6515:
with the north and south poles of the Earth's magnetic field. If the needle is moved by an outside force, torque is exerted on the magnetic dipole of the needle by the Earth's magnetic field, causing it to move back into alignment. The magnetic potential energy of the needle is highest when its field is in the same direction as the Earth's magnetic field. Two magnets will have potential energy in relation to each other and the distance between them, but this also depends on their orientation. If the opposite poles are held apart, the potential energy will be higher the further they are apart and lower the closer they are. Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart.
980:
5950:, 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.
2533:
5440:
49:
5003:
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.
5546:. 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.
2220:
4980:, as work is required to elevate objects against Earth's gravity. The potential energy due to elevated positions is called gravitational potential energy, and is evidenced by water in an elevated reservoir or kept behind a dam. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount.
1266:. 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.
8878:
993:
8902:
4571:
2780:
8914:
6052:
8890:
3209:
2528:{\displaystyle {\begin{aligned}\int _{\gamma }\nabla \Phi (\mathbf {r} )\cdot d\mathbf {r} &=\int _{a}^{b}\nabla \Phi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt,\\&=\int _{a}^{b}{\frac {d}{dt}}\Phi (\mathbf {r} (t))dt=\Phi (\mathbf {r} (b))-\Phi (\mathbf {r} (a))=\Phi \left(\mathbf {x} _{B}\right)-\Phi \left(\mathbf {x} _{A}\right).\end{aligned}}}
3223:
4984:
2869:
4243:
4099:
7748:
2565:
3546:
1712:
2120:
5002:
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
5983:
obtained from the potential energy of descent may be used to start ascending the next grade such as what happens when a road is undulating and has frequent dips. The commercialization of stored energy (in the form of rail cars raised to higher elevations) that is then converted to electrical energy
5011:
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
4994:
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
6546:
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
6514:
is nonzero. Magnetic potential energy is the form of energy related not only to the distance between magnetic materials, but also to the orientation, or alignment, of those materials within the field. For example, the needle of a compass has the lowest magnetic potential energy when it is aligned
4942:
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
7745:
3903:
1231:
is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy. Common notations for potential energy are
3179:
6843:
4212:
1169:, is the work of the Coulomb force during rearrangement of configurations of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their configuration.
3314:
1101:
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
4566:{\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot ({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t})\,dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}r{\dot {r}}dt={\frac {GMm}{r(t_{2})}}-{\frac {GMm}{r(t_{1})}}.}
1530:
1993:
6170:
5088:
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
2775:{\displaystyle {\begin{aligned}\int _{\gamma }\mathbf {F} \cdot d\mathbf {r} &=\int _{a}^{b}\mathbf {F} \cdot \mathbf {v} \,dt,\\&=-\int _{a}^{b}{\frac {d}{dt}}U(\mathbf {r} (t))\,dt=U(\mathbf {x} _{A})-U(\mathbf {x} _{B}).\end{aligned}}}
4791:
1803:
5978:
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
5699:
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
1582:
6339:
6231:
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.
1888:
4682:
2856:
5642:
5537:
6047:
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).
3031:
6506:
1412:
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
5788:
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.
1982:
1894:
is computed by evaluating the scalar field at the start point A and the end point B of the curve. This means the work integral does not depend on the path between A and B and is said to be independent of the path.
6704:
4933:
4108:
3808:
5974:
are an entertaining way to utilize potential energy – chains are used to move a car up an incline (building up gravitational potential energy), to then have that energy converted into kinetic energy as it falls.
5923:
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
6433:
334:
358:
1438:
4094:{\displaystyle W=-\int _{\mathbf {r} (t_{1})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.}
2570:
2225:
6078:
1435:
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
7071:
6029:
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.
5356:
4709:
65:
in the bent limb of the bow. When the string is released, the force between the string and the arrow does work on the arrow. The potential energy in the bow limbs is transformed into the
7244:
3633:
5690:
1721:
6010:. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform
6254:
1353:
a
Scottish engineer and physicist in 1853 as part of a specific effort to develop terminology. He chose the term as part of the pair "actual" vs "potential" going back to work by
6695:
6382:
5223:
5084:
3715:
6199:
4820:
3837:
3541:{\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} \,dt=-\int _{0}^{t}kxv_{x}\,dt=-\int _{0}^{t}kx{\frac {dx}{dt}}dt=\int _{x(t_{0})}^{x(t)}kx\,dx={\frac {1}{2}}kx^{2}}
7320:, and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.
5434:
1357:. In his 1867 discussion of the same topic Rankine describes potential energy as ‘energy of configuration’ in contrast to actual energy as 'energy of activity'. Also in 1867,
1206:
4578:
2924:
1337:
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.
1333:
266:
5760:
2791:
1229:
5551:
2115:{\displaystyle {\nabla W}=-{\nabla U}=-\left({\frac {\partial U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}\right)=\mathbf {F} ,}
7287:
6990:
6958:
6616:
5157:
6863:
5542:
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
7123:
7097:
6926:
6900:
5914:
5872:
5846:
5820:
5786:
5734:
7698:
7267:
7163:
7143:
6636:
1812:
1024:
6449:
1916:
4863:
7675:
6551:
is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into
3745:
1707:{\displaystyle \mathbf {F} ={\nabla U'}=\left({\frac {\partial U'}{\partial x}},{\frac {\partial U'}{\partial y}},{\frac {\partial U'}{\partial z}}\right).}
613:
6223:
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
6002:
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
3181:
where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve
732:
7788:
5446:
7887:
3740:
1551:) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces.
705:
7991:
5242:
and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can
6396:
1254:
Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring
8611:
7770:
Packing Some Power: Energy
Technology: Better ways of storing energy are needed if electricity systems are to become cleaner and more efficient
5937:
3650:
in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the
1536:
is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory,
979:
8655:
8330:
7929:
7586:
7559:
7385:
7350:
1049:
held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term
7696:
Pumped
Hydroelectric Energy Storage and Spatial Diversity of Wind Resources as Methods of Improving Utilization of Renewable Energy Sources
1397:
does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from
5822:
in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with
8797:
687:
7808:
2784:
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
3174:{\displaystyle W=\int _{t_{1}}^{t_{2}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\,dt=\int _{t_{1}}^{t_{2}}F_{z}v_{z}\,dt=F_{z}\Delta z.}
7948:
7866:
7836:
7636:
7532:
7501:
6995:
5294:
5281:
1017:
6638:, corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass
1293:
is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus
1913:
is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is
8738:
7168:
6838:{\displaystyle \phi =-\left({\frac {GM}{r}}+{\frac {Gm}{r}}\right)=-{\frac {G(M+m)}{r}}=-{\frac {GMm}{\mu r}}={\frac {U}{\mu }}}
4207:{\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},}
3580:
1138:
force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the
5985:
5943:
5647:
2961:
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
353:
290:
8918:
8955:
8748:
7964:
7852:
7165:
can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B is
5012:
object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant
3667:
1071:
348:
137:
2170:
can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve
608:
6653:
5187:
5040:
3673:
1134:
There are various types of potential energy, each associated with a particular type of force. For example, the work of an
1091:
6006:
within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a
5366:
is an arbitrary constant dependent on the choice of datum from which potential is measured. Choosing the convention that
603:
8278:
5543:
5247:
1010:
997:
758:
681:
31:
8096:
3654:
between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into
446:
8837:
7984:
7695:
677:
478:
1361:
introduced "kinetic energy" as the opposite of "potential energy", asserting that all actual energy took the form of
5880:, and would make calculations involving sums or differences of potential energies beyond what is possible with the
1062:
897:
786:
712:
572:
505:
8940:
8743:
8601:
8263:
8208:
8188:
8183:
7756:
7672:
6224:
6038:
4999:. When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact.
