Since kinetic energy can never be negative, there is a maximum potential energy and a maximum height, which an object with the given total energy cannot exceed: If we use the gravitational potential energy reference point of zero at y0,y0, we can rewrite the gravitational potential energy U as mgy. Find x(t) for a particle moving with a constant mechanical energy E > 0 and a potential energy U(x) = \(\frac{1}{2}\)kx2, when the particle starts from rest at time t = 0. For a spring-mass system, the graph will be a parabola as it depends on the square of the position. Example: Look at the given picture below. We will simplify our procedure for one-dimensional motion only. The second derivative. First, lets look at an object, freely falling vertically, near the surface of Earth, in the absence of air resistance. The maximum speed v0 gives the initial velocity necessary to reach ymax, the maximum height, and v0 represents the final velocity, after falling from ymax. You can see how the total energy is divided between kinetic and potential energy as the objects height changes. The ___ is the negative of the slope when we visualize the potential energy as a function of the object's position in a graph. Science Physics The graph below shows the potential energy U of a system as one object in the system moves along the x-axis and the rest of the system does not move. The particle in this example can oscillate in the allowed region about either of the two stable equilibrium points we found, but it does not have enough energy to escape from whichever potential well it happens to initially be in. On either side of stable equilibrium points, there is a force that points back to equilibrium. First, lets look at an object, freely falling vertically, near the surface of Earth, in the absence of air resistance. Jun 29, 2022 OpenStax. Imagine the marble has been displaced by a few centimeters on a flat, horizontal surface for an example of this. 6 - Potential energy as a function of the position to find equilibrium points. Legal. Questions [edit | edit source] 1. The line at energy E represents the constant mechanical energy of the object, whereas the kinetic and potential energies, KAKA and UA,UA, are indicated at a particular height yA.yA. They both have a height from the ground and because of their positions they have energy or potential to do work. (a) Is the motion of the particle confined to any regions on the x-axis, and if so, what are they? OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. Rotational Kinetic Energy Calculator. By compressing the spring or stretching it you load a potential energy to it. Sign up to highlight and take notes. We follow the same steps as we did in Example 8.9. For example, a stretched spring, when released, starts moving towards its natural position and starts acquiring speed. 1999-2022, Rice University. Points of unstable equilibrium are located in a potential energy graph as local maximums. The line at energy E represents the constant mechanical energy of the object, whereas the kinetic and potential energies, [latex]{K}_{A}[/latex] and [latex]{U}_{A},[/latex] are indicated at a particular height [latex]{y}_{A}. [/latex] Do this part of the problem for each reference point. Period Pendulum (Pendulum Length) Period Pendulum Calculator (Pendulum Period) Poisson's Ratio Calculator (Modulus) Poisson's Ratio Calculator (Strain) Potential Energy Calculator. Potential energy comes in four fundamental types, one for each of the fundamental forces, and several subtypes Gravitational potential energy is the energy associated with the arrangement of masses the energy of a mass in a gravitational field Electromagnetic potential energy is the energy associated with the arrangement of charges Elastic potential energy is defined as the energy stored within a material (e.g. (b) If the object is released from rest at point \(\text{B}\) with a small force applied, can it reach point \(\text{A}\) or \(\text{C}\)? The potential energy is the energy related to the position of an object. Situation1. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The total potential energy of the system decreases for the exothermic reaction as the system releases energy to the surroundings. Similarly, if the potential energy is decreasing, then the force is positive. Your graph should look like a double potential well, with the zeros determined by solving the equation [latex]U(x)=0[/latex], and the extremes determined by examining the first and second derivatives of U(x), as shown in Figure. Normal Force Calculator. Distance=Area=F.X (distance) We can find energy of the objects from their Force vs. We see that around unstable equilibrium points, the forces point away from the equilibrium point. For this reason, as well as the shape of the potential energy curve, U(x) is called an infinite potential well. As for the object in vertical free fall, you can deduce the physically allowable range of motion and the maximum values of distance and speed, from the limits on the kinetic energy, 0KE.0KE. Similarly, if the potential energy is decreasing, then the force is positive. If the force on either side of an