u00bdkx<sup>2<\/sup><\/div>\n<p>Each symbol has a precise meaning and a fixed SI unit:<\/p>\n<ul>\n<li><strong>EPE<\/strong> \u2014 the elastic potential energy stored, measured in <strong>joules (J)<\/strong>.<\/li>\n<li><strong>k<\/strong> \u2014 the <strong>spring constant<\/strong> (also called the force constant), a measure of stiffness, measured in <strong>newtons per metre (N\/m)<\/strong>. A larger <em>k<\/em> means a stiffer spring.<\/li>\n<li><strong>x<\/strong> \u2014 the <strong>displacement<\/strong> (the extension or compression) from the spring&#8217;s natural length, measured in <strong>metres (m)<\/strong>.<\/li>\n<\/ul>\n<p>A quick unit check confirms the formula gives energy: (N\/m) \u00d7 (m<sup>2<\/sup>) = N\u00b7m = J. Everything lands in joules, exactly as it should.<\/p>\n<p>You can plug numbers straight in, or compute it instantly with our <a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/hookes-law\">Hooke&#8217;s Law calculator<\/a>, which solves for the spring force, the constant, the extension, or the stored energy. The same quantity is derived step by step in <a href=\"https:\/\/openstax.org\/books\/university-physics-volume-1\/pages\/8-1-potential-energy-of-a-system\" target=\"_blank\" rel=\"noopener\">OpenStax University Physics<\/a> if you want the full textbook treatment.<\/p>\n<h2>How a Spring Stores Energy<\/h2>\n<p>Where does the \u00bd come from? It is not a fudge factor \u2014 it falls straight out of how the spring pushes back.<\/p>\n<p>Hooke&#8217;s law says the force needed to stretch a spring grows with the stretch: <em>F = kx<\/em>. At the very start, when <em>x<\/em> is tiny, the force is almost nothing. By the time you have pulled it to its full extension, the force is at its maximum value, <em>kx<\/em>. The force is not constant \u2014 it climbs in a straight line from zero up to <em>kx<\/em>.<\/p>\n<p>Because work is force times distance, and the force here is changing, you cannot just multiply the final force by the distance. You must use the <em>average<\/em> force. Since the force rises evenly from 0 to <em>kx<\/em>, its average is exactly halfway: \u00bdkx. Multiply that average force by the distance moved, <em>x<\/em>, and you get the stored energy:<\/p>\n<div class=\"pf-formula\">W = (average force) \u00d7 distance = (\u00bdkx) \u00d7 x = \u00bdkx<sup>2<\/sup><\/div>\n<p>There is a neat geometric way to see the same result. Plot force against extension and you get a straight line. The work done \u2014 and therefore the energy stored \u2014 is the <strong>area under that line<\/strong>. That area is a triangle, and a triangle&#8217;s area is \u00bd \u00d7 base \u00d7 height = \u00bd \u00d7 x \u00d7 kx = \u00bdkx<sup>2<\/sup>. The \u00bd is simply the \u00bd in the area of a triangle.<\/p>\n<p>This stored energy equals the <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/work-done-in-physics\/\">work done<\/a> against the spring while deforming it. No energy is lost in an ideal spring, so every joule you put in is recoverable.<\/p>\n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/elastic-potential-energy-force-versus-extension-spring-straight.webp\" width=\"432\" height=\"316\" alt=\"Force versus extension graph for a spring. A straight line rises from the origin with slope k, and the shaded triangular area beneath it equals one-half k x squared, the elastic potential energy\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:216px;display:block;margin:0 auto;\" \/><\/figure>\n<p style=\"text-align:center;font-size:13px;color:#142139;font-style:italic;margin-top:6px;\">The shaded triangle under the force\u2013extension line is the work done on the spring, which equals the stored elastic potential energy, \u00bdkx<sup>2<\/sup>.<\/p>\n<p>Want to feel the relationship between stiffness, stretch, and stored energy directly? The interactive lab below lets you drag the spring and watch the force, the energy, and the shaded area update live.