2131:
1179:
1143:
1079:
651:
221:
200:
7293:, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section).
2887:
1525:{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}})}
1296:
8945:
8672:
8016:
7724:
7301:
5793:
4954:
positions of its components only, so the reference state can also be expressed in terms of relative positions.
3216:
3203:
1075:
927:
771:
62:
1120:. A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a
8832:
8817:
8213:
8193:
7317:
6698:
6595:
4963:
917:
877:
641:
6165:{\displaystyle \mathbf {F} _{e}=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}
5953:
Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.
5130:
8345:
8241:
8168:
8158:
8039:
7977:
7707:
6552:
6365:
6067:
Electrostatic potential energy between two bodies in space is obtained from the force exerted by a charge
5947:
3856:
922:
739:
61:
on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into
6175:
4796:
3813:
8285:
8076:
7297:
5439:
5256:
between the two bodies. Using that definition, the gravitational potential energy of a system of masses
3651:
2147:
1162:
1109:
932:
907:
593:
411:
6547:
the missing mass, but it often escapes from the system, where it is not measured). The energy from the
6227:
is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the
4947:
forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience.
4786:{\displaystyle \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}
1890:
This shows that when forces are derivable from a scalar field, the work of those forces along a curve
8733:
8706:
8684:
8591:
8524:
8273:
8203:
8148:
6532:
5395:
4967:
3639:
1147:
1135:
952:
947:
912:
820:
816:
808:
798:
588:
581:
337:
1559:
In this section the relationship between work and potential energy is presented in more detail. The
8950:
8812:
8755:
8677:
8650:
8335:
8198:
8173:
6217:
5373:(i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making
4977:
4838:
1165:
is called intermolecular potential energy. Chemical potential energy, such as the energy stored in
727:
668:
646:
391:
386:
381:
281:
8901:
5874:, would result in potential energy being positive, but infinitely large for all nonzero values of
8894:
8792:
8770:
8760:
8596:
7464:
7313:
7309:
6591:
6536:
6524:
6343:
6026:
5023:). In this case, a simple expression for gravitational potential energy can be derived using the
4944:
1798:{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{C}\nabla U'\cdot d\mathbf {x} ,}
1173:
1151:
1103:
857:
598:
473:
441:
401:
195:
164:
7444:
48:
7377:
7371:
6334:{\displaystyle \Delta U_{AB}({\mathbf {r} })=-\int _{A}^{B}\mathbf {F_{e}} \cdot d\mathbf {r} }
5984:
when needed by an electrical grid, is being undertaken in the United States in a system called
5739:
232:
8694:
8519:
8380:
8340:
8310:
8295:
8268:
8178:
8081:
8021:
8011:
7944:
7925:
7862:
7832:
7632:
7582:
7555:
7549:
7528:
7497:
7427:
7415:
7381:
7346:
6007:
3719:
The negative sign follows the convention that work is gained from a loss of potential energy.
2860:
Examples of work that can be computed from potential functions are gravity and spring forces.
1211:
867:
824:
781:
776:
717:
493:
483:
376:
7367:
7338:
8853:
8827:
8822:
8802:
8787:
8628:
8618:
8398:
8061:
7805:
7456:
6580:
6575:
does not depend on the path between these points, then the work of this force measured from
6055:
5925:
5020:
4704:
3647:
2559:
2167:
1806:
1402:
1158:
1122:
962:
942:
887:
882:
828:
803:
658:
516:
436:
126:
7403:
William John
Macquorn Rankine (1853) "On the general law of the transformation of energy",
7272:
6963:
6931:
6601:
5916:
at infinity is by far the more preferable choice, even if the idea of negative energy in a
4995:
table, this potential energy goes to accelerate the mass of the book and is converted into
1377:
mv. Once this hypothesis became widely accepted, the term "actual energy" gradually faded.
8882:
8863:
8777:
8544:
8459:
8315:
8305:
8290:
8248:
8153:
8103:
8054:
7812:
7752:
7732:
7728:
7702:
7679:
7305:
6848:
6564:
6360:
6044:
6019:
5997:
5034:
3212:
1386:
1350:
1259:
1114:; such a field is described by vectors at every point in space, which is in-turn called a
1083:
1054:
957:
902:
852:
847:
766:
210:
7524:
7518:
7102:
7076:
6905:
6879:
5942:
Gravitational potential energy has a number of practical uses, notably the generation of
5893:
5851:
5825:
5799:
5765:
5713:
5099:). The upward force required while moving at a constant velocity is equal to the weight,
1126:. The potential energy is related to, and can be obtained from, this potential function.
17:
8858:
8807:
8782:
8586:
8551:
8484:
8474:
8390:
8370:
8365:
8350:
8320:
8300:
8258:
8236:
8218:
8118:
8108:
8044:
7941:
Physics for
Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics
7856:
7792:
7613:
7252:
7148:
7128:
6621:
6528:
6385:
6228:
6015:
5980:
5971:
4996:
3655:
1346:
1087:
984:
892:
793:
510:
456:
216:
169:
158:
66:
58:
8934:
8906:
8765:
8725:
8689:
8645:
8571:
8534:
8452:
8375:
8143:
8138:
8091:
8071:
7826:
7774:
7657:
7652:
7468:
7460:
6873:
6438:
6003:
5957:
5704:
over another, there seem to be only two reasonable choices for the distance at which
4971:
2877:
1560:
1139:
872:
699:
54:
5165:
is the potential energy of the object relative to its being on the Earth's surface,
8662:
8606:
8556:
8539:
8360:
8133:
8066:
6866:
6011:
5917:
4677:{\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{2}}}.}
3643:
1987:
1166:
1116:
937:
862:
551:
431:
2851:{\displaystyle P(t)=-{\nabla U}\cdot \mathbf {v} =\mathbf {F} \cdot \mathbf {v} .}
7576:
7494:
The
Science of Energy – a Cultural History of Energy Physics in Victorian Britain
8667:
8633:
8507:
8497:
8440:
8423:
8408:
8355:
8253:
7601:
5965:
5881:
5637:{\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}
5532:{\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}
5120:
4232:
are the radial and tangential unit vectors directed relative to the vector from
3840:
1358:
189:
2884:
For small height changes, gravitational potential energy can be computed using
2126:
is said to be "derivable from a potential". This also necessarily implies that
8701:
8623:
8561:
8529:
8469:
8163:
6540:
722:
396:
6539:
provide the potential energy for certain kinds of radioactive decay, such as
5436:
pairs of two bodies, the potential energy of the system of those two bodies.
8711:
8638:
8512:
8492:
8464:
8403:
7602:
The
Feynman Lectures on Physics Vol. I Ch. 13: Work and Potential Energy (A)
2873:
1354:
1058:
744:
6051:
3248:
direction. The work of this spring on a body moving along the space curve
2934:
is the local gravitational field (9.8 metres per second squared on Earth),
2998:, such as the track of a roller coaster is calculated using its velocity,
1883:{\displaystyle W=U'(\mathbf {x} _{\text{B}})-U'(\mathbf {x} _{\text{A}}).}
27:
Energy held by an object because of its position relative to other objects
8447:
8428:
8413:
8113:
7769:
6584:
6501:{\displaystyle U=-{\frac {1}{2}}\int \mathbf {M} \cdot \mathbf {B} \,dV,}
5961:
5243:
5239:
5124:
1406:
663:
546:
521:
6992:
so that the total work done in moving from A to B and returning to A is
6351:
for voltage) is equal to the electric potential energy per unit charge.