equilibrium point has a direction opposite from that direction of position change, the equilibrium is termed unstable, and this implies that U(x) has a relative maximum there. Description Use this worksheet to make high quality graphs. Fig. You can read off the same type of information from the potential energy diagram in this case, as in the case for the body in vertical free fall, but since the spring potential energy describes a variable force, you can learn more from this graph. That's point A on the figure to the right. Fig. By definition, if the potential energy is increasing then ___. If we let go, the mass initially has zero kinetic energy, +7.5 J of potential energy, and +7.5 J of mechanical energy (recall: ME = KE . At point H, the object is moving in the positive x-direction and the mechanical energy of the system is 5.0 J. Discussion topics include forces, free-body diagrams, force analysis with components, changes in speed and direction, position-time graphs, velocity-time graphs, changes in kinetic and potential energy, and the period-length relationship. At a turning point, the potential energy equals the mechanical energy and the kinetic energy is zero, indicating that the direction of the velocity reverses there. It would naturally come to rest at the bottom if no other external forces move the marble or bowl around. By plotting the potential energy as a function of position, we can learn various physical properties of a system. The particle in this example can oscillate in the allowed region about either of the two stable equilibrium points we found, but it does not have enough energy to escape from whichever potential well it happens to initially be in. Potential energy is not simply a measure of the work an object may do with respect to gravity, but more generally it is a measure of the work an object can do as a function of its position or configuration (meaning that different parts of the spring have moved by different amounts). You can read off the same type of information from the potential energy diagram in this case, as in the case for the body in vertical free fall, but since the spring potential energy describes a variable force, you can learn more from this graph. When the cart crests the hill,. We can find the total kinetic energy of the object after 14m from the graph; we use area under it to find energy. First, lets look at an object, freely falling vertically, near the surface of Earth, in the absence of air resistance. Interpreting a Potential Energy Graph 8,711 views Nov 20, 2013 63 Dislike Share Save CB physics 116 subscribers This is the second part. At ground level, y0 = 0, the potential energy is zero, and the kinetic energy and the speed are maximum: \[v_{0} = \pm \sqrt{\frac{2E}{m}} \ldotp\]. then you must include on every digital page view the following attribution: Use the information below to generate a citation. The potential energy of the object increases momentarily, before returning to its value at equilibrium. Energy must be added to the system in order to reach the transition state. This is like a one-dimensional system, whose mechanical energy E is a constant and whose potential energy, with respect to zero energy at zero displacement from the springs unstretched length, x = 0, is U(x) = \(\frac{1}{2}\)kx2. What is the particles initial velocity? (e) Repeat part (d) if [latex]v=2.0\,\text{m/s}\,\text{at}\,x=0. This transition state is represented as a maximum in the potential energy as a function of the reaction coordinate graph. You can find the values of (a) the allowed regions along the x-axis, for the given value of the mechanical energy, from the condition that the kinetic energy cant be negative, and (b) the equilibrium points and their stability from the properties of the force (stable for a relative minimum and unstable for a relative maximum of potential energy). The graph shows the potential energy U as a function of position x. In the image below, we see the potential energy graph for a system that has stable and unstable equilibrium points. (a) Determine the positions of points \(\text{A}\) and \(\text{B}\), the equilibrium points. As the atoms approach one another, the electrons concentrate between the nuclei, and attraction occurs. The potential energy of the object is unchanged after it is displaced. (a) Is the motion of the particle confined to any regions on the x-axis, and if so, what are they? Here the gravitational potential energy is defined as the energy possessed by an object by virtue of its position relative to others. The relationship between the potential energy and force tells us a lot about the stability of the system. