{"id":427,"date":"2017-11-14T13:47:41","date_gmt":"2017-11-14T13:47:41","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/chapter\/potential-energy-of-a-system\/"},"modified":"2017-11-18T01:29:46","modified_gmt":"2017-11-18T01:29:46","slug":"potential-energy-of-a-system","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/chapter\/potential-energy-of-a-system\/","title":{"raw":"Potential Energy of a System","rendered":"Potential Energy of a System"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Relate the difference of potential energy to work done on a particle for a system without friction or air drag<\/li>\n<li>Explain the meaning of the zero of the potential energy function for a system<\/li>\n<li>Calculate and apply the gravitational potential energy for an object near Earth\u2019s surface and the elastic potential energy of a mass-spring system<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165039051689\">In <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>, we saw that the work done on an object by the constant gravitational force, near the surface of Earth, over any displacement is a function only of the difference in the positions of the end-points of the displacement. This property allows us to define a different kind of energy for the system than its kinetic energy, which is called <span>potential energy<\/span>. We consider various properties and types of potential energy in the following subsections.<\/p>\n<div class=\"bc-section section\" id=\"fs-id1165035692628\">\n<h3>Potential Energy Basics<\/h3>\n<p id=\"fs-id1165035682820\">In <a href=\"\/contents\/305e881e-18f9-4ccd-b96d-e7a1c4db654d\" class=\"target-chapter\">Motion in Two and Three Dimensions<\/a>, we analyzed the motion of a projectile, like kicking a football in <a href=\"#CNX_UPhysics_08_01_football\" class=\"autogenerated-content\">(Figure)<\/a>. For this example, let\u2019s ignore friction and air resistance. As the football rises, the work done by the gravitational force on the football is negative, because the ball\u2019s displacement is positive vertically and the force due to gravity is negative vertically. We also noted that the ball slowed down until it reached its highest point in the motion, thereby decreasing the ball\u2019s kinetic energy. This loss in kinetic energy translates to a gain in gravitational potential energy of the football-Earth system.<\/p>\n<p id=\"fs-id1165035972000\">As the football falls toward Earth, the work done on the football is now positive, because the displacement and the gravitational force both point vertically downward. The ball also speeds up, which indicates an increase in kinetic energy. Therefore, energy is converted from gravitational potential energy back into kinetic energy.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_football\">\n<div class=\"bc-figcaption figcaption\">As a football starts its descent toward the wide receiver, gravitational potential energy is converted back into kinetic energy.<\/div>\n<p><span id=\"fs-id1165035906458\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_football.jpg\" alt=\"An illustration of a football\u2019s trajectory and energy. The kicker kicks the ball, doing work on it and giving it maximum kinetic energy. The potential energy is minimum. This is point one. On the way up, at point two, the kinetic energy of the ball decreases and its potential energy decreases. At the highest point, point three, the kinetic energy of the ball  is at its minimum and its potential energy is maximum. As the ball descends, point four, the kinetic energy increases and the potential energy decreases. The receiver catches the ball at the same height above the ground as it was kicked, at point five. The kinetic energy equals maximum, potential energy is minimum.\"><\/span><\/p><\/div>\n<p id=\"fs-id1165039251424\">Based on this scenario, we can define the difference of potential energy from point <em>A<\/em> to point <em>B<\/em> as the negative of the work done:<\/p>\n<div id=\"fs-id1165039074593\" class=\"equation-callout\">\n<div id=\"fs-id1165039502282\">[latex]\\text{\u0394}{U}_{AB}={U}_{B}-{U}_{A}=\\text{\u2212}{W}_{AB}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165039238293\">This formula explicitly states a <span>potential energy difference<\/span>, not just an absolute potential energy. Therefore, we need to define potential energy at a given position in such a way as to state standard values of potential energy on their own, rather than potential energy differences. We do this by rewriting the potential energy function in terms of an arbitrary constant,<\/p>\n<div id=\"fs-id1165039391866\" class=\"equation-callout\">\n<div id=\"fs-id1165039502050\">[latex]\\text{\u0394}U=U\\left(\\stackrel{\\to }{r}\\right)-U\\left({\\stackrel{\\to }{r}}_{0}\\right).[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165035641022\">The choice of the potential energy at a starting location of [latex]{\\stackrel{\\to }{r}}_{0}[\/latex] is made out of convenience in the given problem. Most importantly, whatever choice is made should be stated and kept consistent throughout the given problem. There are some well-accepted choices of initial potential energy. For example, the lowest height in a problem is usually defined as zero potential energy, or if an object is in space, the farthest point away from the system is often defined as zero potential energy. Then, the potential energy, with respect to zero at [latex]{\\stackrel{\\to }{r}}_{0},[\/latex] is just [latex]U\\left(\\stackrel{\\to }{r}\\right).[\/latex]<\/p>\n<p id=\"fs-id1165039324524\">As long as there is no friction or air resistance, the change in kinetic energy of the football equals the change in gravitational potential energy of the football. This can be generalized to any potential energy:<\/p>\n<div id=\"fs-id1165039099001\">[latex]\\text{\u0394}{K}_{AB}=\\text{\u0394}{U}_{AB}.[\/latex]<\/div>\n<p id=\"fs-id1165039503553\">Let\u2019s look at a specific example, choosing zero potential energy for gravitational potential energy at convenient points.<\/p>\n<div id=\"fs-id1165035686790\" class=\"textbox examples\">\n<p id=\"fs-id1165035772252\"><span>Basic Properties of Potential Energy<\/span><br>\nA particle moves along the <em>x<\/em>-axis under the action of a force given by [latex]F=\\text{\u2212}a{x}^{2}[\/latex], where [latex]a=3\\phantom{\\rule{0.2em}{0ex}}{\\text{N\/m}}^{2}[\/latex]. (a) What is the difference in its potential energy as it moves from [latex]{x}_{A}=1\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] to [latex]{x}_{B}=2\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex]? (b) What is the particle\u2019s potential energy at [latex]x=1\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] with respect to a given 0.5 J of potential energy at [latex]x=0[\/latex]?<\/p>\n<p id=\"fs-id1165035849830\"><span>Strategy<\/span><br>\n(a) The difference in potential energy is the negative of the work done, as defined by <a href=\"#fs-id1165039502282\" class=\"autogenerated-content\">(Figure)<\/a>. The work is defined in the previous chapter as the dot product of the force with the distance. Since the particle is moving forward in the <em>x<\/em>-direction, the dot product simplifies to a multiplication ([latex]\\stackrel{^}{i}\u00b7\\stackrel{^}{i}=1[\/latex]). To find the total work done, we need to integrate the function between the given limits. After integration, we can state the work or the potential energy. (b) The potential energy function, with respect to zero at [latex]x=0[\/latex], is the indefinite integral encountered in part (a), with the constant of integration determined from <a href=\"#fs-id1165039099001\" class=\"autogenerated-content\">(Figure)<\/a>. Then, we substitute the <em>x<\/em>-value into the function of potential energy to calculate the potential energy at [latex]x=1\\phantom{\\rule{0.2em}{0ex}}\\text{m}.[\/latex]<\/p>\n<p><span>Solution<\/span><\/p>\n<ol type=\"a\">\n<li>The work done by the given force as the particle moves from coordinate <em>x<\/em> to [latex]x+dx[\/latex] in one dimension is\n<div><\/div>\n<div id=\"fs-id1165039443277\" class=\"unnumbered\">[latex]dW=\\stackrel{\\to }{F}\u00b7d\\stackrel{\\to }{r}=Fdx=\\text{\u2212}a{x}^{2}dx.[\/latex]<\/div>\n<div><\/div>\n<p>Substituting this expression into <a href=\"#fs-id1165039502282\" class=\"autogenerated-content\">(Figure)<\/a>, we obtain<\/p>\n<div><\/div>\n<div id=\"fs-id1165035633026\" class=\"unnumbered\">[latex]\\text{\u0394}U=\\text{\u2212}W=\\underset{{x}_{1}}{\\overset{{x}_{2}}{\\int }}a{x}^{2}dx=\\frac{1}{3}\\left(3\\phantom{\\rule{0.2em}{0ex}}{\\text{N\/m}}^{2}\\right){{x}^{2}|}_{1\\phantom{\\rule{0.2em}{0ex}}\\text{m}}^{2\\phantom{\\rule{0.2em}{0ex}}\\text{m}}=7\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<\/li>\n<li>The indefinite integral for the potential energy function in part (a) is\n<div><\/div>\n<div id=\"fs-id1165035699939\" class=\"unnumbered\">[latex]U\\left(x\\right)=\\frac{1}{3}a{x}^{3}+\\text{const}.,[\/latex]<\/div>\n<div><\/div>\n<p>and we want the constant to be determined by<\/p>\n<div><\/div>\n<div id=\"fs-id1165039462260\" class=\"unnumbered\">[latex]U\\left(0\\right)=0.5\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<div><\/div>\n<p>Thus, the potential energy with respect to zero at [latex]x=0[\/latex] is just<\/p>\n<div><\/div>\n<div id=\"fs-id1165039303300\" class=\"unnumbered\">[latex]U\\left(x\\right)=\\frac{1}{3}a{x}^{3}+0.5\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<div><\/div>\n<p>Therefore, the potential energy at [latex]x=1\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] is<\/p>\n<div><\/div>\n<div id=\"fs-id1165039078983\" class=\"unnumbered\">[latex]U\\left(1\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)=\\frac{1}{3}\\left(3\\phantom{\\rule{0.2em}{0ex}}{\\text{N\/m}}^{2}\\right){\\left(1\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)}^{3}+0.5\\phantom{\\rule{0.2em}{0ex}}\\text{J}=1.5\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165039187844\"><span>Significance<\/span><br>\nIn this one-dimensional example, any function we can integrate, independent of path, is conservative. Notice how we applied the definition of potential energy difference to determine the potential energy function with respect to zero at a chosen point. Also notice that the potential energy, as determined in part (b), at [latex]x=1\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] is [latex]U\\left(1\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)=1\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex] and at [latex]x=2\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] is [latex]U\\left(2\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)=8\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex]; their difference is the result in part (a).<\/p>\n<\/div>\n<div id=\"fs-id1165035756834\" class=\"check-understanding\">\n<div id=\"fs-id1165035695796\">\n<div id=\"fs-id1165039288706\">\n<p id=\"fs-id1165035615978\"><strong>Check Your Understanding<\/strong> In <a href=\"#fs-id1165035686790\" class=\"autogenerated-content\">(Figure)<\/a>, what are the potential energies of the particle at [latex]x=1\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] and [latex]x=2\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] with respect to zero at [latex]x=1.5\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex]? Verify that the difference of potential energy is still 7 J.<\/p>\n<\/div>\n<div id=\"fs-id1165035924241\">\n<p id=\"fs-id1165039493913\">[latex]\\left(4.63\\phantom{\\rule{0.2em}{0ex}}\\text{J}\\right)-\\left(-2.38\\phantom{\\rule{0.2em}{0ex}}\\text{J}\\right)=7.00\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1165039107955\">\n<h3>Systems of Several Particles<\/h3>\n<p>In general, a system of interest could consist of several particles. The difference in the potential energy of the system is the negative of the work done by gravitational or elastic forces, which, as we will see in the next section, are conservative forces. The potential energy difference depends only on the initial and final positions of the particles, and on some parameters that characterize the interaction (like mass for gravity or the spring constant for a Hooke\u2019s law force).<\/p>\n<p id=\"fs-id1165039471899\">It is important to remember that potential energy is a property of the interactions between objects in a chosen system, and not just a property of each object. This is especially true for electric forces, although in the examples of potential energy we consider below, parts of the system are either so big (like Earth, compared to an object on its surface) or so small (like a massless spring), that the changes those parts undergo are negligible if included in the system.<\/p>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Types of Potential Energy<\/h3>\n<p id=\"fs-id1165039045528\">For each type of interaction present in a system, you can label a corresponding type of potential energy. The total potential energy of the system is the sum of the potential energies of all the types. (This follows from the additive property of the dot product in the expression for the work done.) Let\u2019s look at some specific examples of types of potential energy discussed in <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>. First, we consider each of these forces when acting separately, and then when both act together.