<\/p>\n<div class=\"pf-sim-slot\"><div class=\"pf-sim-slot-header\"><span class=\"icon-dot\"><\/span><span class=\"label\">Elastic Potential Energy Lab<\/span><\/div><div class=\"pf-sim-slot-body\">\n<style>\n.pf-sim-frame{width:100%;border:none;height:600px}\n@media(max-width:760px){.pf-sim-frame{height:1000px}}\n<\/style>\n<iframe src=\"\/labs\/elastic-potential-energy.html?embed=1\" class=\"pf-sim-frame\" loading=\"lazy\"><\/iframe>\n<\/div><\/div>\n<h2>Why Doubling the Stretch Quadruples the Energy<\/h2>\n<p>Here is the single most misunderstood thing about elastic potential energy. The <em>force<\/em> in a spring grows in step with the stretch \u2014 pull twice as far, feel twice the force. But the <em>energy<\/em> does not. Energy depends on <em>x<\/em> squared, so pull twice as far and you store <strong>four<\/strong> times the energy.<\/p>\n<p>That little exponent changes everything. Triple the stretch and the stored energy goes up ninefold. Stretch a spring to ten times its original displacement and it holds a hundred times the energy. The numbers climb fast.<\/p>\n<p>The table below fixes the spring (k = 200 N\/m) and only changes how far it is stretched. Watch the energy column race ahead of the extension column.<\/p>\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=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Extension x (m)<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">x<sup>2<\/sup> (m<sup>2<\/sup>)<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">EPE = \u00bd \u00d7 200 \u00d7 x<sup>2<\/sup> (J)<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Energy vs the 0.10 m row<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.05<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.0025<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.25<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">\u00bc \u00d7<\/td><\/tr>\n<tr><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.10<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.0100<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">1.0<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">1 \u00d7 (baseline)<\/td><\/tr>\n<tr><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.20<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.0400<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">4.0<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">4 \u00d7<\/td><\/tr>\n<tr><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.30<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.0900<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">9.0<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">9 \u00d7<\/td><\/tr>\n<tr><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.40<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">0.1600<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">16.0<\/td><td style=\"border:1px solid #D9CFB8;padding:10px;\">16 \u00d7<\/td><\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>The same <em>x<sup>2<\/sup><\/em> has a second consequence. Squaring a negative number gives a positive result, so compressing a spring by a certain distance stores exactly the same energy as stretching it by that distance. The spring does not care which way you push it \u2014 only how far.<\/p>\n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/elastic-potential-energy-symmetric-u-shaped-parabola-u.webp\" width=\"396\" height=\"264\" alt=\"A symmetric U-shaped parabola of elastic potential energy U equals one-half k x squared against displacement. A compression to the left and an extension of equal size to the right reach the same height, showing they store equal energy\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:198px;display:block;margin:0 auto;\" \/><\/figure>\n<p style=\"text-align:center;font-size:13px;color:#142139;font-style:italic;margin-top:6px;\">The energy curve is a symmetric parabola: a push and a pull of the same size sit at the same height, storing equal energy.<\/p>\n<h2>Real-World Examples of Elastic Potential Energy<\/h2>\n<p>Once you know what to look for, stored elastic energy turns up everywhere \u2014 usually a moment before something springs, launches, or bounces.<\/p>\n<h3>A drawn bow<\/h3>\n<p>Pulling the string bends the limbs of the bow and loads them with elastic potential energy. Release the string and that energy converts almost entirely into the <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/kinetic-energy-formula\/\">kinetic energy<\/a> of the arrow. A heavier draw and a longer pull both mean more stored energy \u2014 and a faster arrow.