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) [latex]x=4.0\,\text{m}[/latex] as the reference point. Repulsive, attractive, electromagnetic force, strong nuclear force. For this reason, as well as the shape of the potential energy curve, U(x) is called an infinite potential well. The following graph is a sketch of the potential energy function. The graph below shows the relation between force (F) and x (the change in length) of a spring. X=-3m - shows the direction of compression. In the graph, we see that when the object reaches \(y_max\), the potential energy equals the total energy of the system, meaning that the kinetic energy at this moment will be zero. Substitute the potential energy in (Equation 8.14) and integrate using an integral solver found on a web search: From the initial conditions at [latex]t=0,[/latex] the initial kinetic energy is zero and the initial potential energy is [latex]\frac{1}{2}k{x}_{0}{}^{2}=E,[/latex] from which you can see that [latex]{x}_{0}\text{/}\sqrt{(2E\text{/}k)}=\pm 1[/latex] and [latex]{\text{sin}}^{-1}(\pm )=\pm {90}^{0}. When we visualize the potential energy as a function of the object's position in a graph, we find that the force is the negative of the slope, \(\Delta U=-F\Delta x\). . Potential energy is the energy that an object has due to its position concerning other things, internal tensions, electric charge, or other factors. The change in length 1 (x) = 1 cm = 1/100 m = 0.01 m. The change in length 2 = 8 cm = 8/100 m = 0.08 m Fig. (c) Suppose a particle of mass m moving with this potential energy has a velocity [latex]{v}_{a}[/latex] when its position is [latex]x=a[/latex]. Create the most beautiful study materials using our templates. Substitute the potential energy U into Equation 8.4.9 and factor out the constants, like m or k. Integrate the function and solve the resulting expression for position, which is now a function of time. Point \(\text{B}\) is a point of unstable equilibrium, so the force applied in the correct direction could move the object away such that it gets to point \(\text{A}\). Imagine the marble is made to rest on the lip of the bowl in a position of unstable equilibrium. We saw earlier that the negative of the slope of the potential energy is the spring force, which in this case is also the net force, and thus is proportional to the acceleration. would be represented by a ___ line in the graph. Given that U(x) = k[1-e-x2]The graph of U(x) is shown in fig. Interpreting a one-dimensional potential energy diagram allows you to obtain qualitative, and some quantitative, information about the motion of a particle. In the second picture we stretch the spring by the amount of X. Points below the curve refer to potential energy, while points above the curve refer to kinetic energy. In the graph shown in Figure 8.10, the x-axis is the height above the ground y and the y-axis is the objects energy. We say that it has become potential energy in the spring. Lets examine the behavior of the springs in two situations. Science Physics The graph below shows the potential energy U of a system as one object in the system moves along the x-axis and the rest of the system does not move. (a) What is the force on the particle at [latex]x=2.0,5.0,8.0,\,\text{and}[/latex] 12 m? The potential energy of the object increases momentarily, before returning to its value at equilibrium. The negative of the slope, on either side of the equilibrium point, gives a force pointing back to the equilibrium point, [latex]F=\pm kx,[/latex] so the equilibrium is termed stable and the force is called a restoring force. Suppose we place a 1 kg mass 0.75 m above the height that has been selected as y = 0. As for the object in vertical free fall, you can deduce the physically allowable range of motion and the maximum values of distance and speed, from the limits on the kinetic energy, 0 K E. Therefore, K = 0 and U = E at a turning point, of which there are two for the elastic spring potential energy, \[x_{max} = \pm \sqrt{\frac{2E}{k}} \ldotp\]. fuel and explosives have Chemical PE a coiled spring or a drawn bow also have PE due to their state [/latex], [latex]\begin{array}{ccc}\hfill {U}_{0}& =\hfill & 0=E-{K}_{0},\hfill \\ \hfill E& =\hfill & {K}_{0}=\frac{1}{2}m{v}_{0}{}^{2},\hfill \\ \hfill {v}_{0}& =\hfill & \pm \sqrt{2E\text{/}m}.\hfill \end{array}[/latex], [latex]{x}_{\text{max}}=\pm \sqrt{2E\text{/}k}. The gliders motion is confined to the region between the turning points, xmaxxxmax.xmaxxxmax. If we release the box spring does work and pushes the box back. A reaction coordinate graph shows how the energy of a system changes during a chemical reaction. Video 8.4: Potential Energy Graphs - YouTube 0:00 / 6:21 Video 8.4: Potential Energy Graphs 40,817 views Sep 12, 2013 Like Dislike Share Save Grant Volle 205 subscribers Interpreting. If we know the expression for the potential energy we can determine the force applied. Potential energy can be divided into many types; Gravitational potential energy, Elastic Potential energy, Electric Potential Energy etc. [/latex] Solving this for A matches results in the problem. Example: In the pictures given below, if the potential energy of the ball in the first picture is P find the potential energy of the ball in second situation in terms of P. You can see that there are two allowed regions for the motion [latex](E\gt U)[/latex] and three equilibrium points (slope [latex]dU\text{/}dx=0),[/latex] of which the central one is unstable [latex]({d}^{2}U\text{/}d{x}^{2} \lt 0),[/latex] and the other two are stable [latex]({d}^{2}U\text{/}d{x}^{2} \gt 0). When you are on the ground floor, you have low potential energy. A steel ball