3208:
1977:{\displaystyle W=U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}}).}
1287:
of free fall is approximately constant, so the weight force of the ball
8581:
8123:
4928:{\displaystyle U(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r}}.}
3226:
3222:
1345:
From around 1840 scientists sought to define and understand energy and
1263:
1038:
636:
489:
406:
95:
4983:
3803:{\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {r}} ,}
2868:
8128:
8049:
8000:
6583:
field. In this case, the force can be defined as the negative of the
5884:
system. Since physicists abhor infinities in their calculations, and
3229:
is one of humankind's oldest applications of elastic potential energy
1405:
field. In this case, the force can be defined as the negative of the
1155:
1046:
695:
541:
451:
132:
7721:
7289:
anywhere one likes. One may set it to be zero at the surface of the
6018:, and electrical energy can be converted to chemical energy through
4860:
in the electrostatic force field is given by the potential function
7523:. Upper Saddle River, New Jersey: Pearson Education, Inc. pp.
6579:
assigns a scalar value to every other point in space and defines a
5232:
However, over large variations in distance, the approximation that
1401:
assigns a scalar value to every other point in space and defines a
1053:
was introduced by the 19th-century
Scottish engineer and physicist
7290:
6050:
5438:
4982:
3221:
3207:
2867:
1255:
1108:. If the force acting on a body varies over space, then one has a
1095:
531:
526:
468:
100:
7426:, ... (London, England: Charles Griffin and Co., 1881), part II,
6508:
where the integral can be over all space or, equivalently, where
3646:
or a catapult) that is deformed under tension or compression (or
1421:), called a "potential", that can be evaluated at the two points
8502:
8435:
8418:
8086:
5116:
5105:, of an object, so the work done in lifting it through a height
3872:
then the work of gravity on this mass as it moves from position
536:
499:
120:
7973:
2945:
In classical physics, gravity exerts a constant downward force
6548:
4988:
4976:
Gravitational energy is the potential energy associated with
7249:
In practical terms, this means that one can set the zero of
5379:
negative; for why this is physically reasonable, see below.
4687:
Potential energy for electrostatic forces between two bodies
3662:
Potential energy for gravitational forces between two bodies
3548:
For convenience, consider contact with the spring occurs at
6341:
typically given in J for Joules. A related quantity called
4240:. Use this to simplify the formula for work of gravity to,
7424:
Miscellaneous
Scientific Papers: by W. J. Macquorn Rankine
6428:{\displaystyle U=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}
4987:
Gravitational force keeps the planets in orbit around the
4950:
Typically the potential energy of a system depends on the
7969:
7789:
Ski Lifts Help Open $ 25 Billion Market for
Storing Power
7296:
A conservative force can be expressed in the language of
1057:, although it has links to the ancient Greek philosopher
329:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
6567:. If the work done by a force on a body that moves from
6523:
Nuclear potential energy is the potential energy of the
4943:
tending to that limit is finite, such as in the case of
3244:
that is proportional to its deformation in the axial or
1389:. If the work done by a force on a body that moves from
7627:
Guth, Alan (1997). "Appendix A, Gravitational Energy".
3638:
Elastic potential energy is the potential energy of an
7673:
Pumped storage in Switzerland – an outlook beyond 2000
5398:
1718:′ must be this case, work along the curve is given by
235:
7275:
7255:
7171:
7151:
7131:
7105:
7079:
6998:
6966:
6934:
6908:
6882:
6851:
6707:
6656:
6624:
6604:
6452:
6399:
6368:
6257:
6178:
6081:
6043:
An object can have potential energy by virtue of its
5896:
5854:
5828:
5802:
5768:
5742:
5716:
5650:
5554:
5449:
5297:
5246:
the gravitational force, whose magnitude is given by
5190:
5133:
5043:
4866:
4799:
4712:
4581:
4246:
4111:
3906:
3816:
3748:
3676:
3583:
3317:
3034:
2938:
is the height above a reference level in metres, and
2890:
2794:
2568:
2223:
1996:
1919:
1815:
1724:
1585:
1441:
1299:
1214:
1182:
293:
3666:
The gravitational potential function, also known as
8846:
8722:
8570:
8483:
8389:
8229:
8030:
7405:
Proceedings of the Philosophical Society of Glasgow
5238:is constant is no longer valid, and we have to use
3727:The gravitational force between two bodies of mass
3635:is called the potential energy of a linear spring.
3555:, then the integral of the product of the distance
107:
94:
74:
41:
7281:
7261:
7238:
7157:
7137:
7117:
7091:
7065:
6984:
6952:
6920:
6894:
6857:
6837:
6689:
6630:
6610:
6531:. The nuclear particles are bound together by the
6500:
6427:
6376:
6333:
6193:
6164:
5908:
5866:
5840:
5814:
5780:
5754:
5728:
5684:
5636:
5531:
5428:
5350:
5217:
5151:
5078:
4927:
4814:
4785:
4676:
4565:
4206:
4093:
3831:
3802:
3709:
3627:
3540:
3173:
2918:
2850:
2774:
2527:
2114:
1976:
1882:
1797:
1706:
1524:
1327:
1223:
1200:
1172:Forces derivable from a potential are also called
328:
260:
6185:
6153:
4806:
4774:
3823:
3791:
6014:to chemical energy through the process known as
7614:"Hyperphysics – Gravitational Potential Energy"
2876:uses the gravitational potential energy of the
7884:Rising Force: The Magic of Magnetic Levitation
7815:, Leonardo-Energy.org website, 30 October 2013
7407:, vol. 3, no. 5, pages 276–280; reprinted in:
7066:{\displaystyle U_{A\to B\to A}=(b-a)+(a-b)=0.}
6960:and the work done going back the other way is
5890:is always non-zero in practice, the choice of
5351:{\displaystyle U=-G{\frac {m_{1}M_{2}}{r}}+K,}
7985:
7487:
7485:
7339:"Fundamental forces and laws: a brief review"
6251:in the electrostatic force field is given by
2166:), evaluation of the work integral using the
1070:Common types of potential energy include the
1018:
8:
7920:Serway, Raymond A.; Jewett, John W. (2010).
7828:Introduction to the theory of ferromagnetism
7711:, December 2007. Accessed: 12 February 2012.
7399:
7397:
5388:, the total potential energy of a system of
4691:The electrostatic force exerted by a charge
2880:to throw projectiles over two hundred meters
1349:. The term "potential energy" was coined by
1161:is called nuclear potential energy; work of
7578:Kinematics and dynamics of planar machinery
7438:
7436:
6872:The work done against gravity by moving an
2146:in space, so the set of forces is called a
1176:. The work done by a conservative force is
7992:
7978:
7970:
7316:vanishes, so every closed form is also an
7239:{\displaystyle U_{A\to B}=(b+c)-(a+c)=b-a}
3628:{\displaystyle U(x)={\frac {1}{2}}kx^{2},}
1025:
1011:
272:
47:
7554:. University Science Books. p. 117.
