Potential energy
Potential energy | |
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gravitational )U = 1⁄2 ⋅ k ⋅ x2 (elastic) |
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Classical mechanics |
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In
Common types of potential energy include the
Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, whose total work is path independent, are called conservative forces. If the force acting on a body varies over space, then one has a force field; such a field is described by vectors at every point in space, which is in-turn called a vector field. A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a scalar potential. The potential energy is related to, and can be obtained from, this potential function.
Overview
There are various types of potential energy, each associated with a particular type of force. For example, the work of an
Forces derivable from a potential are also called conservative forces. The work done by a conservative force is
Potential energy is the energy by virtue of an object's position relative to other objects.[6] Potential energy is often associated with restoring forces such as a spring or the force of gravity. The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall.
Consider a ball whose mass is m and whose height is h. The acceleration g of free fall is approximately constant, so the weight force of the ball mg is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus
The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.
Work and potential energy
Potential energy is closely linked with
If the work for an applied force is independent of the path, then the work done by the force is evaluated from the start to the end of the trajectory of the point of application. This means that there is a function U(x), called a "potential", that can be evaluated at the two points xA and xB to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is
The function U(x) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces.
Derivable from a potential
In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve C takes a special form if the force F is related to a scalar field U′(x) so that
Potential energy U = −U′(x) is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is
In this case, the application of the
Computing potential energy
Given a force field F(x), evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve γ(t) = r(t) from γ(a) = A to γ(b) = B, and computing,
For the force field F, let v = dr/dt, then the gradient theorem yields,
The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity v of the point of application, that is
Examples of work that can be computed from potential functions are gravity and spring forces.[8]
Potential energy for near-Earth gravity
For small height changes, gravitational potential energy can be computed using
In classical physics, gravity exerts a constant downward force F = (0, 0, Fz) on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory r(t) = (x(t), y(t), z(t)), such as the track of a roller coaster is calculated using its velocity, v = (vx, vy, vz), to obtain
Potential energy for a linear spring
A horizontal spring exerts a force F = (−kx, 0, 0) that is proportional to its deformation in the axial or x direction. The work of this spring on a body moving along the space curve s(t) = (x(t), y(t), z(t)), is calculated using its velocity, v = (vx, vy, vz), to obtain
The function
Elastic potential energy is the potential energy of an
Potential energy for gravitational forces between two bodies
The gravitational potential function, also known as
The negative sign follows the convention that work is gained from a loss of potential energy.
Derivation
The gravitational force between two bodies of mass M and m separated by a distance r is given by Newton's law of universal gravitation
Let the mass m move at the velocity v then the work of gravity on this mass as it moves from position r(t1) to r(t2) is given by
This calculation uses the fact that
Potential energy for electrostatic forces between two bodies
The electrostatic force exerted by a charge Q on another charge q separated by a distance r is given by
The work W required to move q from A to any point B in the electrostatic force field is given by the potential function
Reference level
The potential energy is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state; it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of inverse-square law forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience.
Typically the potential energy of a system depends on the relative positions of its components only, so the reference state can also be expressed in terms of relative positions.
Gravitational potential energy
Gravitational energy is the potential energy associated with
Consider a book placed on top of a table. As the book is raised from the floor to the table, some external force works against the gravitational force. If the book falls back to the floor, the "falling" energy the book receives is provided by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into kinetic energy. When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact.
The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. "Height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail.
Local approximation
The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant g = 9.8 m/s2 (
The amount of gravitational potential energy held by an elevated object is equal to the work done against gravity in lifting it. The work done equals the force required to move it upward multiplied with the vertical distance it is moved (remember W = Fd). The upward force required while moving at a constant velocity is equal to the weight, mg, of an object, so the work done in lifting it through a height h is the product mgh. Thus, when accounting only for
Hence, the potential difference is
General formula
However, over large variations in distance, the approximation that g is constant is no longer valid, and we have to use
where K is an arbitrary constant dependent on the choice of datum from which potential is measured. Choosing the convention that K = 0 (i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making U negative; for why this is physically reasonable, see below.
Given this formula for U, the total potential energy of a system of n bodies is found by summing, for all pairs of two bodies, the potential energy of the system of those two bodies.
Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity.
Negative gravitational energy
As with all potential energies, only differences in gravitational potential energy matter for most physical purposes, and the choice of zero point is arbitrary. Given that there is no reasonable criterion for preferring one particular finite r over another, there seem to be only two reasonable choices for the distance at which U becomes zero: and . The choice of at infinity may seem peculiar, and the consequence that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, albeit negative.