<\/p>\n<h3>A trampoline or diving board<\/h3>\n<p>When you land, your weight stretches the mat (or bends the board) and stores energy in the deformation. The surface then snaps back, returning that energy and throwing you upward. Land harder, deform it more, and it launches you higher.<\/p>\n<h3>A car&#8217;s suspension springs<\/h3>\n<p>Hit a bump and the coil springs in a car&#8217;s suspension compress, absorbing the jolt as elastic potential energy instead of passing it straight to the cabin. The spring then releases that energy in a controlled way, smoothing the ride. These springs are stiff \u2014 typical values run to tens of thousands of N\/m.<\/p>\n<h3>A bungee cord<\/h3>\n<p>A bungee jump is a clean swap between two kinds of stored energy. At the top, the jumper has gravitational potential energy. As they fall and the cord stretches taut, that energy is transferred into elastic potential energy in the cord \u2014 which then yanks them back up.<\/p>\n<h3>A wound mainspring<\/h3>\n<p>Winding a mechanical watch or a clockwork toy twists a coiled spring, packing it with elastic potential energy. The spring unwinds slowly, releasing that energy bit by bit to turn the gears \u2014 a tiny, portable energy store.<\/p><figure style=\"margin:32px auto;max-width:640px;text-align:center;\">\n  <img decoding=\"async\" src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/06\/Long.jpeg\" alt=\"Archer at full draw storing elastic potential energy in a bent bow\" loading=\"lazy\" style=\"width:100%;height:auto;border-radius:4px;\" width=\"1460\" height=\"1095\">\n  <figcaption style=\"font-size:13px;color:#1F2E47;font-style:italic;margin-top:8px;\">A bow at full draw stores elastic potential energy that launches the arrow on release.<\/figcaption>\n<\/figure><h2>Common Misconceptions About Elastic Potential Energy<\/h2>\n<p>A handful of specific errors trip up almost every student. Clear these and the topic becomes much easier.<\/p>\n<h3>&#8220;The energy grows in step with the stretch&#8221;<\/h3>\n<p>This is the big one. It is the <em>force<\/em> that grows in step with the stretch (F = kx); the <em>energy<\/em> grows with the stretch squared (\u00bdkx<sup>2<\/sup>). Pull twice as far and you do not double the energy \u2014 you quadruple it. Mixing these up is the most common slip in spring problems.<\/p>\n<h3>&#8220;You can just multiply force by distance&#8221;<\/h3>\n<p>Because the spring force changes as you stretch it, multiplying the final force <em>kx<\/em> by the distance <em>x<\/em> gives <em>kx<sup>2<\/sup><\/em> \u2014 double the right answer. You must use the average force, \u00bdkx, which is where the \u00bd comes from. In practice, forgetting the \u00bd is the fastest way to be exactly twice off.<\/p>\n<h3>&#8220;Compressing a spring stores negative energy&#8221;<\/h3>\n<p>The minus sign in Hooke&#8217;s law (F = \u2212kx) describes the <em>direction<\/em> of the restoring force, not a negative energy. Energy is always positive, because <em>x<sup>2<\/sup><\/em> is always positive. A compressed spring stores just as much usable energy as a stretched one.<\/p>\n<h3>&#8220;\u00bdkx<sup>2<\/sup> works for any material at any stretch&#8221;<\/h3>\n<p>The formula only holds while the object obeys Hooke&#8217;s law \u2014 its linear, elastic region. Stretch a spring past its elastic limit and it deforms permanently; real rubber bands are non-linear and lose energy to heat each cycle. Beyond those limits, \u00bdkx<sup>2<\/sup> is an approximation at best.<\/p>\n<h2>How Elastic Potential Energy Relates to Other Concepts<\/h2>\n<p>Elastic potential energy does not sit on its own \u2014 it is one node in a tightly connected web of mechanics ideas.<\/p>\n<p><strong>Hooke&#8217;s law<\/strong> is its foundation. Hooke&#8217;s law gives the force (F = kx); elastic potential energy is the energy that force stores, equal to the area under the Hooke&#8217;s-law line. Understanding <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/hookes-law\/\">Hooke&#8217;s law<\/a> first makes the \u00bdkx<sup>2<\/sup> formula feel inevitable rather than arbitrary.