7274:
7254:
7176:
7170:
7150:
7130:
7104:
7078:
7073:If the potential is redefined at A to be
7003:
6997:
6965:
6933:
6907:
6881:
6850:
6825:
6799:
6766:
6740:
6722:
6706:
6666:
6655:
6623:
6603:
6488:
6483:
6475:
6462:
6451:
6417:
6409:
6398:
6369:
6367:
6326:
6313:
6308:
6302:
6297:
6278:
6277:
6265:
6256:
6180:
6179:
6177:
6148:
6147:
6139:
6125:
6116:
6100:
6088:
6083:
6080:
5895:
5853:
5827:
5801:
5767:
5741:
5715:
5685:{\displaystyle U=-m\sum G{\frac {M}{r}},}
5669:
5649:
5621:
5611:
5605:
5591:
5581:
5575:
5553:
5516:
5506:
5500:
5486:
5476:
5470:
5448:
5399:
5397:
5327:
5317:
5310:
5296:
5189:
5132:
5067:
5048:
5042:
4907:
4898:
4882:
4865:
4801:
4800:
4798:
4769:
4768:
4760:
4746:
4737:
4721:
4713:
4711:
4663:
4649:
4647:
4630:
4629:
4620:
4601:
4582:
4580:
4548:
4524:
4509:
4485:
4465:
4464:
4453:
4436:
4428:
4423:
4416:
4411:
4394:
4385:
4380:
4368:
4367:
4355:
4350:
4338:
4337:
4322:
4317:
4302:
4285:
4277:
4272:
4265:
4260:
4245:
4195:
4190:
4178:
4177:
4165:
4160:
4148:
4147:
4139:
4129:
4124:
4112:
4110:
4081:
4076:
4068:
4060:
4043:
4035:
4030:
4023:
4018:
4003:
3992:
3984:
3967:
3956:
3944:
3943:
3933:
3921:
3920:
3905:
3818:
3817:
3815:
3786:
3785:
3777:
3760:
3749:
3747:
3686:
3675:
3616:
3599:
3582:
3532:
3515:
3505:
3484:
3474:
3463:
3430:
3418:
3413:
3396:
3390:
3374:
3369:
3352:
3347:
3339:
3333:
3328:
3316:
3156:
3142:
3136:
3126:
3114:
3109:
3102:
3097:
3083:
3078:
3070:
3062:
3057:
3050:
3045:
3033:
2895:
2889:
2840:
2832:
2824:
2813:
2793:
2756:
2751:
2732:
2727:
2710:
2693:
2672:
2666:
2661:
2634:
2629:
2621:
2615:
2610:
2594:
2583:
2577:
2569:
2567:
2508:
2503:
2482:
2477:
2449:
2423:
2391:
2370:
2364:
2359:
2318:
2297:
2282:
2277:
2261:
2247:
2232:
2224:
2222:
2104:
2076:
2053:
2030:
2011:
1997:
1995:
1962:
1957:
1938:
1933:
1918:
1868:
1863:
1839:
1834:
1814:
1787:
1764:
1752:
1741:
1735:
1723:
1671:
1643:
1615:
1594:
1586:
1584:
1513:
1508:
1489:
1484:
1469:
1458:
1452:
1440:
1304:
1298:
1213:
1181:
310:
304:
295:
294:
292:
251:
234:
7888:President and Fellows of Harvard College
7831:(Repr. ed.). Oxford: Clarendon Pr.
6563:Potential energy is closely linked with
5177:is the acceleration due to gravity, and
1385:Potential energy is closely linked with
7343:Textbook of Engineering Physics, Part 1
7329:
6410:
6370:
3079:
3071:
2864:Potential energy for near-Earth gravity
344:
280:
8612:Integrated gasification combined cycle
6201:is a vector of length 1 pointing from
5938:Gravitational potential energy storage
4822:is a vector of length 1 pointing from
4101:The position and velocity of the mass
38:
8656:Radioisotope thermoelectric generator
8331:Quantum chromodynamics binding energy
7924:(8th ed.). Brooks/Cole cengage.
7653:"Energy storage – Packing some power"
7345:. PHI Learning Pvt. Ltd. p. 10.
3741:Newton's law of universal gravitation
2788:of the point of application, that is
1986:In this case, the application of the
707:Newton's law of universal gravitation
7:
8889:
7922:Physics for Scientists and Engineers
6690:{\displaystyle U=-{\frac {GMm}{r}}.}
6377:{\displaystyle {\boldsymbol {\mu }}}
5392:bodies is found by summing, for all
5218:{\displaystyle \Delta U=mg\Delta h.}
5079:{\displaystyle W_{F}=-\Delta U_{F}.}
4575:This calculation uses the fact that
3710:{\displaystyle U=-{\frac {GMm}{r}},}
3283:, is calculated using its velocity,
3198:Potential energy for a linear spring
8913:
8798:World energy supply and consumption
7631:. Perseus Books. pp. 289–293.
7496:. The University of Chicago Press.
6194:{\displaystyle \mathbf {\hat {r}} }
5184:Hence, the potential difference is
4815:{\displaystyle \mathbf {\hat {r}} }
3832:{\displaystyle \mathbf {\hat {r}} }
3233:A horizontal spring exerts a force
688:Mechanics of planar particle motion
296:
7905:. Laxmi Publications. p. 713.
6594:. The associated potential is the
6258:
5749:
5443:Gravitational potential summation
5206:
5191:
5060:
3162:
2814:
2495:
2469:
2443:
2417:
2385:
2291:
2288:
2241:
2238:
2087:
2079:
2064:
2056:
2041:
2033:
2012:
1998:
1770:
1687:
1674:
1659:
1646:
1631:
1618:
1595:
1567:takes a special form if the force
1215:
1192:
25:
6058:formed inside a gas filled sphere
5920:appears to be peculiar at first.
5282:Newtonian constant of gravitation
5115:. Thus, when accounting only for
1805:which can be evaluated using the
8912:
8900:
8888:
8877:
8876:
7861:(2d ed.). New York: Wiley.
6555:, which is radiated into space.
6484:
6476:
6418:
6327:
6314:
6310:
6279:
6182:
6150:
6084:
5429:{\textstyle {\frac {n(n-1)}{2}}}
4803:
4771:
4714:
4381:
4351:
4318:
4191:
4161:
4140:
4125:
4113:
4077:
4069:
4004:
3993:
3945:
3922:
3820:
3788:
3750:
3348:
3340:
2841:
2833:
2825:
2752:
2728:
2694:
2630:
2622:
2595:
2584:
2504:
2478:
2450:
2424:
2392:
2319:
2298:
2262:
2248:
2105:
1958:
1934:
1864:
1835:
1788:
1753:
1742:
1587:
1509:
1485:
1470:
1459:
992:
991:
978:
311:
69:of the arrow as it takes flight.
7943:(5th ed.). W. H. Freeman.
5944:pumped-storage hydroelectricity
5250:, with respect to the distance
5181:is the altitude of the object.
4138:
7221:
7209:
7203:
7191:
7180:
7054:
7042:
7036:
7024:
7013:
7007:
6979:
6967:
6947:
6935:
6784:
6772:
6284:
6274:
6225:electrostatic potential energy
6063:Electrostatic potential energy
5417:
5405:
4958:Gravitational potential energy
4876:
4870:
4554:
4541:
4515:
4502:
4391:
4334:
4328:
4310:
3962:
3949:
3939:
3926:
3668:gravitational potential energy
3593:
3587:
3494:
3488:
3480:
3467:
2804:
2798:
2762:
2747:
2738:
2723:
2707:
2704:
2698:
2690:
2463:
2460:
2454:
2446:
2437:
2434:
2428:
2420:
2405:
2402:
2396:
2388:
2333:
2327:
2311:
2308:
2302:
2294:
2252:
2244:
1968:
1953:
1944:
1929:
1874:
1859:
1845:
1830:
1563:that defines work along curve
1519:
1504:
1495:
1480:
1269:Consider a ball whose mass is
1078:of a deformed spring, and the
1072:gravitational potential energy
248:
242:
1:
7882:Livingston, James D. (2011).
7806:Storing Energy on Rail Tracks
7760:. Accessed: 13 February 2012.