has more potential energy raised above the ground than it has after falling to Earth. You can think of potential energy as kinetic energy waiting to happen. The negative of slope of a potential energy graph is the force that causes the object's motion. Find the potential energy of a particle due to this force when it is at a distance x from the wall, assuming the potential energy at the wall to be zero. Paper back, excellent condition, like newThe Beehive Effect begins by introducing you to the ancient knowledge of hive location and the electromagnetic fields that enhance a hive to levels of maximum potential. We note in this expression that the quantity of the total energy divided by the weight (mg) is located at the maximum height of the particle, or ymax. A potential energy vs position graph is shown for a 1 kg particle moving along the x axis. Graphs of potential energy as a function of position are useful in understanding the properties of a chemical bond between two atoms. potential energy, stored energy that depends upon the relative position of various parts of a system. Where are you the most stable? At the bottom of the potential well, x = 0, U = 0 and the kinetic energy is a maximum, K = E, so vmax = \(\sqrt{\frac{2E}{m}}\). We know that the particle is at rest, \(K_C=0\). For example, apples on the tree, or compressed spring or a stone thrown from any height with respect to ground are examples of potential energy. This energy is said to be stored inside the object. What is its speed at [latex]x=2.0\,\text{m? What effect does doubling the height have on potential energy? are licensed under a, Coordinate Systems and Components of a Vector, Position, Displacement, and Average Velocity, Finding Velocity and Displacement from Acceleration, Relative Motion in One and Two Dimensions, Potential Energy and Conservation of Energy, Rotation with Constant Angular Acceleration, Relating Angular and Translational Quantities, Moment of Inertia and Rotational Kinetic Energy, Gravitational Potential Energy and Total Energy, Comparing Simple Harmonic Motion and Circular Motion. In the first picture, system including a spring and a box is at rest. The object can never be at a location where the potential energy curve is above the total energy line. The local minimum in the curve represents the distance where attractive and repulsive forces are balanced. At ground level, [latex]{y}_{0}=0[/latex], the potential energy is zero, and the kinetic energy and the speed are maximum: The maximum speed [latex]\pm {v}_{0}[/latex] gives the initial velocity necessary to reach [latex]{y}_{\text{max}},[/latex] the maximum height, and [latex]\text{}{v}_{0}[/latex] represents the final velocity, after falling from [latex]{y}_{\text{max}}. Example: Find the Kinetic Energy of the object at 14m from the given graph below. Spring potential energy. The electric potential at a place in an electric field is the amount of effort required to transport a unit positive charge from infinity to that point, whereas electric potential energy is the amount of energy required to move a charge against the electric field. From the initial conditions at t=0,t=0, the initial kinetic energy is zero and the initial potential energy is 12kx02=E,12kx02=E, from which you can see that x0/(2E/k)=1x0/(2E/k)=1 and sin1()=900.sin1()=900. This is true for any (positive) value of E because the potential energy is unbounded with respect to x. The work done by the pulling force F p is in positive as it has . An object will be in motion and still have potential energy. In a Potential Energy vs Position graph, the total mechanical energy of the systemwould be represented by a ___ line in the graph. We will look at which factors effects the magnitude of potential energy or which does not effect. Solving for y results in. Of course the thinner spring is more compressed than the thicker one where the quantity of compression shows the loaded potential energy. Find [latex]x(t)[/latex] for the mass-spring system in Figure if the particle starts from [latex]{x}_{0}=0[/latex] at [latex]t=0. When we think of potential energy, often the first thing that comes to mind is an object high in the air and. The difference between the maximum and the energy of the ___ at the beginning of the reaction is called the, More about Potential Energy Graphs and Motion, Charged Particle in Uniform Electric Field, Electric Field Between Two Parallel Plates, Magnetic Field of a Current-Carrying Wire, Mechanical Energy in Simple Harmonic Motion, Galileo's Leaning Tower of Pisa Experiment, Electromagnetic Radiation and Quantum Phenomena, Centripetal Acceleration and Centripetal Force, Total Internal Reflection in Optical Fibre. There are two basic things to know about potential energy diagrams: equilibrium points and accessibility. Now, work is the transfer of energy. This is most easily accomplished for a one-dimensional system, whose potential energy can be plotted in one two-dimensional graphfor example, U(x) versus xon a piece of paper or a computer program. StudySmarter is commited to creating, free, high quality explainations, opening education to all. Fig. The total energy of the system is a constant horizontal line. At these points, the rate of change of the potential energy with distance will also be zero. By the end of this section, you will be able to: Often, you can get a good deal of useful information about the dynamical behavior of a mechanical system just by interpreting a graph of its potential energy as a function of position, called a potential energy diagram. That, after all, is the value of potential energy diagrams. 