<\/p>\n<p><strong>Kinetic energy<\/strong> is its partner in motion. When a spring releases, its stored \u00bdkx<sup>2<\/sup> becomes \u00bdmv<sup>2<\/sup> of movement. A spring-loaded launcher is just elastic potential energy turning into <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/kinetic-energy-formula\/\">kinetic energy<\/a> \u2014 and notice both formulas carry a square, which is no coincidence.<\/p>\n<p><strong>Simple harmonic motion<\/strong> is what you get when you let a mass on a spring run free. Energy sloshes endlessly between elastic potential energy at the turning points and kinetic energy at the centre \u2014 a cycle explored in depth in <a href=\"https:\/\/openstax.org\/books\/university-physics-volume-1\/pages\/15-2-energy-in-simple-harmonic-motion\" target=\"_blank\" rel=\"noopener\">OpenStax University Physics \u00a715.2<\/a>. That continuous trade is the engine of <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/simple-harmonic-motion\/\">simple harmonic motion<\/a>.<\/p>\n<p>It also sits beside <strong>gravitational potential energy<\/strong> as the other everyday form of stored energy. A bungee cord converts one into the other; the difference is that gravitational PE grows linearly with height (mgh), while elastic PE grows with the square of displacement.<\/p>\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=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Energy type<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Formula<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Grows with<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Stored when\u2026<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:10px;text-align:left;background:#142139;color:#FAF6EE;\">Everyday example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">Elastic potential<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">\u00bdkx<sup>2<\/sup><\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">displacement<sup>2<\/sup> (x<sup>2<\/sup>)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">a spring or elastic object is deformed<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">drawn bow, compressed car spring<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">Kinetic<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">\u00bdmv<sup>2<\/sup><\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">speed<sup>2<\/sup> (v<sup>2<\/sup>)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">an object is moving<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">a rolling ball, a moving car<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">Gravitational potential<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">mgh<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">height h (linear)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">a mass is raised<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:10px;\">water behind a dam, a lifted weight<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h2>Worked Problems<\/h2>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 1<\/div><div class=\"pf-problem-question\">A spring with a spring constant of k = 200 N\/m is stretched by x = 0.10 m. How much elastic potential energy is stored?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Use the elastic potential energy formula. EPE = \u00bdkx<sup>2<\/sup>.<\/p>\n<p>Step 2: Substitute with units. EPE = \u00bd \u00d7 (200 N\/m) \u00d7 (0.10 m)<sup>2<\/sup>.<\/p>\n<p>Step 3: Solve. (0.10 m)<sup>2<\/sup> = 0.010 m<sup>2<\/sup>, so EPE = \u00bd \u00d7 200 \u00d7 0.010 = 1.0 J.<\/p>\n<p><strong>Answer: 1.0 J<\/strong><\/p>\n<\/div><\/details><\/div>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 2<\/div><div class=\"pf-problem-question\">The same spring (k = 200 N\/m) is now stretched twice as far, to x = 0.20 m. How much energy is stored, and how does it compare with Problem 1?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Apply EPE = \u00bdkx<sup>2<\/sup> again.<\/p>\n<p>Step 2: Substitute. EPE = \u00bd \u00d7 (200 N\/m) \u00d7 (0.20 m)<sup>2<\/sup> = \u00bd \u00d7 200 \u00d7 0.040.<\/p>\n<p>Step 3: Solve. EPE = 4.0 J.