<\/p>\n<div class=\"bc-section section\" id=\"fs-id1165039369440\">\n<h4>Gravitational potential energy near Earth\u2019s surface<\/h4>\n<p id=\"fs-id1165039285376\">The system of interest consists of our planet, Earth, and one or more particles near its surface (or bodies small enough to be considered as particles, compared to Earth). The gravitational force on each particle (or body) is just its weight <em>mg<\/em> near the surface of Earth, acting vertically down. According to Newton\u2019s third law, each particle exerts a force on Earth of equal magnitude but in the opposite direction. Newton\u2019s second law tells us that the magnitude of the acceleration produced by each of these forces on Earth is <em>mg<\/em> divided by Earth\u2019s mass. Since the ratio of the mass of any ordinary object to the mass of Earth is vanishingly small, the motion of Earth can be completely neglected. Therefore, we consider this system to be a group of single-particle systems, subject to the uniform gravitational force of Earth.<\/p>\n<p id=\"fs-id1165039209828\">In <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>, the work done on a body by Earth\u2019s uniform gravitational force, near its surface, depended on the mass of the body, the acceleration due to gravity, and the difference in height the body traversed, as given by <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796#fs-id1165035740928\" class=\"autogenerated-content\">(Figure)<\/a>. By definition, this work is the negative of the difference in the gravitational potential energy, so that difference is<\/p>\n<div id=\"fs-id1165035676115\">[latex]\\text{\u0394}{U}_{\\text{grav}}=\\text{\u2212}{W}_{\\text{grav},AB}=mg\\left({y}_{B}-{y}_{A}\\right).[\/latex]<\/div>\n<p id=\"fs-id1165039090814\">You can see from this that the <span class=\"no-emphasis\">gravitational potential energy<\/span> function, near Earth\u2019s surface, is<\/p>\n<div id=\"fs-id1165039084074\" class=\"equation-callout\">\n<div id=\"fs-id1165039441699\">[latex]U\\left(y\\right)=mgy+\\text{const}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165039107630\">You can choose the value of the constant, as described in the discussion of <a href=\"#fs-id1165039502050\" class=\"autogenerated-content\">(Figure)<\/a>; however, for solving most problems, the most convenient constant to choose is zero for when [latex]y=0,[\/latex] which is the lowest vertical position in the problem.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_jumper\">\n<div class=\"bc-figcaption figcaption\">Don\u2019t jump\u2014you have so much potential (gravitational potential energy, that is). (credit: Andy Spearing)<\/div>\n<p><span id=\"fs-id1165039242348\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_jumper.jpg\" alt=\"A photograph of a person jumping off a high cliff into a lake.\"><\/span><\/p><\/div>\n<div id=\"fs-id1165038964832\" class=\"textbox examples\">\n<p id=\"fs-id1165035785797\"><span>Gravitational Potential Energy of a Hiker<\/span><br>\nThe summit of Great Blue Hill in Milton, MA, is 147 m above its base and has an elevation above sea level of 195 m (<a href=\"#CNX_UPhysics_08_01_GrBlueHill\" class=\"autogenerated-content\">(Figure)<\/a>). (Its Native American name, <em>Massachusett<\/em>, was adopted by settlers for naming the Bay Colony and state near its location.) A 75-kg hiker ascends from the base to the summit. What is the gravitational potential energy of the hiker-Earth system with respect to zero gravitational potential energy at base height, when the hiker is (a) at the base of the hill, (b) at the summit, and (c) at sea level, afterward?<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_GrBlueHill\">\n<div class=\"bc-figcaption figcaption\">Sketch of the profile of Great Blue Hill, Milton, MA. The altitudes of the three levels are indicated.<\/div>\n<p><span id=\"fs-id1165039085986\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_GrBlueHill.jpg\" alt=\"Sketch of the profile of Great Blue Hill, Milton, MA. The summit is 195 meters above sea level. The base of the hill is 147 meters below the summit.\"><\/span><\/p><\/div>\n<p id=\"fs-id1165039302945\"><span>Strategy<\/span><br>\nFirst, we need to pick an origin for the <em>y<\/em>-axis and then determine the value of the constant that makes the potential energy zero at the height of the base. Then, we can determine the potential energies from <a href=\"#fs-id1165039441699\" class=\"autogenerated-content\">(Figure)<\/a>, based on the relationship between the zero potential energy height and the height at which the hiker is located.<\/p>\n<p id=\"fs-id1165039075284\"><span>Solution<\/span><\/p>\n<ol id=\"fs-id1165035618779\" type=\"a\">\n<li>Let\u2019s choose the origin for the <em>y<\/em>-axis at base height, where we also want the zero of potential energy to be. This choice makes the constant equal to zero and\n<div><\/div>\n<div id=\"fs-id1165035704066\" class=\"unnumbered\">[latex]U\\left(\\text{base}\\right)=U\\left(0\\right)=0.[\/latex]<\/div>\n<\/li>\n<li>At the summit, [latex]y=147\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex], so\n<div><\/div>\n<div id=\"fs-id1165039092156\" class=\"unnumbered\">[latex]U\\left(\\text{summit}\\right)=U\\left(147\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)=mgh=\\left(75\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}9.8\\phantom{\\rule{0.2em}{0ex}}\\text{N}\\right)\\left(147\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)=108\\phantom{\\rule{0.2em}{0ex}}\\text{kJ}.[\/latex]<\/div>\n<\/li>\n<li>At sea level, [latex]y=\\left(147-195\\right)\\text{m}=-48\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex], so\n<div><\/div>\n<div id=\"fs-id1165039297023\" class=\"unnumbered\">[latex]U\\left(\\text{sea-level}\\right)=\\left(75\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}9.8\\phantom{\\rule{0.2em}{0ex}}\\text{N}\\right)\\left(-48\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)=-35.3\\phantom{\\rule{0.2em}{0ex}}\\text{kJ}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165039442596\"><span>Significance<\/span><br>\nBesides illustrating the use of <a href=\"#fs-id1165035676115\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1165039441699\" class=\"autogenerated-content\">(Figure)<\/a>, the values of gravitational potential energy we found are reasonable. The gravitational potential energy is higher at the summit than at the base, and lower at sea level than at the base. Gravity does work on you on your way up, too! It does negative work and not quite as much (in magnitude), as your muscles do. But it certainly does work. Similarly, your muscles do work on your way down, as negative work. The numerical values of the potential energies depend on the choice of zero of potential energy, but the physically meaningful differences of potential energy do not. [Note that since <a href=\"#fs-id1165039502050\" class=\"autogenerated-content\">(Figure)<\/a> is a difference, the numerical values do not depend on the origin of coordinates.]<\/p>\n<\/div>\n<div id=\"fs-id1165036083411\" class=\"check-understanding\">\n<div id=\"fs-id1165035926252\">\n<div id=\"fs-id1165033367269\">\n<p><strong>Check Your Understanding<\/strong> What are the values of the gravitational potential energy of the hiker at the base, summit, and sea level, with respect to a sea-level zero of potential energy?<\/p>\n<\/div>\n<div id=\"fs-id1165035849800\">\n<p id=\"fs-id1165036156053\">35.3 kJ, 143 kJ, 0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1165039266683\">\n<h4>Elastic potential energy<\/h4>\n<p id=\"fs-id1165036159199\">In <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>, we saw that the work done by a perfectly elastic spring, in one dimension, depends only on the spring constant and the squares of the displacements from the unstretched position, as given in <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796#fs-id1165039209368\" class=\"autogenerated-content\">(Figure)<\/a>. This work involves only the properties of a Hooke\u2019s law interaction and not the properties of real springs and whatever objects are attached to them. Therefore, we can define the difference of <span class=\"no-emphasis\">elastic potential energy<\/span> for a spring force as the negative of the work done by the spring force in this equation, before we consider systems that embody this type of force. Thus,<\/p>\n<div id=\"fs-id1165035743019\">[latex]\\text{\u0394}U=\\text{\u2212}{W}_{AB}=\\frac{1}{2}k\\left({x}_{B}^{2}-{x}_{A}^{2}\\right),[\/latex]<\/div>\n<p id=\"fs-id1165036082561\">where the object travels from point <em>A<\/em> to point <em>B<\/em>. The potential energy function corresponding to this difference is<\/p>\n<div id=\"fs-id1165039074371\" class=\"equation-callout\">\n<div id=\"fs-id1165039398969\">[latex]U\\left(x\\right)=\\frac{1}{2}k{x}^{2}+\\text{const}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165039399552\">If the spring force is the only force acting, it is simplest to take the zero of potential energy at [latex]x=0[\/latex], when the spring is at its unstretched length. Then, the constant is <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a> is zero. (Other choices may be more convenient if other forces are acting.)<\/p>\n<div id=\"fs-id1165035980783\" class=\"textbox examples\">\n<p id=\"fs-id1165035618194\"><span>Spring Potential Energy<\/span><br>\nA system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4 N\/cm. (a) How much elastic potential energy does the spring contribute when its length is 23 cm? (b) How much more potential energy does it contribute if its length increases to 26 cm?<\/p>\n<p id=\"fs-id1165036152598\"><span>Strategy<\/span><br>\nWhen the spring is at its unstretched length, it contributes nothing to the potential energy of the system, so we can use <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a> with the constant equal to zero. The value of <em>x<\/em> is the length minus the unstretched length. When the spring is expanded, the spring\u2019s displacement or difference between its relaxed length and stretched length should be used for the <em>x<\/em>-value in calculating the potential energy of the spring.<\/p>\n<p id=\"fs-id1165039336747\"><span>Solution<\/span><\/p>\n<ol id=\"fs-id1165039450221\" type=\"a\">\n<li>The displacement of the spring is [latex]x=23\\phantom{\\rule{0.2em}{0ex}}\\text{cm}-20\\phantom{\\rule{0.2em}{0ex}}\\text{cm}=3\\phantom{\\rule{0.2em}{0ex}}\\text{cm}[\/latex], so the contributed potential energy is [latex]U=\\frac{1}{2}k{x}^{2}=\\frac{1}{2}\\left(4\\phantom{\\rule{0.2em}{0ex}}\\text{N\/cm}\\right){\\left(3\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)}^{2}=0.18\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex].<\/li>\n<li>When the spring\u2019s displacement is [latex]x=26\\phantom{\\rule{0.2em}{0ex}}\\text{cm}-20\\phantom{\\rule{0.2em}{0ex}}\\text{cm}=6\\phantom{\\rule{0.2em}{0ex}}\\text{cm}[\/latex], the potential energy is [latex]U=\\frac{1}{2}k{x}^{2}=\\frac{1}{2}\\left(4\\phantom{\\rule{0.2em}{0ex}}\\text{N\/cm}\\right){\\left(6\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)}^{2}=0.72\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex], which is a 0.54-J increase over the amount in part (a).<\/li>\n<\/ol>\n<p id=\"fs-id1165039096164\"><span>Significance<\/span><br>\nCalculating the elastic potential energy and potential energy differences from <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a> involves solving for the potential energies based on the given lengths of the spring. Since <em>U<\/em> depends on [latex]{x}^{2}[\/latex], the potential energy for a compression (negative <em>x<\/em>) is the same as for an extension of equal magnitude.<\/p>\n<\/div>\n<div id=\"fs-id1165035834925\" class=\"check-understanding\">\n<div id=\"fs-id1165039472655\">\n<div id=\"fs-id1165035712092\">\n<p id=\"fs-id1165035944692\"><strong>Check Your Understanding<\/strong> When the length of the spring in <a href=\"#fs-id1165035980783\" class=\"autogenerated-content\">(Figure)<\/a> changes from an initial value of 22.0 cm to a final value, the elastic potential energy it contributes changes by [latex]-0.0800\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex] Find the final length.<\/p>\n<\/div>\n<div id=\"fs-id1165035723425\">\n<p id=\"fs-id1165036153427\">22.8 cm. Using 0.02 m for the initial displacement of the spring (see above), we calculate the final displacement of the spring to be 0.028 m; therefore the length of the spring is the unstretched length plus the displacement, or 22.8 cm.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1165035939827\">\n<h4>Gravitational and elastic potential energy<\/h4>\n<p id=\"fs-id1165035687891\">A simple system embodying both gravitational and elastic types of potential energy is a one-dimensional, vertical <span class=\"no-emphasis\">mass-spring system<\/span>. This consists of a massive particle (or block), hung from one end of a perfectly elastic, massless spring, the other end of which is fixed, as illustrated in <a href=\"#CNX_UPhysics_08_01_VertM-S\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_VertM-S\">\n<div class=\"bc-figcaption figcaption\">A vertical mass-spring system, with the <em>y<\/em>-axis pointing upwards. The mass is initially at an equilibrium position and pulled downward to [latex]{y}_{\\text{pull}}.[\/latex] An oscillation begins, centered at the equilibrium position.