7736:. Accessed: 12 February 2012.
7685:. Accessed: 13 February 2012.
7520:Chemistry The Central Science
7368:"Energy, Work and Metabolism"
7099:and the potential at B to be
6697:The gravitational potential (
5695:Negative gravitational energy
1990:to the work function yields,
1714:This means that the units of
1571:is related to a scalar field
1092:International System of Units
1090:. The unit for energy in the
614:Koopman–von Neumann mechanics
7722:Pumped Hydroelectric Storage
7443:Roche, John (1 March 2003).
7414:, series 4, vol. 5, no. 30,
5986:Advanced Rail Energy Storage
5544:gravitational binding energy
682:Non-inertial reference frame
32:Potential Energy (The Flash)
7548:John Robert Taylor (2005).
7517:Brown, Theodore L. (2006).
7449:European Journal of Physics
7445:"What is potential energy?"
6559:Forces and potential energy
5248:Newton's law of gravitation
5171:is the mass of the object,
1201:{\displaystyle W=-\Delta U}
609:Appell's equation of motion
479:Inertial frame of reference
30:For the Flash episode, see
8972:
7461:10.1088/0143-0807/24/2/359
7366:McCall, Robert P. (2010).
6590:For example, gravity is a
6384:in an externally produced
6036:
5995:
5935:
4961:
3201:
2930:is the mass in kilograms,
2919:{\displaystyle U_{g}=mgh,}
2154:Computing potential energy
1555:Derivable from a potential
1328:{\displaystyle U_{g}=mgh.}
261:{\textstyle \int F(r)\,dr}
29:
8872:
8602:Fossil fuel power station
8264:Electric potential energy
8209:Thermodynamic temperature
8189:Thermodynamic free energy
8184:Thermodynamic equilibrium
8007:
7965:What is potential energy?
7903:Comprehensive Physics XII
7858:Classical electrodynamics
7757:Hawaiian Electric Company
7629:The Inflationary Universe
7373:Physics of the Human Body
6355:Magnetic potential energy
6347:(commonly denoted with a
6039:Electric potential energy
6033:Electric potential energy
5992:Chemical potential energy
5755:{\displaystyle r=\infty }
2942:is the energy in joules.
2132:conservative vector field
1381:Work and potential energy
1144:electric potential energy
1080:electric potential energy
46:
8673:Concentrated solar power
7901:Kumar, Narinder (2004).
7825:Aharoni, Amikam (1996).
7795:online, 6 September 2012
7727:5 September 2012 at the
7337:Jain, Mahesh C. (2009).
6646:separated by a distance
6587:of the potential field.
6519:Nuclear potential energy
4699:separated by a distance
3735:separated by a distance
3217:elastic potential energy
3204:Elastic potential energy
1409:of the potential field.
1224:{\displaystyle \Delta U}
1076:elastic potential energy
772:Rotating reference frame
604:Hamilton–Jacobi equation
63:elastic potential energy
18:Nuclear potential energy
8214:Volume (thermodynamics)
8194:Thermodynamic potential
8097:Mass–energy equivalence
7492:Smith, Crosbie (1998).
6701:) of the two bodies is
6596:gravitational potential
5382:Given this formula for
4964:Gravitational potential
713:Newton's laws of motion
573:Newton's laws of motion
57:, when the archer does
8169:Quantum thermodynamics
8159:Laws of thermodynamics
8040:Conservation of energy
7708:University of Colorado
7412:Philosophical Magazine
7283:
7263:
7240:
7159:
7139:
7119:
7093:
7067:
6986:
6954:
6922:
6896:
6859:
6839:
6691:
6632:
6612:
6553:electromagnetic energy
6502:
6429:
6378:
6335:
6195:
6166:
6059:
5910:
5868:
5842:
5816:
5782:
5756:
5730:
5686:
5638:
5539:
5533:
5430:
5352:
5219:
5153:
5080:
4991:
4929:
4816:
4787:
4678:
4567:
4208:
4095:
3857:gravitational constant
3833:
3804:
3711:
3642:object (for example a
3629:
3542:
3230:
3219:
3175:
2920:
2881:
2852:
2776:
2529:
2116:
1978:
1884:
1799:
1708:
1526:
1329:
1225:
1202:
740:Simple harmonic motion
653:Euler's laws of motion
447:D'Alembert's principle
330:
262:
8956:Mechanical quantities
8286:Interatomic potential
8077:Energy transformation
7939:Tipler, Paul (2004).
7811:12 April 2014 at the
7701:1 August 2014 at the
7678:17 March 2012 at the
7418:(February 1853); and
7376:. JHU Press. p.
7298:differential geometry
7284:
7282:{\displaystyle \phi }
7264:
7241:
7160:
7140:
7120:
7094:
7068:
6987:
6985:{\displaystyle (a-b)}
6955:
6953:{\displaystyle (b-a)}
6923:
6897:
6860:
6840:
6692:
6633:
6613:
6611:{\displaystyle \phi }
6503:
6430:
6393:has potential energy
6379:
6336:
6196:
6167:
6054:
5936:Further information:
5911:
5869:
5843:
5817:
5783:
5757:
5731:
5687:
5639:
5534:
5442:
5431:
5353:
5220:
5154:
5152:{\displaystyle U=mgh}
5081:
4986:
4930:
4817:
4788:
4679:
4568:
4209:
4096:
3866:move at the velocity
3834:
3805:
3712:
3652:electromagnetic force
3630:
3543:
3225:
3215:are used for storing
3211:
3176:
2921:
2871:
2853:
2777:
2530:
2117:
1979:
1885:
1800:
1709:
1527:
1330:
1226:
1203:
1163:intermolecular forces
594:Hamiltonian mechanics
412:Statistical mechanics
331:
263:
8734:Efficient energy use
8707:Airborne wind energy
8685:Solar thermal energy
8592:Electricity delivery
8204:Thermodynamic system
8149:Irreversible process
7751:7 April 2014 at the
7575:Burton Paul (1979).
7273:
7253:
7169:
7149:
7145:is a constant (i.e.
7129:
7103:
7077:
6996:
6964:
6932:
6906:
6880:
6858:{\displaystyle \mu }
6849:
6705:
6654:
6622:
6602:
6533:strong nuclear force
6450:
6397:
6366:
6255:
6176:
6079:
5894:
5852:
5826:
5800:
5766:
5740:
5714:
5648:
5552:
5447:
5396:
5295:
5188:
5131:
5041:
4968:Gravitational energy
4864:
4797:
4710:
4579:
4244:
4109:
3904:
3841:a vector of length 1
3814:
3746:
3674:
3581:
3315:
3032:
2888:
2792:
2566:
2537:For the force field
2221:
2158:Given a force field
1994:
1917:
1813:
1722:
1583:
1439:
1297:
1275:and whose height is
1212:
1180:
1148:strong nuclear force
817:Angular acceleration
809:Rotational frequency
589:Lagrangian mechanics
582:Analytical mechanics
338:Second law of motion
291:
233:
8756:Energy conservation
8678:Photovoltaic system
8651:Nuclear power plant
8336:Quantum fluctuation
8199:Thermodynamic state
8174:Thermal equilibrium
7853:Jackson, John David
7551:Classical Mechanics
7422:W. J. Millar, ed.,
7118:{\displaystyle b+c}
7092:{\displaystyle a+c}
6921:{\displaystyle U=b}
6895:{\displaystyle U=a}
6598:, often denoted by
6537:Weak nuclear forces
6307:
6218:vacuum permittivity
5956:It is also used by
5909:{\displaystyle U=0}
5867:{\displaystyle r=0}
5841:{\displaystyle U=0}
5815:{\displaystyle r=0}
5781:{\displaystyle U=0}
5729:{\displaystyle r=0}
5127:, the equation is:
5037:, and the equation
5007:Local approximation
4978:gravitational force
4839:vacuum permittivity
4435:
4284:
4042:
3966:
3498:
3423:
3379:
3338:
3121:
3069:
2671:
2620:
2369:
2287:
1281:. The acceleration
1174:conservative forces
1105:conservative forces
669:Harmonic oscillator
647:Equations of motion
282:Classical mechanics
276:Part of a series on
8793:Sustainable energy
8771:Energy development
8761:Energy consumption
8597:Energy engineering
7314:de Rham cohomology
7279:
7259:
7236:
7155:
7135:
7115:
7089:
7063:
6982:
6950:
6918:
6892:
6876:from point A with
6874:infinitesimal mass
6855:
6835:
6687:
6628:
6608:
6592:conservative force
6498:
6425:
6374:
6344:electric potential
6331:
6293:
6191:
6162:
6075:which is given by
6071:on another charge
6060:
6027:chemical potential
5960:for lifting up an
5946:. For example, in
5928:for more on this.