4 - Visual representation of how forces point back to equilibrium around a point of stable equilibrium. Its velocity and therefore kinetic energy is zero at that point, which means that the total energy is equal to the potential energy. First, we take the derivative of the potential energy with respect to the position, $$\begin{align*}\frac{\operatorname dU}{\operatorname dx}&=1-3{(2x-3)}^2(2),\\\frac{\operatorname dU}{\operatorname dx}&=-24x^2+72x-53.\end{align*}$$. At the bottom of the potential well, [latex]x=0,U=0[/latex] and the kinetic energy is a maximum, [latex]K=E,\,\text{so}\,{v}_{\text{max}}=\pm \sqrt{2E\text{/}m}.[/latex]. We see that gravitational potential energy depends on the weight and height of the object. 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's law force). The mechanical energy of the object is conserved, E = K + U, and the potential energy, with respect to zero at ground level, is U(y) = mgy, which is a straight line through the origin with slope mg . The difference between the maximum and the energy of the reactant at the beginning of the reaction is called the activation energy \(E_act\). This corresponds to Hooke's Law, which experimentally proves the description of the motion for a spring-mass system. For example, take a look at the point \(y_A\). Solving for y results in. Create and find flashcards in record time. 2.2 Coordinate Systems and Components of a Vector, 3.1 Position, Displacement, and Average Velocity, 3.3 Average and Instantaneous Acceleration, 3.6 Finding Velocity and Displacement from Acceleration, 4.5 Relative Motion in One and Two Dimensions, 8.2 Conservative and Non-Conservative Forces, 8.4 Potential Energy Diagrams and Stability, 10.2 Rotation with Constant Angular Acceleration, 10.3 Relating Angular and Translational Quantities, 10.4 Moment of Inertia and Rotational Kinetic Energy, 10.8 Work and Power for Rotational Motion, 13.1 Newtons Law of Universal Gravitation, 13.3 Gravitational Potential Energy and Total Energy, 15.3 Comparing Simple Harmonic Motion and Circular Motion, 17.4 Normal Modes of a Standing Sound Wave, 1.4 Heat Transfer, Specific Heat, and Calorimetry, 2.3 Heat Capacity and Equipartition of Energy, 4.1 Reversible and Irreversible Processes, 4.4 Statements of the Second Law of Thermodynamics, 8 Potential Energy and Conservation of Energy. where \(k\) is the spring constant that determines the stiffness of the spring in Newtons per meter, \(\frac{\mathrm N}{\mathrm m}\), and \(x\) is the object's displacement from the equilibrium position in meters \(\mathrm m\). Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . \(\frac{\operatorname dU}{\operatorname dx}\) is positive. The transition state is represented as a ___ in the potential energy as a function of the reaction coordinate graph. This tool estimates the potential energy on the basis of three values. However, from the slope of this potential energy curve, you can also deduce information about the force on the glider and its acceleration. The main difference is that the kinetic energy and potential energy are always interchanging such that the total energy of the system is constant. If \(\frac{\operatorname dU}{\operatorname dx}\) is positive, ___. At the top of a building that is a thousand meters tall, or just above the surface on the ground floor? The "kinks" in the graph occur at (1. The horizontal line is the total energy of the system. Kinetic by OpenStax offers access to innovative study tools designed to help you maximize your learning potential. (b) Are there any equilibrium points, and if so, where are they and are they stable or unstable? These two examples of gravitational and spring potential energy are calculated differently. The potential energy for a particle undergoing one-dimensional motion along the x-axis is U(x) = 2(x4 x2), where U is in joules and x is in meters. Six evenly-spaced points along the x-axis are labeled. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo The function is zero at the origin, becomes negative as x increases in the positive or negative directions ([latex]{x}^{2}[/latex] is larger than [latex]{x}^{4}[/latex] for [latex]x\lt 1[/latex]), and then becomes positive at sufficiently large [latex]|x|[/latex]. As I said before area under the force vs distance graph gives us the work and energy is the capability of doing work. Want to cite, share, or modify this book? In stable equilibrium points, forced point back to the equilibrium point. About This Article If the particle has a total energy of 4.0 J, do the following: a) indicate points or regions where on the graph it speeds up. It is represented by a horizontal line on the graph, as know that the potential energy \(U\) and the