<\/div>\n<p><span id=\"fs-id1165039475145\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_VertM-S.jpg\" alt=\"A vertical mass spring system is illustrated. The upper end of the spring is attached to the ceiling. A block of mass m is attached to the lower end.  The spring is drawn at two positions. On the left, the mass is in the equilibrium position. To  the right of this, the spring is drawn with the mass pulled down a distance y sub pull. This position of the mass is labeled as h equal to zero. A graph of y as a function of X is shown to the rightly the illustrations, with y equals zero aligned with the equilibrium position in the illustrations. The plot is sinusoidal, with the minimum y at x=0 and even with the lower mass position in the illustrations.\"><\/span><\/p><\/div>\n<p id=\"fs-id1165039472389\">First, let\u2019s consider the potential energy of the system. Assuming the spring is massless, the system of the block and Earth gains and loses potential energy. We need to define the constant in the potential energy function of <a href=\"#fs-id1165039441699\" class=\"autogenerated-content\">(Figure)<\/a>. Often, the ground is a suitable choice for when the gravitational potential energy is zero; however, in this case, the lowest point or when [latex]h=0[\/latex] is a convenient location for zero gravitational potential energy. Note that this choice is arbitrary, and the problem can be solved correctly even if another choice is picked.<\/p>\n<p id=\"fs-id1165039400355\">We must also define the elastic potential energy of the system and the corresponding constant, as detailed in <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a>. The equilibrium location is the most suitable mathematically to choose for where the potential energy of the spring is zero.<\/p>\n<p>Therefore, based on this convention, each potential energy and kinetic energy can be written out for three critical points of the system: (1) the lowest pulled point, (2) the equilibrium position of the spring, and (3) the highest point achieved. We note that the total energy of the system is conserved, so any total energy in this chart could be matched up to solve for an unknown quantity. The results are shown in <a href=\"#fs-id1165039443587\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"fs-id1165039443587\" summary=\"The table has four rows and four columns. The first row and first column are a header row and column. The column headings are Gravitational P. E., Elastic P. E., and Kinetic P. E. The row headings are (3) Highest Point, (2) Equilibrium, and  (3) Lowest Point. The Gravitational P. E. Column has 2 m g y sub pull at the highest point, m g y sub pull at the equilibrium point, and zero at the lowest point. The Elastic   P. E. column has one half k y sub pull squared at the highest point, zero at the equilibrium point, and one half k y sub pull squared at the lowest point. The Kinetic P. E. Column has zero at the highest point, one half m v squared at the equilibrium point, and zero at the lowest point.\">\n<caption><span>Components of Energy in a Vertical Mass-Spring System<\/span><\/caption>\n<thead>\n<tr valign=\"top\">\n<th><\/th>\n<th>Gravitational P.E.<\/th>\n<th>Elastic P.E.<\/th>\n<th>Kinetic E.<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>(3) Highest Point<\/td>\n<td>[latex]2mg{y}_{\\text{pull}}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}k{y}^{2}{}_{\\text{pull}}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>(2) Equilibrium<\/td>\n<td>[latex]mg{y}_{\\text{pull}}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}m{v}^{2}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>(1) Lowest Point<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}k{y}^{2}{}_{\\text{pull}}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_Bungee\">\n<div class=\"bc-figcaption figcaption\">A <span class=\"no-emphasis\">bungee jumper<\/span> transforms gravitational potential energy at the start of the jump into elastic potential energy at the bottom of the jump.<\/div>\n<p><span id=\"fs-id1165036147313\"><img src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_Bungee.jpg\" alt=\"A photograph of a bungee jumper.\"><\/span><\/p><\/div>\n<div id=\"fs-id1165039485752\" class=\"textbox examples\">\n<p id=\"fs-id1165039398773\"><span>Potential Energy of a Vertical Mass-Spring System<\/span><br>\nA block weighing [latex]12\\phantom{\\rule{0.2em}{0ex}}\\text{N}[\/latex] is hung from a spring with a spring constant of [latex]6.0\\phantom{\\rule{0.2em}{0ex}}\\text{N}\\text{\/}\\text{m}[\/latex], as shown in <a href=\"#CNX_UPhysics_08_01_VertM-S\" class=\"autogenerated-content\">(Figure)<\/a>. The block is pulled down an additional [latex]5.0\\phantom{\\rule{0.2em}{0ex}}\\text{cm}[\/latex] from its equilibrium position and released. (a) What is the difference in just the spring potential energy, from an initial equilibrium position to its pulled-down position? (b) What is the difference in just the gravitational potential energy from its initial equilibrium position to its pulled-down position? (c) What is the kinetic energy of the block as it passes through the equilibrium position from its pulled-down position?<\/p>\n<p id=\"fs-id1165036146944\"><span>Strategy<\/span><br>\nIn parts (a) and (b), we want to find a difference in potential energy, so we can use <a href=\"#fs-id1165035743019\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1165035676115\" class=\"autogenerated-content\">(Figure)<\/a>, respectively. Each of these expressions takes into consideration the change in the energy relative to another position, further emphasizing that potential energy is calculated with a reference or second point in mind. By choosing the conventions of the lowest point in the diagram where the gravitational potential energy is zero and the equilibrium position of the spring where the elastic potential energy is zero, these differences in energies can now be calculated. In part (c), we take a look at the differences between the two potential energies. The difference between the two results in kinetic energy, since there is no friction or drag in this system that can take energy from the system.<\/p>\n<p id=\"fs-id1165039237055\"><span>Solution<\/span><\/p>\n<ol id=\"fs-id1165036084542\" type=\"a\">\n<li>Since the gravitational potential energy is zero at the lowest point, the change in gravitational potential energy is\n<div><\/div>\n<div id=\"fs-id1165035648589\" class=\"unnumbered\">[latex]\\text{\u0394}{U}_{\\text{grav}}=mgy-0=\\left(12\\phantom{\\rule{0.2em}{0ex}}\\text{N}\\right)\\left(5.0\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)=0.60\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<\/li>\n<li>The equilibrium position of the spring is defined as zero potential energy. Therefore, the change in elastic potential energy is\n<div><\/div>\n<div id=\"fs-id1165035906727\" class=\"unnumbered\">[latex]\\text{\u0394}{U}_{\\text{elastic}}=0-\\frac{1}{2}k{y}_{\\text{pull}}^{2}=\\text{\u2212}\\left(\\frac{1}{2}\\right)\\left(6.0\\frac{\\text{N}}{\\text{m}}\\right){\\left(5.0\\phantom{\\rule{0.2em}{0ex}}\\text{cm}\\right)}^{2}=-0.75\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<\/li>\n<li>The block started off being pulled downward with a relative potential energy of [latex]0.75\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex] The gravitational potential energy required to rise [latex]5.0\\phantom{\\rule{0.2em}{0ex}}\\text{cm is}\\phantom{\\rule{0.2em}{0ex}}0.60\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex]. The energy remaining at this equilibrium position must be kinetic energy. We can solve for this gain in kinetic energy from <a href=\"#fs-id1165039502050\" class=\"autogenerated-content\">(Figure)<\/a>,\n<div><\/div>\n<div id=\"fs-id1165039440067\" class=\"unnumbered\">[latex]\\text{\u0394}K=\\text{\u2212}\\left(\\text{\u0394}{U}_{\\text{elastic}}+\\text{\u0394}{U}_{\\text{grav}}\\right)=\\text{\u2212}\\left(-0.75\\phantom{\\rule{0.2em}{0ex}}\\text{J}+0.60\\phantom{\\rule{0.2em}{0ex}}\\text{J}\\right)=0.15\\phantom{\\rule{0.2em}{0ex}}\\text{J}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165036080825\"><span>Significance<\/span><br>\nEven though the potential energies are relative to a chosen zero location, the solutions to this problem would be the same if the zero energy points were chosen at different locations.<\/p>\n<\/div>\n<div id=\"fs-id1165035687581\" class=\"check-understanding\">\n<div id=\"fs-id1165039237492\">\n<div id=\"fs-id1165039047234\">\n<p id=\"fs-id1165035867745\"><strong>Check Your Understanding<\/strong> Suppose the mass in <a href=\"#fs-id1165039485752\" class=\"autogenerated-content\">(Figure)<\/a> is in equilibrium, and you pull it down another 3.0 cm, making the pulled-down distance a total of [latex]8.0\\phantom{\\rule{0.2em}{0ex}}\\text{cm}.[\/latex] The elastic potential energy of the spring increases, because you\u2019re stretching it more, but the gravitational potential energy of the mass decreases, because you\u2019re lowering it. Does the total potential energy increase, decrease, or remain the same?<\/p>\n<\/div>\n<div id=\"fs-id1165039422522\">\n<p id=\"fs-id1165039510367\">It increases because you had to exert a downward force, doing positive work, to pull the mass down, and that\u2019s equal to the change in the total potential energy.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039461263\" class=\"media-2\">\n<p id=\"fs-id1165033371194\">View this <a href=\"https:\/\/openstaxcollege.org\/l\/21conenerskat\">simulation<\/a> to learn about conservation of energy with a skater! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy and friction as he moves. You can also take the skater to different planets or even space!<\/p>\n<\/div>\n<p id=\"fs-id1165039078790\">A sample chart of a variety of energies is shown in <a href=\"#fs-id1165036086155\" class=\"autogenerated-content\">(Figure)<\/a> to give you an idea about typical energy values associated with certain events. Some of these are calculated using kinetic energy, whereas others are calculated by using quantities found in a form of potential energy that may not have been discussed at this point.<\/p>\n<table id=\"fs-id1165036086155\" summary=\"The table has two columns and 14 rows. The first row is a header row with headings \u201cObject or Phenomenon\u201d and \u201cEnergy in Joules.\u201d Row 2 has \u201cBig Bang\u201d and 10 to the 68. Row 3 has \u201cAnnual world energy use\u201d and 4.0 times 10 to the 20. Row 4 has \u201cLarge fusion bomb (9 megaton)\u201d and 3.8 times 10 to the 16. Row 5 has \u201cHiroshima-size fission bomb (10 kiloton)\u201d and 4.2 times 10 to the 13. Row  6 has \u201c1 barrel crude oil\u201d and 5.9 times 10 to the 9. Row 7 has \u201c1 ton TNT\u201d and 4.2   times 10 to the 9. Row 8 has \u201c1 gallon of gasoline\u201d  times 10 to the 8. Row 9 has \u201cDaily adult food intake (recommended)\u201d and 1.2 times 10 to the 7. Row 10 \u201c1000 k g car at 90 k m per h\u201d and 3.1 times 10 to the 5. Row 11 has \u201cTennis ball at 100 k m per h\u201d and 22. Row 12 has \u201cMosquito, 10 to the minus 2 gram at 0.5 meters p0er second\u201d and 1.3 times 10 to the minus 6. Row 13 has \u201csingle electron in a T V tune beam\u201d and 4.0 times 10 to the minus 15. Row 14 has \u201cEnergy to break one D N A strand\u201d and 10 to the \u201319.\">\n<caption><span>Energy of Various Objects and Phenomena<\/span><\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Object\/phenomenon<\/th>\n<th>Energy in joules<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Big Bang<\/td>\n<td>[latex]{10}^{68}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Annual world energy use<\/td>\n<td>[latex]4.0\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{20}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Large fusion bomb (9 megaton)<\/td>\n<td>[latex]3.8\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{16}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Hiroshima-size fission bomb (10 kiloton)<\/td>\n<td>[latex]4.2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{13}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1 barrel crude oil<\/td>\n<td>[latex]5.9\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{9}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1 ton TNT<\/td>\n<td>[latex]4.2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{9}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1 gallon of gasoline<\/td>\n<td>[latex]1.2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{8}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Daily adult food intake (recommended)<\/td>\n<td>[latex]1.2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{7}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1000-kg car at 90 km\/h<\/td>\n<td>[latex]3.1\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{5}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Tennis ball at 100 km\/h<\/td>\n<td>[latex]22[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Mosquito [latex]\\left({10}^{-2}\\phantom{\\rule{0.2em}{0ex}}\\text{g at 0.5 m\/s}\\right)[\/latex]<\/td>\n<td>[latex]1.3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-6}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Single electron in a TV tube beam<\/td>\n<td>[latex]4.0\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-15}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Energy to break one DNA strand<\/td>\n<td>[latex]{10}^{-19}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\" id=\"fs-id1165039411801\">\n<h3>Summary<\/h3>\n<ul id=\"fs-id1165039512018\">\n<li>For a single-particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.