5906:
5864:
5838:
5812:
5778:
5752:
5726:
5682:
5634:
5540:
5529:
5426:
5348:
5215:
5149:
5076:
4992:
4945:inverse-square law
4925:
4812:
4783:
4695:on another charge
4674:
4563:
4407:
4256:
4204:
4091:
4014:
3916:
3829:
3800:
3707:
3625:
3538:
3459:
3409:
3365:
3324:
3231:
3220:
3171:
3093:
3041:
2916:
2882:
2848:
2772:
2770:
2657:
2606:
2525:
2523:
2355:
2273:
2112:
1974:
1880:
1795:
1704:
1522:
1325:
1221:
1198:
1152:weak nuclear force
1074:of an object, the
985:Physics portal
599:Routhian mechanics
474:Frame of reference
326:
258:
8928:
8927:
8695:Solar power tower
8341:Quantum potential
8179:Thermal reservoir
8082:Energy transition
7931:978-1-4390-4844-3
7787:Downing, Louise.
7694:Levine, Jonah G.
7588:978-0-13-516062-6
7581:. Prentice-Hall.
7561:978-1-891389-22-1
7387:978-0-8018-9455-8
7352:978-81-203-3862-3
7262:{\displaystyle U}
7158:{\displaystyle c}
7138:{\displaystyle c}
6833:
6820:
6791:
6753:
6735:
6682:
6631:{\displaystyle V}
6470:
6239:required to move
6188:
6156:
6145:
6123:
6025:The similar term
6008:chemical reaction
5677:
5627:
5597:
5522:
5492:
5424:
5337:
4920:
4905:
4848:required to move
4809:
4777:
4766:
4744:
4669:
4657:
4638:
4595:
4558:
4519:
4473:
4459:
4376:
4346:
4308:
4186:
4156:
4066:
3990:
3826:
3794:
3783:
3702:
3607:
3523:
3448:
2685:
2383:
2217:, and computing,
2094:
2071:
2048:
1965:
1941:
1898:Potential energy
1871:
1842:
1694:
1666:
1638:
1516:
1492:
1035:
1034:
782:Centrifugal force
777:Centripetal force
733:Euler's equations
718:Relative velocity
494:Moment of inertia
324:
298:
271:
270:
53:In the case of a
16:(Redirected from
8963:
8941:Energy (physics)
8916:
8915:
8904:
8892:
8891:
8880:
8879:
8854:Carbon footprint
8788:Renewable energy
8629:Hydroelectricity
8619:Geothermal power
8062:Energy condition
7994:
7987:
7980:
7971:
7954:
7935:
7907:
7906:
7898:
7892:
7891:
7879:
7873:
7872:
7849:
7843:
7842:
7822:
7816:
7802:
7796:
7785:
7779:
7767:
7761:
7743:
7737:
7718:
7712:
7692:
7686:
7669:
7663:
7662:
7649:
7643:
7642:
7624:
7618:
7617:
7610:
7604:
7599:
7593:
7592:
7572:
7566:
7565:
7545:
7539:
7538:
7514:
7508:
7507:
7489:
7480:
7479:
7477:
7475:
7440:
7431:
7401:
7392:
7391:
7363:
7357:
7356:
7334:
7288:
7286:
7285:
7280:
7268:
7266:
7265:
7260:
7245:
7243:
7242:
7237:
7187:
7186:
7164:
7162:
7161:
7156:
7144:
7142:
7141:
7136:
7124:
7122:
7121:
7116:
7098:
7096:
7095:
7090:
7072:
7070:
7069:
7064:
7020:
7019:
6991:
6989:
6988:
6983:
6959:
6957:
6956:
6951:
6927:
6925:
6924:
6919:
6902:to point B with
6901:
6899:
6898:
6893:
6864:
6862:
6861:
6856:
6844:
6842:
6841:
6836:
6834:
6826:
6821:
6819:
6811:
6800:
6792:
6787:
6767:
6759:
6755:
6754:
6749:
6741:
6736:
6731:
6723:
6696:
6694:
6693:
6688:
6683:
6678:
6667:
6637:
6635:
6634:
6629:
6617:
6615:
6614:
6609:
6581:scalar potential
6513:
6507:
6505:
6504:
6499:
6487:
6479:
6471:
6463:
6445:
6434:
6432:
6431:
6426:
6421:
6413:
6392:
6386:magnetic B-field
6383:
6381:
6380:
6375:
6373:
6359:The energy of a
6340:
6338:
6337:
6332:
6330:
6319:
6318:
6317:
6306:
6301:
6283:
6282:
6273:
6272:
6200:
6198:
6197:
6192:
6190:
6189:
6181:
6171:
6169:
6168:
6163:
6158:
6157:
6149:
6146:
6144:
6143:
6134:
6126:
6124:
6122:
6121:
6120:
6101:
6093:
6092:
6087:
5926:inflation theory
5915:
5913:
5912:
5907:
5889:
5879:
5873:
5871:
5870:
5865:
5847:
5845:
5844:
5839:
5821:
5819:
5818:
5813:
5787:
5785:
5784:
5779:
5762:. The choice of
5761:
5759:
5758:
5753:
5735:
5733:
5732:
5727:
5709:
5691:
5689:
5688:
5683:
5678:
5670:
5643:
5641:
5640:
5635:
5633:
5629:
5628:
5626:
5625:
5616:
5615:
5606:
5598:
5596:
5595:
5586:
5585:
5576:
5538:
5536:
5535:
5530:
5528:
5524:
5523:
5521:
5520:
5511:
5510:
5501:
5493:
5491:
5490:
5481:
5480:
5471:
5435:
5433:
5432:
5427:
5425:
5420:
5400:
5391:
5387:
5378:
5372:
5365:
5357:
5355:
5354:
5349:
5338:
5333:
5332:
5331:
5322:
5321:
5311:
5288:
5279:
5273:
5264:
5255:
5237:
5224:
5222:
5221:
5216:
5176:
5170:
5164:
5158:
5156:
5155:
5150:
5114:
5108:
5104:
5098:
5085:
5083:
5082:
5077:
5072:
5071:
5053:
5052:
5032:
5021:standard gravity
5018:
4934:
4932:
4931:
4926:
4921:
4916:
4908:
4906:
4904:
4903:
4902:
4883:
4821:
4819:
4818:
4813:
4811:
4810:
4802:
4792:
4790:
4789:
4784:
4779:
4778:
4770:
4767:
4765:
4764:
4755:
4747:
4745:
4743:
4742:
4741:
4722:
4717:
4683:
4681:
4680:
4675:
4670:
4668:
4667:
4658:
4650:
4648:
4640:
4639:
4631:
4628:
4627:
4609:
4608:
4596:
4594:
4583:
4572:
4570:
4569:
4564:
4559:
4557:
4553:
4552:
4536:
4525:
4520:
4518:
4514:
4513:
4497:
4486:
4475:
4474:
4466:
4460:
4458:
4457:
4448:
4437:
4434:
4433:
4432:
4422:
4421:
4420:
4390:
4389:
4384:
4378:
4377:
4369:
4360:
4359:
4354:
4348:
4347:
4339:
4327:
4326:
4321:
4309:
4307:
4306:
4297:
4286:
4283:
4282:
4281:
4271:
4270:
4269:
4213:
4211:
4210:
4205:
4200:
4199:
4194:
4188:
4187:
4179:
4170:
4169:
4164:
4158:
4157:
4149:
4143:
4134:
4133:
4128:
4116:
4100:
4098:
4097:
4092:
4080:
4072:
4067:
4065:
4064:
4055:
4044:
4041:
4040:
4039:
4029:
4028:
4027:
4007:
3996:
3991:
3989:
3988:
3979:
3968:
3965:
3961:
3960:
3948:
3942:
3938:
3937:
3925:
3899:
3885:
3871:
3838:
3836:
3835:
3830:
3828:
3827:
3819:
3809:
3807:
3806:
3801:
3796:
3795:
3787:
3784:
3782:
3781:
3772:
3761:
3753:
3716:
3714:
3713:
3708:
3703:
3698:
3687:
3634:
3632:
3631:
3626:
3621:
3620:
3608:
3600:
3554:
3547:
3545:
3544:
3539:
3537:
3536:
3524:
3516:
3497:
3483:
3479:
3478:
3449:
3447:
3439:
3431:
3422:
3417:
3395:
3394:
3378:
3373:
3351:
3343:
3337:
3332:
3310:
3282:
3243:
3193:
3180:
3178:
3177:
3172:
3161:
3160:
3141:
3140:
3131:
3130:
3120:
3119:
3118:
3108:
3107:
3106:
3082:
3074:
3068:
3067:
3066:
3056:
3055:
3054:
3027:
2997:
2960:
2925:
2923:
2922:
2917:
2900:
2899:
2857:
2855:
2854:
2849:
2844:
2836:
2828:
2820:
2781:
2779:
2778:
2773:
2771:
2761:
2760:
2755:
2737:
2736:
2731:
2697:
2686:
2684:
2673:
2670:
2665:
2647:
2633:
2625:
2619:
2614:
2598:
2587:
2582:
2581:
2560:gradient theorem
2557:
2534:
2532:
2531:
2526:
2524:
2517:
2513:
2512:
2507:
2491:
2487:
2486:
2481:
2453:
2427:
2395:
2384:
2382:
2371:
2368:
2363:
2348:
2326:
2322:
2301:
2286:
2281:
2265:
2251:
2237:
2236:
2216:
2202:
2188:
2168:gradient theorem
2138:defines a force
2134:. The potential
2121:
2119:
2118:
2113:
2108:
2100:
2096:
2095:
2093:
2085:
2077:
2072:
2070:
2062:
2054:
2049:
2047:
2039:
2031:
2018:
2004:
1983:
1981:
1980:
1975:
1967:
1966:
1963:
1961:
1943:
1942:
1939:
1937:
1912:
1889:
1887:
1886:
1881:
1873:
1872:
1869:
1867:
1858:
1844:
1843:
1840:
1838:
1829:
1807:gradient theorem
1804:
1802:
1801:
1796:
1791:
1780:
1769:
1768:
1756:
1745:
1740:
1739:
1713:
1711:
1710:
1705:
1700:
1696:
1695:
1693:
1685:
1684:
1672:
1667:
1665:
1657:
1656:
1644:
1639:
1637:
1629:
1628:
1616:
1606:
1605:
1590:
1531:
1529:
1528:
1523:
1518:
1517:
1514:
1512:
1494:
1493:
1490:
1488:
1473:
1462:
1457:
1456:
1403:scalar potential
1376:
1374:
1373:
1370:
1367:
1334:
1332:
1331:
1326:
1309:
1308:
1292:
1286:
1280:
1274:
1262:or the force of
1230:
1228:
1227:
1222:
1207:
1205:
1204:
1199:
1123:scalar potential
1051:potential energy
1043:potential energy
1027:
1020:
1013:
1000:
995:
994:
987:
983:
982:
888:Johann Bernoulli
883:Daniel Bernoulli
804:Tangential speed
708:
684:
659:Fictitious force
654:
506:Mechanical power
496:
437:Angular momentum
335:
333:
332:
327:
325:
323:
315:
314:
305:
300:
299:
273:
267:
265:
264:
259:
186:
185:
181:
155:
154:
150:
111:other quantities
109:Derivations from
51:
42:Potential energy
39:
21:
8971:
8970:
8966:
8965:
8964:
8962:
8961:
8960:
8946:Forms of energy
8931:
8930:
8929:
8924:
8868:
8864:Waste-to-energy
8842:
8778:Energy security
8724:
8718:
8574:
8566:
8545:Natural uranium
8479:
8460:Mechanical wave
8391:Energy carriers
8385:
8225:
8154:Isolated system
8032:
8026:
8003:
7998:
7961:
7951:
7938:
7932:
7919:
7916:
7911:
7910:
7900:
7899:
7895:
7881:
7880:
7876:
7869:
7851:
7850:
7846:
7839:
7824:
7823:
7819:
7813:Wayback Machine
7804:Kernan, Aedan.
7803:
7799:
7786:
7782:
7768:
7764:
7753:Wayback Machine
7744:
7740:
7733:Duke University
7729:Wayback Machine
7720:Yang, Chi-Jen.
7719:
7715:
7703:Wayback Machine
7693:
7689:
7680:Wayback Machine
7671:Jacob, Thierry.
7670:
7666:
7661:. 3 March 2011.