The
The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where the total energy of the universe can meaningfully be considered; see
Uses
Gravitational potential energy has a number of practical uses, notably the generation of pumped-storage hydroelectricity. For example, in Dinorwig, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. The process is not completely efficient and some of the original energy from the surplus electricity is in fact lost to friction.[12][13][14][15][16]
Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.
It is also used by counterweights for lifting up an elevator, crane, or sash window.
Another practical use is utilizing gravitational potential energy to descend (perhaps coast) downhill in transportation such as the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline. In some cases the
Chemical potential energy
Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of
The similar term chemical potential is used to indicate the potential of a substance to undergo a change of configuration, be it in the form of a chemical reaction, spatial transport, particle exchange with a reservoir, etc.
Electric potential energy
An object can have potential energy by virtue of its electric charge and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy).
Electrostatic potential energy
Electrostatic potential energy between two bodies in space is obtained from the force exerted by a charge Q on another charge q which is given by
If the electric charge of an object can be assumed to be at rest, then it has potential energy due to its position relative to other charged objects. The electrostatic potential energy is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work that must be done to move it from an infinite distance away to its present location, adjusted for non-electrical forces on the object. This energy will generally be non-zero if there is another electrically charged object nearby.
The work W required to move q from A to any point B in the electrostatic force field is given by
Magnetic potential energy
The energy of a magnetic moment in an externally produced magnetic B-field B has potential energy[20]
The magnetization M in a field is
Nuclear potential energy
Nuclear potential energy is the potential energy of the
Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them can have less mass than if they were individually free, in which case this mass difference can be liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the
Forces and potential energy
Potential energy is closely linked with
For example, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by or , corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass M and m separated by a distance r is
The work done against gravity by moving an infinitesimal mass from point A with to point B with is and the work done going back the other way is so that the total work done in moving from A to B and returning to A is
In practical terms, this means that one can set the zero of and anywhere one likes. One may set it to be zero at the surface of the Earth, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section).
A conservative force can be expressed in the language of
Notes
- ISBN 978-81-203-3862-3.
- ISBN 978-0-8018-9455-8.
- ^ William John Macquorn Rankine (1853) "On the general law of the transformation of energy", Proceedings of the Philosophical Society of Glasgow, vol. 3, no. 5, pages 276–280; reprinted in: (1) Philosophical Magazine, series 4, vol. 5, no. 30, pp. 106–117 (February 1853); and (2) W. J. Millar, ed., Miscellaneous Scientific Papers: by W. J. Macquorn Rankine, ... (London, England: Charles Griffin and Co., 1881), part II, pp. 203–208.
- ISBN 0-226-76420-6.
- S2CID 250895349. Retrieved 15 February 2023.
- ISBN 0-13-109686-9.
- ISBN 978-1-891389-22-1.
- ISBN 978-0-13-516062-6.
- ^ The Feynman Lectures on Physics Vol. I Ch. 13: Work and Potential Energy (A)
- ^ "Hyperphysics – Gravitational Potential Energy".
- ISBN 0-201-14942-7.
- ^ "Energy storage – Packing some power". The Economist. 3 March 2011.
- ^ Jacob, Thierry.Pumped storage in Switzerland – an outlook beyond 2000 Archived 17 March 2012 at the Wayback Machine Stucky. Accessed: 13 February 2012.
- ^ Levine, Jonah G. Pumped Hydroelectric Energy Storage and Spatial Diversity of Wind Resources as Methods of Improving Utilization of Renewable Energy Sources Archived 1 August 2014 at the Wayback Machine page 6, University of Colorado, December 2007. Accessed: 12 February 2012.
- ^ Yang, Chi-Jen. Pumped Hydroelectric Storage Archived 5 September 2012 at the Wayback Machine Duke University. Accessed: 12 February 2012.
- Hawaiian Electric Company. Accessed: 13 February 2012.
- ^ Packing Some Power: Energy Technology: Better ways of storing energy are needed if electricity systems are to become cleaner and more efficient, The Economist, 3 March 2012
- ^ Downing, Louise. Ski Lifts Help Open $25 Billion Market for Storing Power, Bloomberg News online, 6 September 2012
- ^ Kernan, Aedan. Storing Energy on Rail Tracks Archived 12 April 2014 at the Wayback Machine, Leonardo-Energy.org website, 30 October 2013
- ISBN 0-19-851791-2.
- ISBN 0-471-43132-X.
- ^ Livingston, James D. (2011). Rising Force: The Magic of Magnetic Levitation. President and Fellows of Harvard College. p. 152.
- ^ Kumar, Narinder (2004). Comprehensive Physics XII. Laxmi Publications. p. 713.
References
- Serway, Raymond A.; Jewett, John W. (2010). Physics for Scientists and Engineers (8th ed.). Brooks/Cole cengage. ISBN 978-1-4390-4844-3.
- Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.