<\/li>\n<li>Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location.<\/li>\n<li>The potential energies for Earth\u2019s constant gravity, near its surface, and for a Hooke\u2019s law force are linear and quadratic functions of position, respectively.<\/li>\n<\/ul>\n<\/div>\n<div class=\"review-conceptual-questions\" id=\"fs-id1165039349944\">\n<h3>Conceptual Questions<\/h3>\n<div id=\"fs-id1165039088707\">\n<div id=\"fs-id1165035870929\">\n<p id=\"fs-id1165036158227\">The kinetic energy of a system must always be positive or zero. Explain whether this is true for the potential energy of a system.<\/p>\n<\/div>\n<div id=\"fs-id1165036083174\">\n<p id=\"fs-id1165039119722\">The potential energy of a system can be negative because its value is relative to a defined point.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035863734\">\n<div id=\"fs-id1165039495833\">\n<p id=\"fs-id1165035862588\">The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer drives from it, starting just before the swimmer steps on the board until just after his feet leave it.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035664421\">\n<div id=\"fs-id1165036075740\">\n<p id=\"fs-id1165039420504\">Describe the gravitational potential energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown.<\/p>\n<\/div>\n<div id=\"fs-id1165039508390\">\n<p id=\"fs-id1165035639538\">If the reference point of the ground is zero gravitational potential energy, the javelin first increases its gravitational potential energy, followed by a decrease in its gravitational potential energy as it is thrown until it hits the ground. The overall change in gravitational potential energy of the javelin is zero unless the center of mass of the javelin is lower than from where it is initially thrown, and therefore would have slightly less gravitational potential energy.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035609975\">\n<div id=\"fs-id1165035676190\">\n<p id=\"fs-id1165039434233\">A couple of soccer balls of equal mass are kicked off the ground at the same speed but at different angles. Soccer ball A is kicked off at an angle slightly above the horizontal, whereas ball B is kicked slightly below the vertical. How do each of the following compare for ball A and ball B? (a) The initial kinetic energy and (b) the change in gravitational potential energy from the ground to the highest point? If the energy in part (a) differs from part (b), explain why there is a difference between the two energies.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035640115\">\n<div id=\"fs-id1165035971333\">\n<p id=\"fs-id1165039243934\">What is the dominant factor that affects the speed of an object that started from rest down a frictionless incline if the only work done on the object is from gravitational forces?<\/p>\n<\/div>\n<div id=\"fs-id1165039502736\">\n<p id=\"fs-id1165035734500\">the vertical height from the ground to the object<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035951008\">\n<div id=\"fs-id1165039210730\">\n<p id=\"fs-id1165035703366\">Two people observe a leaf falling from a tree. One person is standing on a ladder and the other is on the ground. If each person were to compare the energy of the leaf observed, would each person find the following to be the same or different for the leaf, from the point where it falls off the tree to when it hits the ground: (a) the kinetic energy of the leaf; (b) the change in gravitational potential energy; (c) the final gravitational potential energy?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-problems\" id=\"fs-id1165039426252\">\n<h3>Problems<\/h3>\n<div id=\"fs-id1165035974202\">\n<div id=\"fs-id1165036084222\">\n<p id=\"fs-id1165035707085\">Using values from <a href=\"#fs-id1165036086155\" class=\"autogenerated-content\">(Figure)<\/a>, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous X-rays. Later-model tube TVs had shielding that absorbed X-rays before they escaped and exposed viewers.)<\/p>\n<\/div>\n<div id=\"fs-id1165039303143\">\n<p id=\"fs-id1165039328857\">40,000<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039195103\">\n<div id=\"fs-id1165039274912\">\n<p id=\"fs-id1165039045866\">If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year\u2019s supply of energy (using data from <a href=\"#fs-id1165039443587\" class=\"autogenerated-content\">(Figure)<\/a>)?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035678778\">\n<div id=\"fs-id1165033368936\">\n<p id=\"fs-id1165039506976\">A camera weighing 10 N falls from a small drone hovering [latex]20\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] overhead and enters free fall. What is the gravitational potential energy change of the camera from the drone to the ground if you take a reference point of (a) the ground being zero gravitational potential energy? (b) The drone being zero gravitational potential energy? What is the gravitational potential energy of the camera (c) before it falls from the drone and (d) after the camera lands on the ground if the reference point of zero gravitational potential energy is taken to be a second person looking out of a building [latex]30\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] from the ground?<\/p>\n<\/div>\n<div id=\"fs-id1165039026114\">\n<p id=\"fs-id1165035992436\">[latex]\\text{a.}\\phantom{\\rule{0.2em}{0ex}}-200\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{b.}\\phantom{\\rule{0.2em}{0ex}}-200\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{c.}\\phantom{\\rule{0.2em}{0ex}}-100\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{d.}\\phantom{\\rule{0.2em}{0ex}}-300\\phantom{\\rule{0.2em}{0ex}}\\text{J}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165038995791\">\n<div id=\"fs-id1165038963823\">\n<p id=\"fs-id1165039305629\">Someone drops a [latex]50-\\text{g}[\/latex] pebble off of a docked cruise ship, [latex]70.0\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] from the water line. A person on a dock [latex]3.0\\phantom{\\rule{0.2em}{0ex}}\\text{m}[\/latex] from the water line holds out a net to catch the pebble. (a) How much work is done on the pebble by gravity during the drop? (b) What is the change in the gravitational potential energy during the drop? If the gravitational potential energy is zero at the water line, what is the gravitational potential energy (c) when the pebble is dropped? (d) When it reaches the net? What if the gravitational potential energy was [latex]30.0[\/latex] Joules at water level? (e) Find the answers to the same questions in (c) and (d).<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039000504\">\n<div id=\"fs-id1165036159547\">\n<p id=\"fs-id1165039376541\">A cat\u2019s crinkle ball toy of mass [latex]15\\phantom{\\rule{0.2em}{0ex}}\\text{g}[\/latex] is thrown straight up with an initial speed of [latex]3\\phantom{\\rule{0.2em}{0ex}}\\text{m\/s}[\/latex]. Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball as it leaves the hand? (b) How much work is done by the gravitational force during the ball\u2019s rise to its peak? (c) What is the change in the gravitational potential energy of the ball during the rise to its peak? (d) If the gravitational potential energy is taken to be zero at the point where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy be when it leaves the hand? (f) What is the maximum height the ball reaches?<\/p>\n<\/div>\n<div id=\"fs-id1165035962196\">\n<p id=\"fs-id1165039218494\">[latex]\\text{a.}\\phantom{\\rule{0.2em}{0ex}}0.068\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{b.}\\phantom{\\rule{0.2em}{0ex}}-0.068\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{c.}\\phantom{\\rule{0.2em}{0ex}}0.068\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{d.}\\phantom{\\rule{0.2em}{0ex}}0.068\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{e.}\\phantom{\\rule{0.2em}{0ex}}-0.068\\phantom{\\rule{0.2em}{0ex}}\\text{J};\\phantom{\\rule{0.2em}{0ex}}\\text{f.}\\phantom{\\rule{0.2em}{0ex}}46\\phantom{\\rule{0.2em}{0ex}}\\text{cm}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165039070256\">\n<dt>potential energy<\/dt>\n<dd id=\"fs-id1165035651980\">function of position, energy possessed by an object relative to the system considered<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039314150\">\n<dt>potential energy difference<\/dt>\n<dd id=\"fs-id1165035824264\">negative of the work done acting between two points in space<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Relate the difference of potential energy to work done on a particle for a system without friction or air drag<\/li>\n<li>Explain the meaning of the zero of the potential energy function for a system<\/li>\n<li>Calculate and apply the gravitational potential energy for an object near Earth\u2019s surface and the elastic potential energy of a mass-spring system<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165039051689\">In <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>, we saw that the work done on an object by the constant gravitational force, near the surface of Earth, over any displacement is a function only of the difference in the positions of the end-points of the displacement. This property allows us to define a different kind of energy for the system than its kinetic energy, which is called <span>potential energy<\/span>. We consider various properties and types of potential energy in the following subsections.<\/p>\n<div class=\"bc-section section\" id=\"fs-id1165035692628\">\n<h3>Potential Energy Basics<\/h3>\n<p id=\"fs-id1165035682820\">In <a href=\"\/contents\/305e881e-18f9-4ccd-b96d-e7a1c4db654d\" class=\"target-chapter\">Motion in Two and Three Dimensions<\/a>, we analyzed the motion of a projectile, like kicking a football in <a href=\"#CNX_UPhysics_08_01_football\" class=\"autogenerated-content\">(Figure)<\/a>. For this example, let\u2019s ignore friction and air resistance. As the football rises, the work done by the gravitational force on the football is negative, because the ball\u2019s displacement is positive vertically and the force due to gravity is negative vertically. We also noted that the ball slowed down until it reached its highest point in the motion, thereby decreasing the ball\u2019s kinetic energy. This loss in kinetic energy translates to a gain in gravitational potential energy of the football-Earth system.<\/p>\n<p id=\"fs-id1165035972000\">As the football falls toward Earth, the work done on the football is now positive, because the displacement and the gravitational force both point vertically downward. The ball also speeds up, which indicates an increase in kinetic energy. Therefore, energy is converted from gravitational potential energy back into kinetic energy.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_football\">\n<div class=\"bc-figcaption figcaption\">As a football starts its descent toward the wide receiver, gravitational potential energy is converted back into kinetic energy.<\/div>\n<p><span id=\"fs-id1165035906458\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_football.jpg\" alt=\"An illustration of a football\u2019s trajectory and energy. The kicker kicks the ball, doing work on it and giving it maximum kinetic energy. The potential energy is minimum. This is point one. On the way up, at point two, the kinetic energy of the ball decreases and its potential energy decreases. At the highest point, point three, the kinetic energy of the ball  is at its minimum and its potential energy is maximum. As the ball descends, point four, the kinetic energy increases and the potential energy decreases. The receiver catches the ball at the same height above the ground as it was kicked, at point five. The kinetic energy equals maximum, potential energy is minimum.