<\/p>\n<p>Step 4: Compare. Doubling the stretch (0.10 m \u2192 0.20 m) raised the energy from 1.0 J to 4.0 J \u2014 a factor of four, because energy depends on x<sup>2<\/sup>.<\/p>\n<p><strong>Answer: 4.0 J \u2014 four times the energy of Problem 1<\/strong><\/p>\n<\/div><\/details><\/div>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 3<\/div><div class=\"pf-problem-question\">A spring with k = 250 N\/m stores 5.0 J of elastic potential energy. By how much is it stretched?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Start from EPE = \u00bdkx<sup>2<\/sup> and rearrange for x: x = \u221a(2 \u00d7 EPE \u00f7 k).<\/p>\n<p>Step 2: Substitute. x = \u221a(2 \u00d7 5.0 J \u00f7 250 N\/m) = \u221a(10 \u00f7 250).<\/p>\n<p>Step 3: Solve. x = \u221a0.040 = 0.20 m.<\/p>\n<p><strong>Answer: 0.20 m<\/strong><\/p>\n<\/div><\/details><\/div>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 4<\/div><div class=\"pf-problem-question\">A spring stores 12 J of energy when stretched 0.40 m. Find its spring constant k.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Rearrange EPE = \u00bdkx<sup>2<\/sup> for k: k = 2 \u00d7 EPE \u00f7 x<sup>2<\/sup>.<\/p>\n<p>Step 2: Substitute. k = (2 \u00d7 12 J) \u00f7 (0.40 m)<sup>2<\/sup> = 24 \u00f7 0.16.<\/p>\n<p>Step 3: Solve. k = 150 N\/m.<\/p>\n<p><strong>Answer: 150 N\/m<\/strong><\/p>\n<\/div><\/details><\/div>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 5<\/div><div class=\"pf-problem-question\">A toy launcher has a spring of k = 400 N\/m, compressed by x = 0.05 m. It fires a 0.020 kg (20 g) ball horizontally. Assuming all the stored energy becomes kinetic energy, find the ball&#039;s launch speed.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Find the stored elastic potential energy. EPE = \u00bdkx<sup>2<\/sup> = \u00bd \u00d7 400 \u00d7 (0.05)<sup>2<\/sup> = \u00bd \u00d7 400 \u00d7 0.0025 = 0.50 J.<\/p>\n<p>Step 2: Set it equal to kinetic energy (energy is conserved). \u00bdmv<sup>2<\/sup> = 0.50 J.<\/p>\n<p>Step 3: Rearrange for v. v = \u221a(2 \u00d7 0.50 \u00f7 0.020) = \u221a(1.0 \u00f7 0.020) = \u221a50.<\/p>\n<p>Step 4: Solve. v \u2248 7.1 m\/s.<\/p>\n<p><strong>Answer: \u2248 7.1 m\/s<\/strong><\/p>\n<\/div><\/details><\/div>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 6<\/div><div class=\"pf-problem-question\">A 2.0 kg mass is hung from a spring and stretches it 0.10 m before coming to rest. (a) Find the spring constant. (b) Find the elastic potential energy stored at this stretch. Take g = 9.81 m\/s^2.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: At rest, the spring force balances the weight: kx = mg.<\/p>\n<p>Step 2: Solve for k. k = mg \u00f7 x = (2.0 \u00d7 9.81) \u00f7 0.10 = 19.62 \u00f7 0.10 = 196 N\/m (to 3 s.f.).<\/p>\n<p>Step 3: Find the stored energy. EPE = \u00bdkx<sup>2<\/sup> = \u00bd \u00d7 196.2 \u00d7 (0.10)<sup>2<\/sup> = \u00bd \u00d7 196.2 \u00d7 0.010 \u2248 0.98 J.<\/p>\n<p>Step 4: Sanity check. The weight dropped through 0.10 m, releasing mgh = 2.0 \u00d7 9.81 \u00d7 0.10 \u2248 1.96 J of gravitational PE \u2014 yet only \u2248 0.98 J (exactly half) is stored in the spring. The other half went into kinetic energy as the mass sped up on its way down, which is why a real mass overshoots and oscillates rather than stopping gently.<\/p>\n<p><strong>Answer: (a) \u2248 196 N\/m; (b) \u2248 0.98 J<\/strong><\/p>\n<\/div><\/details><\/div>\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 7<\/div><div class=\"pf-problem-question\">A vertical spring with k = 800 N\/m is compressed 0.15 m and launches a 0.10 kg ball straight up. Ignoring air resistance and the spring&#039;s mass, how high above the launch point does the ball rise? Take g = 9.81 m\/s^2.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Find the stored elastic potential energy. EPE = \u00bdkx<sup>2<\/sup> = \u00bd \u00d7 800 \u00d7 (0.15)<sup>2<\/sup> = \u00bd \u00d7 800 \u00d7 0.0225 = 9.0 J.<\/p>\n<p>Step 2: At the highest point, all of it has become gravitational PE: mgh = 9.0 J.<\/p>\n<p>Step 3: Rearrange for h. h = 9.0 \u00f7 (m \u00d7 g) = 9.0 \u00f7 (0.10 \u00d7 9.81) = 9.0 \u00f7 0.981.<\/p>\n<p>Step 4: Solve. h \u2248 9.2 m.