kinetic energy \(K\) are interchanging values such that the total mechanical energy \(E\) remains constant. While performing an S.H.M., the particle possesses speed (hence kinetic energy) at all the positions except at the extreme positions. For systems whose motion is in more than one dimension, the motion needs to be studied in three-dimensional space. However, in second picture the box compresses the spring and loads it with potential energy. Potential Energy Definition and Mathematics of Work Calculating the Amount of Work Done by Forces Potential Energy Kinetic Energy Mechanical Energy Power An object can store energy as the result of its position. When an object is located at one of these positions or in one of these regions it is said to be in a state of equilibrium: stable, unstable, dynamic, and static (or neutral). [/latex], Thermal Expansion in Two and Three Dimensions, Vapor Pressure, Partial Pressure, and Daltons Law, Heat Capacity of an Ideal Monatomic Gas at Constant Volume, Chapter 3 The First Law of Thermodynamics, Quasi-static and Non-quasi-static Processes, Chapter 4 The Second Law of Thermodynamics, Create and interpret graphs of potential energy, Explain the connection between stability and potential energy, To find the equilibrium points, we solve the equation. We see that local minimums indicate locations of stable equilibrium. A test charge with twice the quantity of charge would possess twice the potential energy at a given location; yet its electric potential at . The relationship between the potential energy and force, \(F=-\frac{\operatorname dU}{\operatorname dx}\), tells us a lot about the stability of the system. At a stable equilibrium point, on either side of the equilibrium point, there is a force that points back to equilibrium. Work=Force. Potential Energy Diagram Worksheet STEM Road Map: A Framework for Integrated STEM Education is the first resource to offer an integrated STEM curricula encompassing the entire K-12 spectrum, with complete grade-level learning based on a spiraled approach to building conceptual understanding. A 4.0-kg particle moving along the x-axis is acted upon by the force whose functional form appears below. Six evenly-spaced points along the x-axis are labeled. A mysterious constant force of 10 N acts horizontally on everything. The purple ball has kinetic energy due to its velocity. The conservation of mechanical energy and the relations between kinetic energy and speed, and potential energy and force, enable you to deduce much information about the qualitative behavior of the motion of a particle, as well as some quantitative information, from a graph of its potential energy. (b) It is possible that if the object is released from rest at point \(\text{B}\) it can reach point \(\text{A}\). Explore different tracks and view the kinetic energy, potential energy and friction as she moves. The total energy of the system is a constant horizontal line. Now you can solve for x: \[x(t) = \sqrt{\left(\dfrac{2E}{k}\right)} \sin \Big[\left(\sqrt{\dfrac{k}{m}}\right)t \pm 90^{o} \Big] = \pm \sqrt{\left(\dfrac{2E}{k}\right)} \cos \Big[ \left(\sqrt{\dfrac{k}{m}}\right)t \Big] \ldotp\]. If you are redistributing all or part of this book in a print format, That, after all, is the value of potential energy diagrams. Potential energy is stored energy while kinetic energy is the energy of motion. What is its speed at B, where [latex]{x}_{B}=-2.0\,\text{m?}[/latex]. The energy at this distance is called the bond energy. c) indicate where the kinetic energy of the particle is a maximum. Points and are examples of unstable equilibrium points. At ground level, y 0 = 0, the potential energy is zero, and the kinetic energy and the speed are maximum: (8.5.4) U 0 = 0 = E K 0, (8.5.5) E = K 0 = 1 2 m v 0 2, (8.5.6) v 0 = 2 E m. The maximum speed v 0 gives the initial velocity necessary to reach y max, the maximum height, and v 0 represents the final velocity, after falling from y max. 0 = 8.85 10 12 C 2 / J m. For charges with the same sign, E has a + sign and tends to get smaller as r increases. For example, apples on the tree, or compressed spring or a stone thrown from any height with respect to ground are examples of potential energy. as a function of time t and displacement x. [/latex] What is the particles initial velocity? A particle of mass 0.50 kg moves along the x-axis with a potential energy whose dependence on x is shown below. The potential energy U(x) associated with F(x) is graphed in Fig. The function is zero at the origin, becomes negative as x increases in the positive or negative directions (x2 is larger than x4 for x < 1), and then becomes positive at sufficiently large |x|. Ep=1/2kx (a) Sketch a graph of the potential energy function [latex]U(x)=k{x}^{2}\text{/}2+A{e}^{\text{}\alpha {x}^{2}},[/latex] where [latex]k,A,\,\text{and}\,\alpha[/latex] are constants. We follow the same steps as we did in (Example 8.9). As I read the graph, the potential energy at x=2 is PE=-7.5 J (plus or minus .1) You really should write down some equations rather than just explaining in (too few) words what you did. 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