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1165039251424\">Based on this scenario, we can define the difference of potential energy from point <em>A<\/em> to point <em>B<\/em> as the negative of the work done:<\/p>\n<div id=\"fs-id1165039074593\" class=\"equation-callout\">\n<div id=\"fs-id1165039502282\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b8b81c0ee35b47d481f43c0744a928bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#123;&#85;&#125;&#95;&#123;&#66;&#125;&#45;&#123;&#85;&#125;&#95;&#123;&#65;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#87;&#125;&#95;&#123;&#65;&#66;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"192\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1165039238293\">This formula explicitly states a <span>potential energy difference<\/span>, not just an absolute potential energy. Therefore, we need to define potential energy at a given position in such a way as to state standard values of potential energy on their own, rather than potential energy differences. We do this by rewriting the potential energy function in terms of an arbitrary constant,<\/p>\n<div id=\"fs-id1165039391866\" class=\"equation-callout\">\n<div id=\"fs-id1165039502050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7b3996386909509c994cf0425dfab684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#85;&#61;&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#114;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#114;&#125;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"177\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1165035641022\">The choice of the potential energy at a starting location of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4d3f9537efe8b79f357e54988b05b642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#114;&#125;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -3px;\" \/> is made out of convenience in the given problem. Most importantly, whatever choice is made should be stated and kept consistent throughout the given problem. There are some well-accepted choices of initial potential energy. For example, the lowest height in a problem is usually defined as zero potential energy, or if an object is in space, the farthest point away from the system is often defined as zero potential energy. Then, the potential energy, with respect to zero at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8ecd46067d7737f2e1385419ae7f6d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#114;&#125;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -4px;\" \/> is just <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e582eaa6b44f8970dd5338265269fc1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#114;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"58\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1165039324524\">As long as there is no friction or air resistance, the change in kinetic energy of the football equals the change in gravitational potential energy of the football. This can be generalized to any potential energy:<\/p>\n<div id=\"fs-id1165039099001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d3da366e73a1876de300f9e5bcab4790_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#75;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#65;&#66;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"99\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1165039503553\">Let\u2019s look at a specific example, choosing zero potential energy for gravitational potential energy at convenient points.<\/p>\n<div id=\"fs-id1165035686790\" class=\"textbox examples\">\n<p id=\"fs-id1165035772252\"><span>Basic Properties of Potential Energy<\/span><br \/>\nA particle moves along the <em>x<\/em>-axis under the action of a force given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c446f60b0e67f9c6fcb7868413f091f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: 0px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5763369ba2dcf8e604f039842e554133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"89\" style=\"vertical-align: -4px;\" \/>. (a) What is the difference in its potential energy as it moves from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-933417a5c17907512573a8068c0dc9d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#65;&#125;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"72\" style=\"vertical-align: -3px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7787b503dcac7d807ea00f6042acf121_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#66;&#125;&#61;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"73\" style=\"vertical-align: -3px;\" \/>? (b) What is the particle\u2019s potential energy at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-fc88758930c187e26372c531d74da6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/> with respect to a given 0.5 J of potential energy at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<p id=\"fs-id1165035849830\"><span>Strategy<\/span><br \/>\n(a) The difference in potential energy is the negative of the work done, as defined by <a href=\"#fs-id1165039502282\" class=\"autogenerated-content\">(Figure)<\/a>. The work is defined in the previous chapter as the dot product of the force with the distance. Since the particle is moving forward in the <em>x<\/em>-direction, the dot product simplifies to a multiplication (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b70464034fce9546fe09bb2b0f9a8826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#105;&#125;&middot;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#94;&#125;&#123;&#105;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/>). To find the total work done, we need to integrate the function between the given limits. After integration, we can state the work or the potential energy. (b) The potential energy function, with respect to zero at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, is the indefinite integral encountered in part (a), with the constant of integration determined from <a href=\"#fs-id1165039099001\" class=\"autogenerated-content\">(Figure)<\/a>. Then, we substitute the <em>x<\/em>-value into the function of potential energy to calculate the potential energy at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5ad8e34229b2d44c2ace65da6033f4bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: -1px;\" \/><\/p>\n<p><span>Solution<\/span><\/p>\n<ol type=\"a\">\n<li>The work done by the given force as the particle moves from coordinate <em>x<\/em> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a75b1cad3711f214c3429da4124c80c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#100;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -2px;\" \/> in one dimension is\n<div><\/div>\n<div id=\"fs-id1165039443277\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ae725f5f6a0f2f4a67fdf9025b1a5c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#87;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#70;&#125;&middot;&#100;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#111;&#32;&#125;&#123;&#114;&#125;&#61;&#70;&#100;&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#100;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"220\" style=\"vertical-align: -2px;\" \/><\/div>\n<div><\/div>\n<p>Substituting this expression into <a href=\"#fs-id1165039502282\" class=\"autogenerated-content\">(Figure)<\/a>, we obtain<\/p>\n<div><\/div>\n<div id=\"fs-id1165035633026\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e6d5111ddf2669c06e0cbf31fde85ae8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#85;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#87;&#61;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#115;&#101;&#116;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#92;&#105;&#110;&#116;&#32;&#125;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#100;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#124;&#125;&#95;&#123;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#61;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"354\" style=\"vertical-align: -19px;\" \/><\/div>\n<\/li>\n<li>The indefinite integral for the potential energy function in part (a) is\n<div><\/div>\n<div id=\"fs-id1165035699939\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-76c1776ffc475d08df1d0d8707c69670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#116;&#125;&#46;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"173\" style=\"vertical-align: -6px;\" \/><\/div>\n<div><\/div>\n<p>and we want the constant to be determined by<\/p>\n<div><\/div>\n<div id=\"fs-id1165039462260\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0edd1f4919110eadac83b8acc6d57395_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/div>\n<div><\/div>\n<p>Thus, the potential energy with respect to zero at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> is just<\/p>\n<div><\/div>\n<div id=\"fs-id1165039303300\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c50f6bc949efa037fd7ea56d5603e697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#48;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"162\" style=\"vertical-align: -6px;\" \/><\/div>\n<div><\/div>\n<p>Therefore, the potential energy at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-fc88758930c187e26372c531d74da6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/> is<\/p>\n<div><\/div>\n<div id=\"fs-id1165039078983\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e971b71aeb0737f52dd50b53b11cc795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#43;&#48;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#61;&#49;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"344\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165039187844\"><span>Significance<\/span><br \/>\nIn this one-dimensional example, any function we can integrate, independent of path, is conservative. Notice how we applied the definition of potential energy difference to determine the potential energy function with respect to zero at a chosen point. Also notice that the potential energy, as determined in part (b), at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-fc88758930c187e26372c531d74da6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f56144a02abb27cdf01b3dbee31604fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/> and at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8366c3f553d3c29e817c9254e1b7f5d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-dec0b371d9749d217b0ae9e1e99fada4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/>; their difference is the result in part (a).<\/p>\n<\/div>\n<div id=\"fs-id1165035756834\" class=\"check-understanding\">\n<div id=\"fs-id1165035695796\">\n<div id=\"fs-id1165039288706\">\n<p id=\"fs-id1165035615978\"><strong>Check Your Understanding<\/strong> In <a href=\"#fs-id1165035686790\" class=\"autogenerated-content\">(Figure)<\/a>, what are the potential energies of the particle at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-fc88758930c187e26372c531d74da6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8366c3f553d3c29e817c9254e1b7f5d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/> with respect to zero at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c0add0f18c3db6d2566b9ce06968040f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"75\" style=\"vertical-align: -1px;\" \/>? Verify that the difference of potential energy is still 7 J.<\/p>\n<\/div>\n<div id=\"fs-id1165035924241\">\n<p id=\"fs-id1165039493913\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-22a88a9d28fcb103293951b7384dce26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#54;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#46;&#51;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#46;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"218\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1165039107955\">\n<h3>Systems of Several Particles<\/h3>\n<p>In general, a system of interest could consist of several particles. The difference in the potential energy of the system is the negative of the work done by gravitational or elastic forces, which, as we will see in the next section, are conservative forces. The potential energy difference depends only on the initial and final positions of the particles, and on some parameters that characterize the interaction (like mass for gravity or the spring constant for a Hooke\u2019s law force).<\/p>\n<p id=\"fs-id1165039471899\">It is important to remember that potential energy is a property of the interactions between objects in a chosen system, and not just a property of each object. This is especially true for electric forces, although in the examples of potential energy we consider below, parts of the system are either so big (like Earth, compared to an object on its surface) or so small (like a massless spring), that the changes those parts undergo are negligible if included in the system.<\/p>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Types of Potential Energy<\/h3>\n<p id=\"fs-id1165039045528\">For each type of interaction present in a system, you can label a corresponding type of potential energy. The total potential energy of the system is the sum of the potential energies of all the types. (This follows from the additive property of the dot product in the expression for the work done.) Let\u2019s look at some specific examples of types of potential energy discussed in <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>. First, we consider each of these forces when acting separately, and then when both act together.<\/p>\n<div class=\"bc-section section\" id=\"fs-id1165039369440\">\n<h4>Gravitational potential energy near Earth\u2019s surface<\/h4>\n<p id=\"fs-id1165039285376\">The system of interest consists of our planet, Earth, and one or more particles near its surface (or bodies small enough to be considered as particles, compared to Earth). The gravitational force on each particle (or body) is just its weight <em>mg<\/em> near the surface of Earth, acting vertically down. According to Newton\u2019s third law, each particle exerts a force on Earth of equal magnitude but in the opposite direction. Newton\u2019s second law tells us that the magnitude of the acceleration produced by each of these forces on Earth is <em>mg<\/em> divided by Earth\u2019s mass. Since the ratio of the mass of any ordinary object to the mass of Earth is vanishingly small, the motion of Earth can be completely neglected. Therefore, we consider this system to be a group of single-particle systems, subject to the uniform gravitational force of Earth.