<\/p>\n<p><strong>Answer: \u2248 9.2 m<\/strong><\/p>\n<\/div><\/details><\/div>\n<h2>Frequently Asked Questions<\/h2>\n<details class=\"pf-faq-item\"><summary>What is elastic potential energy in simple terms?<\/summary><div class=\"pf-faq-item-answer\">\nElastic potential energy is the energy stored in an object when you stretch, squash, bend, or twist it \u2014 as long as it springs back to its original shape. You put energy in by deforming it, and you get that energy back as motion when you let go. A stretched rubber band and a compressed spring are classic examples.\n<\/div><\/details>\n<details class=\"pf-faq-item\"><summary>What is the formula for elastic potential energy?<\/summary><div class=\"pf-faq-item-answer\">\nThe formula is EPE = \u00bdkx<sup>2<\/sup>. Here, k is the spring constant in newtons per metre (N\/m), x is the distance the spring is stretched or compressed from its natural length in metres (m), and the result is the energy in joules (J). It applies to any object that obeys Hooke&#8217;s law.\n<\/div><\/details>\n<details class=\"pf-faq-item\"><summary>Why is there a \u00bd in \u00bdkx^2?<\/summary><div class=\"pf-faq-item-answer\">\nThe \u00bd appears because the spring force is not constant \u2014 it grows steadily from zero to its maximum value kx as you stretch the spring. The work stored is the average force (\u00bdkx) times the distance (x), giving \u00bdkx<sup>2<\/sup>. Equivalently, it is the triangular area under the force\u2013extension graph.\n<\/div><\/details>\n<details class=\"pf-faq-item\"><summary>Does compressing a spring store the same energy as stretching it?<\/summary><div class=\"pf-faq-item-answer\">\nYes. The formula uses x<sup>2<\/sup>, and squaring removes the sign, so a compression and a stretch of the same size store exactly the same amount of energy. The spring does not care about the direction of the displacement, only its magnitude. This is why the energy-versus-displacement graph is a symmetric parabola.\n<\/div><\/details>\n<details class=\"pf-faq-item\"><summary>What are the units of elastic potential energy?<\/summary><div class=\"pf-faq-item-answer\">\nElastic potential energy is measured in joules (J), the SI unit of all energy. This follows from the formula: the spring constant k is in newtons per metre (N\/m) and x<sup>2<\/sup> is in square metres (m<sup>2<\/sup>), so \u00bdkx<sup>2<\/sup> has units of N\u00b7m, which equals one joule.\n<\/div><\/details>\n<details class=\"pf-faq-item\"><summary>Is elastic potential energy the same as Hooke&#039;s law?<\/summary><div class=\"pf-faq-item-answer\">\nNo, but they are closely linked. Hooke&#8217;s law describes the force a spring exerts (F = kx), while elastic potential energy describes the energy stored (\u00bdkx<sup>2<\/sup>). The stored energy is the area under the Hooke&#8217;s-law force\u2013extension line, so the law gives you the force and the energy formula gives you the result of that force.\n<\/div><\/details>\n","protected":false},"excerpt":{"rendered":"<p>Elastic potential energy is the energy stored when a spring or elastic object is stretched or compressed, given by EPE = \u00bdkx\u00b2. This guide explains the formula, why energy grows with the square of the stretch, and how to use it, with worked examples.<\/p>\n","protected":false},"author":1,"featured_media":338,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[138,140,185,16,139],"class_list":["post-336","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mechanics","tag-elastic-potential-energy","tag-hookes-law","tag-kx-squared","tag-potential-energy","tag-spring-constant"],"_links":{"self":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/336","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=336"}],"version-history":[{"count":9,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/336\/revisions"}],"predecessor-version":[{"id":1689,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/336\/revisions\/1689"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media\/338"}],"wp:attachment":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media?parent=336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/categories?post=336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/tags?post=336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}