<\/p>\n<p id=\"fs-id1165039209828\">In <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>, the work done on a body by Earth\u2019s uniform gravitational force, near its surface, depended on the mass of the body, the acceleration due to gravity, and the difference in height the body traversed, as given by <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796#fs-id1165035740928\" class=\"autogenerated-content\">(Figure)<\/a>. By definition, this work is the negative of the difference in the gravitational potential energy, so that difference is<\/p>\n<div id=\"fs-id1165035676115\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0c546f8331658630daf33493491701a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#114;&#97;&#118;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#87;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#114;&#97;&#118;&#125;&#44;&#65;&#66;&#125;&#61;&#109;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#66;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#65;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"264\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1165039090814\">You can see from this that the <span class=\"no-emphasis\">gravitational potential energy<\/span> function, near Earth\u2019s surface, is<\/p>\n<div id=\"fs-id1165039084074\" class=\"equation-callout\">\n<div id=\"fs-id1165039441699\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1c99a570db3d9aaa9705a1e08562490f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#109;&#103;&#121;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#116;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1165039107630\">You can choose the value of the constant, as described in the discussion of <a href=\"#fs-id1165039502050\" class=\"autogenerated-content\">(Figure)<\/a>; however, for solving most problems, the most convenient constant to choose is zero for when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-cfd9b205a64e8a430ee45bde3912fd46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> which is the lowest vertical position in the problem.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_jumper\">\n<div class=\"bc-figcaption figcaption\">Don\u2019t jump\u2014you have so much potential (gravitational potential energy, that is). (credit: Andy Spearing)<\/div>\n<p><span id=\"fs-id1165039242348\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_jumper.jpg\" alt=\"A photograph of a person jumping off a high cliff into a lake.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165038964832\" class=\"textbox examples\">\n<p id=\"fs-id1165035785797\"><span>Gravitational Potential Energy of a Hiker<\/span><br \/>\nThe summit of Great Blue Hill in Milton, MA, is 147 m above its base and has an elevation above sea level of 195 m (<a href=\"#CNX_UPhysics_08_01_GrBlueHill\" class=\"autogenerated-content\">(Figure)<\/a>). (Its Native American name, <em>Massachusett<\/em>, was adopted by settlers for naming the Bay Colony and state near its location.) A 75-kg hiker ascends from the base to the summit. What is the gravitational potential energy of the hiker-Earth system with respect to zero gravitational potential energy at base height, when the hiker is (a) at the base of the hill, (b) at the summit, and (c) at sea level, afterward?<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_GrBlueHill\">\n<div class=\"bc-figcaption figcaption\">Sketch of the profile of Great Blue Hill, Milton, MA. The altitudes of the three levels are indicated.<\/div>\n<p><span id=\"fs-id1165039085986\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_GrBlueHill.jpg\" alt=\"Sketch of the profile of Great Blue Hill, Milton, MA. The summit is 195 meters above sea level. The base of the hill is 147 meters below the summit.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1165039302945\"><span>Strategy<\/span><br \/>\nFirst, we need to pick an origin for the <em>y<\/em>-axis and then determine the value of the constant that makes the potential energy zero at the height of the base. Then, we can determine the potential energies from <a href=\"#fs-id1165039441699\" class=\"autogenerated-content\">(Figure)<\/a>, based on the relationship between the zero potential energy height and the height at which the hiker is located.<\/p>\n<p id=\"fs-id1165039075284\"><span>Solution<\/span><\/p>\n<ol id=\"fs-id1165035618779\" type=\"a\">\n<li>Let\u2019s choose the origin for the <em>y<\/em>-axis at base height, where we also want the zero of potential energy to be. This choice makes the constant equal to zero and\n<div><\/div>\n<div id=\"fs-id1165035704066\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6d392afb5d3933dc89fb35f83d88edb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#97;&#115;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li>At the summit, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-3d0938d9a818864367f7a91f13a7cadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#52;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -4px;\" \/>, so\n<div><\/div>\n<div id=\"fs-id1165039092156\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-6a53b778beb47913326190f776eb2287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#109;&#109;&#105;&#116;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#109;&#103;&#104;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#46;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#48;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"485\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li>At sea level, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-11a3d8424913eb2e0ab5df023f9fb4eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#55;&#45;&#49;&#57;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#61;&#45;&#52;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"213\" style=\"vertical-align: -4px;\" \/>, so\n<div><\/div>\n<div id=\"fs-id1165039297023\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c1c2c109939fac07ccf6ecfc4e072319_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#97;&#45;&#108;&#101;&#118;&#101;&#108;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#46;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#53;&#46;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"357\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165039442596\"><span>Significance<\/span><br \/>\nBesides illustrating the use of <a href=\"#fs-id1165035676115\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1165039441699\" class=\"autogenerated-content\">(Figure)<\/a>, the values of gravitational potential energy we found are reasonable. The gravitational potential energy is higher at the summit than at the base, and lower at sea level than at the base. Gravity does work on you on your way up, too! It does negative work and not quite as much (in magnitude), as your muscles do. But it certainly does work. Similarly, your muscles do work on your way down, as negative work. The numerical values of the potential energies depend on the choice of zero of potential energy, but the physically meaningful differences of potential energy do not. [Note that since <a href=\"#fs-id1165039502050\" class=\"autogenerated-content\">(Figure)<\/a> is a difference, the numerical values do not depend on the origin of coordinates.]<\/p>\n<\/div>\n<div id=\"fs-id1165036083411\" class=\"check-understanding\">\n<div id=\"fs-id1165035926252\">\n<div id=\"fs-id1165033367269\">\n<p><strong>Check Your Understanding<\/strong> What are the values of the gravitational potential energy of the hiker at the base, summit, and sea level, with respect to a sea-level zero of potential energy?<\/p>\n<\/div>\n<div id=\"fs-id1165035849800\">\n<p id=\"fs-id1165036156053\">35.3 kJ, 143 kJ, 0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1165039266683\">\n<h4>Elastic potential energy<\/h4>\n<p id=\"fs-id1165036159199\">In <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796\" class=\"target-chapter\">Work<\/a>, we saw that the work done by a perfectly elastic spring, in one dimension, depends only on the spring constant and the squares of the displacements from the unstretched position, as given in <a href=\"\/contents\/062e941d-8793-4fe6-8857-ea4285163796#fs-id1165039209368\" class=\"autogenerated-content\">(Figure)<\/a>. This work involves only the properties of a Hooke\u2019s law interaction and not the properties of real springs and whatever objects are attached to them. Therefore, we can define the difference of <span class=\"no-emphasis\">elastic potential energy<\/span> for a spring force as the negative of the work done by the spring force in this equation, before we consider systems that embody this type of force. Thus,<\/p>\n<div id=\"fs-id1165035743019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-782178e995628e7aa0e002d5a747e98a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#85;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#87;&#125;&#95;&#123;&#65;&#66;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#66;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#65;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"211\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1165036082561\">where the object travels from point <em>A<\/em> to point <em>B<\/em>. The potential energy function corresponding to this difference is<\/p>\n<div id=\"fs-id1165039074371\" class=\"equation-callout\">\n<div id=\"fs-id1165039398969\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a7ea5f2645bffe1c9bffb620fca21cfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#116;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"168\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1165039399552\">If the spring force is the only force acting, it is simplest to take the zero of potential energy at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, when the spring is at its unstretched length. Then, the constant is <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a> is zero. (Other choices may be more convenient if other forces are acting.)<\/p>\n<div id=\"fs-id1165035980783\" class=\"textbox examples\">\n<p id=\"fs-id1165035618194\"><span>Spring Potential Energy<\/span><br \/>\nA system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4 N\/cm. (a) How much elastic potential energy does the spring contribute when its length is 23 cm? (b) How much more potential energy does it contribute if its length increases to 26 cm?<\/p>\n<p id=\"fs-id1165036152598\"><span>Strategy<\/span><br \/>\nWhen the spring is at its unstretched length, it contributes nothing to the potential energy of the system, so we can use <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a> with the constant equal to zero. The value of <em>x<\/em> is the length minus the unstretched length. When the spring is expanded, the spring\u2019s displacement or difference between its relaxed length and stretched length should be used for the <em>x<\/em>-value in calculating the potential energy of the spring.<\/p>\n<p id=\"fs-id1165039336747\"><span>Solution<\/span><\/p>\n<ol id=\"fs-id1165039450221\" type=\"a\">\n<li>The displacement of the spring is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-3867469f9eacdcdaeaaa4c31b7bcd066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#45;&#50;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#61;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"202\" style=\"vertical-align: 0px;\" \/>, so the contributed potential energy is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-632dd528663ba7644d2f2853a29d3277_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;&#46;&#49;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"311\" style=\"vertical-align: -6px;\" \/>.<\/li>\n<li>When the spring\u2019s displacement is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2ee645543f1854c88c84a66fe865d0fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#45;&#50;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#61;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"202\" style=\"vertical-align: 0px;\" \/>, the potential energy is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-2fb123409a5f6c3c9001bba12e05a598_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;&#46;&#55;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"311\" style=\"vertical-align: -6px;\" \/>, which is a 0.54-J increase over the amount in part (a).<\/li>\n<\/ol>\n<p id=\"fs-id1165039096164\"><span>Significance<\/span><br \/>\nCalculating the elastic potential energy and potential energy differences from <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a> involves solving for the potential energies based on the given lengths of the spring. Since <em>U<\/em> depends on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>, the potential energy for a compression (negative <em>x<\/em>) is the same as for an extension of equal magnitude.<\/p>\n<\/div>\n<div id=\"fs-id1165035834925\" class=\"check-understanding\">\n<div id=\"fs-id1165039472655\">\n<div id=\"fs-id1165035712092\">\n<p id=\"fs-id1165035944692\"><strong>Check Your Understanding<\/strong> When the length of the spring in <a href=\"#fs-id1165035980783\" class=\"autogenerated-content\">(Figure)<\/a> changes from an initial value of 22.0 cm to a final value, the elastic potential energy it contributes changes by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-957c13f245a9b923c0bcdebd4279f829_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#48;&#56;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: 0px;\" \/> Find the final length.<\/p>\n<\/div>\n<div id=\"fs-id1165035723425\">\n<p id=\"fs-id1165036153427\">22.8 cm. Using 0.02 m for the initial displacement of the spring (see above), we calculate the final displacement of the spring to be 0.028 m; therefore the length of the spring is the unstretched length plus the displacement, or 22.8 cm.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" id=\"fs-id1165035939827\">\n<h4>Gravitational and elastic potential energy<\/h4>\n<p id=\"fs-id1165035687891\">A simple system embodying both gravitational and elastic types of potential energy is a one-dimensional, vertical <span class=\"no-emphasis\">mass-spring system<\/span>. This consists of a massive particle (or block), hung from one end of a perfectly elastic, massless spring, the other end of which is fixed, as illustrated in <a href=\"#CNX_UPhysics_08_01_VertM-S\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_VertM-S\">\n<div class=\"bc-figcaption figcaption\">A vertical mass-spring system, with the <em>y<\/em>-axis pointing upwards. The mass is initially at an equilibrium position and pulled downward to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-897aa61e8fc215cd63530608701be9f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#117;&#108;&#108;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"36\" style=\"vertical-align: -6px;\" \/> An oscillation begins, centered at the equilibrium position.<\/div>\n<p><span id=\"fs-id1165039475145\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_VertM-S.jpg\" alt=\"A vertical mass spring system is illustrated. The upper end of the spring is attached to the ceiling. A block of mass m is attached to the lower end.  The spring is drawn at two positions. On the left, the mass is in the equilibrium position. To  the right of this, the spring is drawn with the mass pulled down a distance y sub pull. This position of the mass is labeled as h equal to zero. A graph of y as a function of X is shown to the rightly the illustrations, with y equals zero aligned with the equilibrium position in the illustrations. The plot is sinusoidal, with the minimum y at x=0 and even with the lower mass position in the illustrations.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1165039472389\">First, let\u2019s consider the potential energy of the system. Assuming the spring is massless, the system of the block and Earth gains and loses potential energy. We need to define the constant in the potential energy function of <a href=\"#fs-id1165039441699\" class=\"autogenerated-content\">(Figure)<\/a>. Often, the ground is a suitable choice for when the gravitational potential energy is zero; however, in this case, the lowest point or when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-f4498f0ceb1b7ca6b196ab84e449c19b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/> is a convenient location for zero gravitational potential energy. Note that this choice is arbitrary, and the problem can be solved correctly even if another choice is picked.<\/p>\n<p id=\"fs-id1165039400355\">We must also define the elastic potential energy of the system and the corresponding constant, as detailed in <a href=\"#fs-id1165039398969\" class=\"autogenerated-content\">(Figure)<\/a>. The equilibrium location is the most suitable mathematically to choose for where the potential energy of the spring is zero.<\/p>\n<p>Therefore, based on this convention, each potential energy and kinetic energy can be written out for three critical points of the system: (1) the lowest pulled point, (2) the equilibrium position of the spring, and (3) the highest point achieved. We note that the total energy of the system is conserved, so any total energy in this chart could be matched up to solve for an unknown quantity. The results are shown in <a href=\"#fs-id1165039443587\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"fs-id1165039443587\" summary=\"The table has four rows and four columns. The first row and first column are a header row and column. The column headings are Gravitational P. E., Elastic P. E., and Kinetic P. E. The row headings are (3) Highest Point, (2) Equilibrium, and  (3) Lowest Point. The Gravitational P. E. Column has 2 m g y sub pull at the highest point, m g y sub pull at the equilibrium point, and zero at the lowest point. The Elastic   P. E. column has one half k y sub pull squared at the highest point, zero at the equilibrium point, and one half k y sub pull squared at the lowest point. The Kinetic P. E. Column has zero at the highest point, one half m v squared at the equilibrium point, and zero at the lowest point.\">\n<caption><span>Components of Energy in a Vertical Mass-Spring System<\/span><\/caption>\n<thead>\n<tr valign=\"top\">\n<th><\/th>\n<th>Gravitational P.E.<\/th>\n<th>Elastic P.E.<\/th>\n<th>Kinetic E.<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>(3) Highest Point<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-48d94a499201ccbf74e45c1c035a22ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#109;&#103;&#123;&#121;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#117;&#108;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d13f7bf25f823fae982082e97de9bffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#117;&#108;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>(2) Equilibrium<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-37d1f8786f091968221a3e1c511180fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#103;&#123;&#121;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#117;&#108;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a5ad94aa83a58eb646f5cf16230ee64d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#109;&#123;&#118;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>(1) Lowest Point<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-d13f7bf25f823fae982082e97de9bffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#117;&#108;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"bc-figure figure\" id=\"CNX_UPhysics_08_01_Bungee\">\n<div class=\"bc-figcaption figcaption\">A <span class=\"no-emphasis\">bungee jumper<\/span> transforms gravitational potential energy at the start of the jump into elastic potential energy at the bottom of the jump.<\/div>\n<p><span id=\"fs-id1165036147313\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-content\/uploads\/sites\/289\/2017\/11\/CNX_UPhysics_08_01_Bungee.jpg\" alt=\"A photograph of a bungee jumper.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165039485752\" class=\"textbox examples\">\n<p id=\"fs-id1165039398773\"><span>Potential Energy of a Vertical Mass-Spring System<\/span><br \/>\nA block weighing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-237bf38e69b6ef6b2d928730ac9c3626_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"33\" style=\"vertical-align: -1px;\" \/> is hung from a spring with a spring constant of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-425ba534fce5baa8a25d7c80ee44aa20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/>, as shown in <a href=\"#CNX_UPhysics_08_01_VertM-S\" class=\"autogenerated-content\">(Figure)<\/a>. The block is pulled down an additional <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-4f767f01a1e253abda18a3c24eaddefb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> from its equilibrium position and released. (a) What is the difference in just the spring potential energy, from an initial equilibrium position to its pulled-down position? (b) What is the difference in just the gravitational potential energy from its initial equilibrium position to its pulled-down position? (c) What is the kinetic energy of the block as it passes through the equilibrium position from its pulled-down position?<\/p>\n<p id=\"fs-id1165036146944\"><span>Strategy<\/span><br \/>\nIn parts (a) and (b), we want to find a difference in potential energy, so we can use <a href=\"#fs-id1165035743019\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1165035676115\" class=\"autogenerated-content\">(Figure)<\/a>, respectively. Each of these expressions takes into consideration the change in the energy relative to another position, further emphasizing that potential energy is calculated with a reference or second point in mind. By choosing the conventions of the lowest point in the diagram where the gravitational potential energy is zero and the equilibrium position of the spring where the elastic potential energy is zero, these differences in energies can now be calculated. In part (c), we take a look at the differences between the two potential energies. The difference between the two results in kinetic energy, since there is no friction or drag in this system that can take energy from the system.<\/p>\n<p id=\"fs-id1165039237055\"><span>Solution<\/span><\/p>\n<ol id=\"fs-id1165036084542\" type=\"a\">\n<li>Since the gravitational potential energy is zero at the lowest point, the change in gravitational potential energy is\n<div><\/div>\n<div id=\"fs-id1165035648589\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-35671bd13e830e38d0693431fde332cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#114;&#97;&#118;&#125;&#125;&#61;&#109;&#103;&#121;&#45;&#48;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;&#54;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"335\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li>The equilibrium position of the spring is defined as zero potential energy. Therefore, the change in elastic potential energy is\n<div><\/div>\n<div id=\"fs-id1165035906727\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-58aae48ee1f53ab381d5462095e3b905_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#97;&#115;&#116;&#105;&#99;&#125;&#125;&#61;&#48;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#107;&#123;&#121;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#117;&#108;&#108;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#46;&#48;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#48;&#46;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"426\" style=\"vertical-align: -8px;\" \/><\/div>\n<\/li>\n<li>The block started off being pulled downward with a relative potential energy of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a9af749040c5b2bb4f3f615bff093faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: 0px;\" \/> The gravitational potential energy required to rise <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-75997ff865580d02be31f56aba3da19f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#54;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"114\" style=\"vertical-align: 0px;\" \/>. The energy remaining at this equilibrium position must be kinetic energy. We can solve for this gain in kinetic energy from <a href=\"#fs-id1165039502050\" class=\"autogenerated-content\">(Figure)<\/a>,\n<div><\/div>\n<div id=\"fs-id1165039440067\" class=\"unnumbered\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-fe4d4060a92626a97a450fc3961ba1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#75;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#97;&#115;&#116;&#105;&#99;&#125;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#123;&#85;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#114;&#97;&#118;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#43;&#48;&#46;&#54;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"404\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165036080825\"><span>Significance<\/span><br \/>\nEven though the potential energies are relative to a chosen zero location, the solutions to this problem would be the same if the zero energy points were chosen at different locations.<\/p>\n<\/div>\n<div id=\"fs-id1165035687581\" class=\"check-understanding\">\n<div id=\"fs-id1165039237492\">\n<div id=\"fs-id1165039047234\">\n<p id=\"fs-id1165035867745\"><strong>Check Your Understanding<\/strong> Suppose the mass in <a href=\"#fs-id1165039485752\" class=\"autogenerated-content\">(Figure)<\/a> is in equilibrium, and you pull it down another 3.0 cm, making the pulled-down distance a total of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-ddf8b059c7706ba0a7081e14164b796c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\" \/> The elastic potential energy of the spring increases, because you\u2019re stretching it more, but the gravitational potential energy of the mass decreases, because you\u2019re lowering it. Does the total potential energy increase, decrease, or remain the same?<\/p>\n<\/div>\n<div id=\"fs-id1165039422522\">\n<p id=\"fs-id1165039510367\">It increases because you had to exert a downward force, doing positive work, to pull the mass down, and that\u2019s equal to the change in the total potential energy.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039461263\" class=\"media-2\">\n<p id=\"fs-id1165033371194\">View this <a href=\"https:\/\/openstaxcollege.org\/l\/21conenerskat\">simulation<\/a> to learn about conservation of energy with a skater! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy and friction as he moves. You can also take the skater to different planets or even space!<\/p>\n<\/div>\n<p id=\"fs-id1165039078790\">A sample chart of a variety of energies is shown in <a href=\"#fs-id1165036086155\" class=\"autogenerated-content\">(Figure)<\/a> to give you an idea about typical energy values associated with certain events. Some of these are calculated using kinetic energy, whereas others are calculated by using quantities found in a form of potential energy that may not have been discussed at this point.<\/p>\n<table id=\"fs-id1165036086155\" summary=\"The table has two columns and 14 rows. The first row is a header row with headings \u201cObject or Phenomenon\u201d and \u201cEnergy in Joules.\u201d Row 2 has \u201cBig Bang\u201d and 10 to the 68. Row 3 has \u201cAnnual world energy use\u201d and 4.0 times 10 to the 20. Row 4 has \u201cLarge fusion bomb (9 megaton)\u201d and 3.8 times 10 to the 16. Row 5 has \u201cHiroshima-size fission bomb (10 kiloton)\u201d and 4.2 times 10 to the 13. Row  6 has \u201c1 barrel crude oil\u201d and 5.9 times 10 to the 9. Row 7 has \u201c1 ton TNT\u201d and 4.2   times 10 to the 9. Row 8 has \u201c1 gallon of gasoline\u201d  times 10 to the 8. Row 9 has \u201cDaily adult food intake (recommended)\u201d and 1.2 times 10 to the 7. Row 10 \u201c1000 k g car at 90 k m per h\u201d and 3.1 times 10 to the 5. Row 11 has \u201cTennis ball at 100 k m per h\u201d and 22. Row 12 has \u201cMosquito, 10 to the minus 2 gram at 0.5 meters p0er second\u201d and 1.3 times 10 to the minus 6. Row 13 has \u201csingle electron in a T V tune beam\u201d and 4.0 times 10 to the minus 15. Row 14 has \u201cEnergy to break one D N A strand\u201d and 10 to the \u201319.\">\n<caption><span>Energy of Various Objects and Phenomena<\/span><\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Object\/phenomenon<\/th>\n<th>Energy in joules<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Big Bang<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-3f594c92da61a87a1d61d965e7371344_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#54;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Annual world energy use<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e38d398de5869c54b459edd23706cfaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Large fusion bomb (9 megaton)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-bafba825c3d49c563aa14c3cdb0d2851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Hiroshima-size fission bomb (10 kiloton)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-738edc1b61ecb16b5e5316b4d68ecef2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1 barrel crude oil<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-24746d99592fa58e6ff4570f6d6778ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1 ton TNT<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0252802a02e532c99c910d292c02d710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1 gallon of gasoline<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-7ed44aa22a2152b41a7a8368f9274976_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Daily adult food intake (recommended)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e60b19dab16f292d7a7755216cf92206_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1000-kg car at 90 km\/h<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-c81ec2c59dd59df6ce6798373d8bac0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Tennis ball at 100 km\/h<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-01cb55a185cee9b8f0aa311ce70a4b82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Mosquito <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-048f95888ab1b44fd94e4cc5ac5a2b84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#32;&#97;&#116;&#32;&#48;&#46;&#53;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"147\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-a8a415a59c8e6cdeee375761a67d9bce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Single electron in a TV tube beam<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-0d54449fe6b152d696038868f3d267c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Energy to break one DNA strand<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5755be2c379ff331b0a9e982eead370b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\" id=\"fs-id1165039411801\">\n<h3>Summary<\/h3>\n<ul id=\"fs-id1165039512018\">\n<li>For a single-particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.<\/li>\n<li>Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location.<\/li>\n<li>The potential energies for Earth\u2019s constant gravity, near its surface, and for a Hooke\u2019s law force are linear and quadratic functions of position, respectively.<\/li>\n<\/ul>\n<\/div>\n<div class=\"review-conceptual-questions\" id=\"fs-id1165039349944\">\n<h3>Conceptual Questions<\/h3>\n<div id=\"fs-id1165039088707\">\n<div id=\"fs-id1165035870929\">\n<p id=\"fs-id1165036158227\">The kinetic energy of a system must always be positive or zero. Explain whether this is true for the potential energy of a system.<\/p>\n<\/div>\n<div id=\"fs-id1165036083174\">\n<p id=\"fs-id1165039119722\">The potential energy of a system can be negative because its value is relative to a defined point.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035863734\">\n<div id=\"fs-id1165039495833\">\n<p id=\"fs-id1165035862588\">The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer drives from it, starting just before the swimmer steps on the board until just after his feet leave it.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035664421\">\n<div id=\"fs-id1165036075740\">\n<p id=\"fs-id1165039420504\">Describe the gravitational potential energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown.<\/p>\n<\/div>\n<div id=\"fs-id1165039508390\">\n<p id=\"fs-id1165035639538\">If the reference point of the ground is zero gravitational potential energy, the javelin first increases its gravitational potential energy, followed by a decrease in its gravitational potential energy as it is thrown until it hits the ground. The overall change in gravitational potential energy of the javelin is zero unless the center of mass of the javelin is lower than from where it is initially thrown, and therefore would have slightly less gravitational potential energy.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035609975\">\n<div id=\"fs-id1165035676190\">\n<p id=\"fs-id1165039434233\">A couple of soccer balls of equal mass are kicked off the ground at the same speed but at different angles. Soccer ball A is kicked off at an angle slightly above the horizontal, whereas ball B is kicked slightly below the vertical. How do each of the following compare for ball A and ball B? (a) The initial kinetic energy and (b) the change in gravitational potential energy from the ground to the highest point? If the energy in part (a) differs from part (b), explain why there is a difference between the two energies.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035640115\">\n<div id=\"fs-id1165035971333\">\n<p id=\"fs-id1165039243934\">What is the dominant factor that affects the speed of an object that started from rest down a frictionless incline if the only work done on the object is from gravitational forces?<\/p>\n<\/div>\n<div id=\"fs-id1165039502736\">\n<p id=\"fs-id1165035734500\">the vertical height from the ground to the object<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035951008\">\n<div id=\"fs-id1165039210730\">\n<p id=\"fs-id1165035703366\">Two people observe a leaf falling from a tree. One person is standing on a ladder and the other is on the ground. If each person were to compare the energy of the leaf observed, would each person find the following to be the same or different for the leaf, from the point where it falls off the tree to when it hits the ground: (a) the kinetic energy of the leaf; (b) the change in gravitational potential energy; (c) the final gravitational potential energy?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-problems\" id=\"fs-id1165039426252\">\n<h3>Problems<\/h3>\n<div id=\"fs-id1165035974202\">\n<div id=\"fs-id1165036084222\">\n<p id=\"fs-id1165035707085\">Using values from <a href=\"#fs-id1165036086155\" class=\"autogenerated-content\">(Figure)<\/a>, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous X-rays. Later-model tube TVs had shielding that absorbed X-rays before they escaped and exposed viewers.)<\/p>\n<\/div>\n<div id=\"fs-id1165039303143\">\n<p id=\"fs-id1165039328857\">40,000<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039195103\">\n<div id=\"fs-id1165039274912\">\n<p id=\"fs-id1165039045866\">If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year\u2019s supply of energy (using data from <a href=\"#fs-id1165039443587\" class=\"autogenerated-content\">(Figure)<\/a>)?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165035678778\">\n<div id=\"fs-id1165033368936\">\n<p id=\"fs-id1165039506976\">A camera weighing 10 N falls from a small drone hovering <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-de14950f4e1f77f370bd31fb44d9fca5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/> overhead and enters free fall. What is the gravitational potential energy change of the camera from the drone to the ground if you take a reference point of (a) the ground being zero gravitational potential energy? (b) The drone being zero gravitational potential energy? What is the gravitational potential energy of the camera (c) before it falls from the drone and (d) after the camera lands on the ground if the reference point of zero gravitational potential energy is taken to be a second person looking out of a building <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5bb765390d3e371ddd7d29d4278eddd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/> from the ground?<\/p>\n<\/div>\n<div id=\"fs-id1165039026114\">\n<p id=\"fs-id1165035992436\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-5ebcebbcbc9b9f33a62579b6261658c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"348\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165038995791\">\n<div id=\"fs-id1165038963823\">\n<p id=\"fs-id1165039305629\">Someone drops a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-1bbd0e0b1f0232a23d9b23cb26882b12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -3px;\" \/> pebble off of a docked cruise ship, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-abb2a66506ed28fdb944b8dd37c6956d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#48;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: 0px;\" \/> from the water line. A person on a dock <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-e80f53bc4a60168e4ef3a5b2b19f963b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> from the water line holds out a net to catch the pebble. (a) How much work is done on the pebble by gravity during the drop? (b) What is the change in the gravitational potential energy during the drop? If the gravitational potential energy is zero at the water line, what is the gravitational potential energy (c) when the pebble is dropped? (d) When it reaches the net? What if the gravitational potential energy was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-64a071aea410471d8ab8aef2e445475f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> Joules at water level? (e) Find the answers to the same questions in (c) and (d).<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039000504\">\n<div id=\"fs-id1165036159547\">\n<p id=\"fs-id1165039376541\">A cat\u2019s crinkle ball toy of mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-62d5a04db0571bebaa167bd4f67a4988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"29\" style=\"vertical-align: -3px;\" \/> is thrown straight up with an initial speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-409340573e2052fefbad897c083654cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/>. Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball as it leaves the hand? (b) How much work is done by the gravitational force during the ball\u2019s rise to its peak? (c) What is the change in the gravitational potential energy of the ball during the rise to its peak? (d) If the gravitational potential energy is taken to be zero at the point where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy be when it leaves the hand? (f) What is the maximum height the ball reaches?<\/p>\n<\/div>\n<div id=\"fs-id1165035962196\">\n<p id=\"fs-id1165039218494\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-content\/ql-cache\/quicklatex.com-39976dc6a83fa083af54fdca6583481c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#48;&#54;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#48;&#46;&#48;&#54;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#48;&#54;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#48;&#54;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#48;&#46;&#48;&#54;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#59;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"510\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165039070256\">\n<dt>potential energy<\/dt>\n<dd id=\"fs-id1165035651980\">function of position, energy possessed by an object relative to the system considered<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039314150\">\n<dt>potential energy difference<\/dt>\n<dd id=\"fs-id1165035824264\">negative of the work done acting between two points in space<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-427","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":418,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/427","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/427\/revisions"}],"predecessor-version":[{"id":428,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/427\/revisions\/428"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/parts\/418"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapters\/427\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/media?parent=427"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/chapter-type?post=427"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/contributor?post=427"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/license?post=427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}