7651:
7650:
7646:
7639:
7626:
7625:
7621:
7612:
7611:
7607:
7600:
7596:
7589:
7574:
7573:
7569:
7562:
7547:
7546:
7542:
7535:
7516:
7515:
7511:
7504:
7491:
7490:
7483:
7473:
7471:
7442:
7441:
7434:
7402:
7395:
7388:
7365:
7364:
7360:
7353:
7336:
7335:
7331:
7326:
7306:Euclidean space
7271:
7270:
7251:
7250:
7172:
7167:
7166:
7147:
7146:
7127:
7126:
7101:
7100:
7075:
7074:
6999:
6994:
6993:
6962:
6961:
6930:
6929:
6904:
6903:
6878:
6877:
6847:
6846:
6812:
6801:
6768:
6742:
6724:
6721:
6717:
6703:
6702:
6699:specific energy
6668:
6652:
6651:
6620:
6619:
6600:
6599:
6585:vector gradient
6561:
6521:
6509:
6448:
6447:
6441:
6395:
6394:
6388:
6364:
6363:
6361:magnetic moment
6357:
6309:
6261:
6253:
6252:
6215:
6174:
6173:
6135:
6127:
6112:
6105:
6082:
6077:
6076:
6065:
6045:electric charge
6041:
6035:
6020:electrochemical
6000:
5998:Chemical energy
5994:
5972:Roller coasters
5940:
5934:
5892:
5891:
5885:
5875:
5850:
5849:
5824:
5823:
5798:
5797:
5764:
5763:
5738:
5737:
5712:
5711:
5705:
5697:
5646:
5645:
5617:
5607:
5587:
5577:
5571:
5567:
5550:
5549:
5512:
5502:
5482:
5472:
5466:
5462:
5445:
5444:
5401:
5394:
5393:
5389:
5383:
5374:
5367:
5361:
5323:
5313:
5312:
5293:
5292:
5284:
5275:
5272:
5266:
5263:
5257:
5251:
5233:
5230:
5228:General formula
5186:
5185:
5172:
5166:
5160:
5129:
5128:
5110:
5109:is the product
5106:
5100:
5090:
5063:
5044:
5039:
5038:
5024:
5013:
5009:
4974:
4962:Main articles:
4960:
4940:
4938:Reference level
4909:
4894:
4887:
4862:
4861:
4836:
4795:
4794:
4756:
4748:
4733:
4726:
4708:
4707:
4689:
4659:
4616:
4597:
4587:
4577:
4576:
4544:
4537:
4526:
4505:
4498:
4487:
4449:
4438:
4424:
4412:
4379:
4349:
4316:
4298:
4287:
4273:
4261:
4242:
4241:
4231:
4222:
4189:
4159:
4123:
4107:
4106:
4056:
4045:
4031:
4019:
3980:
3969:
3952:
3929:
3902:
3901:
3897:
3887:
3883:
3873:
3867:
3812:
3811:
3773:
3762:
3744:
3743:
3725:
3688:
3672:
3671:
3664:
3612:
3579:
3578:
3568:
3549:
3528:
3470:
3440:
3432:
3386:
3313:
3312:
3308:
3301:
3294:
3284:
3249:
3234:
3206:
3200:
3182:
3152:
3132:
3122:
3110:
3098:
3058:
3046:
3030:
3029:
3025:
3018:
3011:
2999:
2962:
2957:
2946:
2891:
2886:
2885:
2866:
2790:
2789:
2769:
2768:
2750:
2726:
2677:
2645:
2644:
2599:
2573:
2564:
2563:
2542:
2522:
2521:
2502:
2498:
2476:
2472:
2375:
2346:
2345:
2317:
2266:
2228:
2219:
2218:
2204:
2190:
2171:
2156:
2142:at every point
2086:
2078:
2063:
2055:
2040:
2032:
2029:
2025:
1992:
1991:
1956:
1932:
1915:
1914:
1899:
1862:
1851:
1833:
1822:
1811:
1810:
1773:
1760:
1731:
1720:
1719:
1686:
1677:
1673:
1658:
1649:
1645:
1630:
1621:
1617:
1614:
1610:
1598:
1581:
1580:
1557:
1540:, from A to B.
1507:
1483:
1448:
1437:
1436:
1434:
1427:
1407:vector gradient
1383:
1371:
1368:
1365:
1364:
1362:
1359:William Thomson
1351:William Rankine
1343:
1300:
1295:
1294:
1288:
1282:
1276:
1270:
1249:
1210:
1209:
1178:
1177:
1132:
1084:electric charge
1055:William Rankine
1031:
990:
977:
976:
969:
968:
967:
842:
834:
833:
813:
767:Circular motion
761:
751:
750:
749:
706:
676:
673:
652:
631:
623:
622:
619:
618:
576:
566:
558:
557:
556:
515:
511:Mechanical work
504:
488:
426:
418:
417:
416:
371:
363:
340:
316:
306:
289:
288:
231:
230:
204:
183:
179:
178:
173:
152:
148:
147:
141:
112:
110:
77:
70:
35:
28:
23:
22:
15:
12:
11:
5:
8969:
8967:
8959:
8958:
8953:
8948:
8943:
8933:
8932:
8926:
8925:
8923:
8922:
8910:
8898:
8886:
8873:
8870:
8869:
8867:
8866:
8861:
8859:Jevons paradox
8856:
8850:
8848:
8844:
8843:
8841:
8840:
8835:
8830:
8825:
8820:
8815:
8810:
8805:
8800:
8795:
8790:
8785:
8783:Energy storage
8780:
8775:
8774:
8773:
8763:
8758:
8753:
8752:
8751:
8746:
8741:
8730:
8728:
8720:
8719:
8717:
8716:
8715:
8714:
8709:
8699:
8698:
8697:
8692:
8682:
8681:
8680:
8675:
8665:
8660:
8659:
8658:
8653:
8643:
8642:
8641:
8636:
8631:
8621:
8616:
8615:
8614:
8609:
8599:
8594:
8589:
8587:Electric power
8584:
8578:
8576:
8568:
8567:
8565:
8564:
8559:
8554:
8549:
8548:
8547:
8537:
8532:
8527:
8522:
8517:
8516:
8515:
8510:
8505:
8495:
8489:
8487:
8485:Primary energy
8481:
8480:
8478:
8477:
8472:
8467:
8462:
8457:
8456:
8455:
8445:
8444:
8443:
8433:
8432:
8431:
8426:
8416:
8411:
8406:
8401:
8395:
8393:
8387:
8386:
8384:
8383:
8378:
8373:
8368:
8363:
8358:
8353:
8348:
8343:
8338:
8333:
8328:
8323:
8318:
8313:
8308:
8303:
8298:
8293:
8288:
8283:
8282:
8281:
8271:
8266:
8261:
8256:
8251:
8246:
8245:
8244:
8233:
8231:
8227:
8226:
8224:
8223:
8222:
8221:
8216:
8211:
8206:
8201:
8196:
8191:
8186:
8181:
8176:
8171:
8166:
8161:
8156:
8151:
8146:
8141:
8136:
8131:
8126:
8121:
8119:Entropic force
8116:
8109:Thermodynamics
8106:
8101:
8100:
8099:
8094:
8084:
8079:
8074:
8069:
8064:
8059:
8058:
8057:
8047:
8042:
8036:
8034:
8028:
8027:
8025:
8024:
8019:
8014:
8008:
8005:
8004:
7999:
7997:
7996:
7989:
7982:
7974:
7968:
7967:
7960:
7959:External links
7957:
7956:
7955:
7949:
7936:
7930:
7915:
7912:
7909:
7908:
7893:
7890:. p. 152.
7874:
7867:
7844:
7837:
7817:
7797:
7793:Bloomberg News
7780:
7778:, 3 March 2012
7762:
7746:Energy Storage
7738:
7713:
7687:
7664:
7644:
7637:
7619:
7605:
7594:
7587:
7567:
7560:
7540:
7533:
7509:
7502:
7481:
7455:(2): 185–196.
7432:
7393:
7386:
7358:
7351:
7328:
7327:
7325:
7322:
7278:
7258:
7235:
7232:
7229:
7226:
7223:
7220:
7217:
7214:
7211:
7208:
7205:
7202:
7199:
7196:
7193:
7190:
7185:
7182:
7179:
7175:
7154:
7134:
7114:
7111:
7108:
7088:
7085:
7082:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7018:
7015:
7012:
7009:
7006:
7002:
6981:
6978:
6975:
6972:
6969:
6949:
6946:
6943:
6940:
6937:
6917:
6914:
6911:
6891:
6888:
6885:
6854:
6832:
6829:
6824:
6818:
6815:
6810:
6807:
6804:
6798:
6795:
6790:
6786:
6783:
6780:
6777:
6774:
6771:
6765:
6762:
6758:
6752:
6748:
6745:
6739:
6734:
6730:
6727:
6720:
6716:
6713:
6710:
6686:
6681:
6677:
6674:
6671:
6665:
6662:
6659:
6627:
6607:
6560:
6557:
6529:atomic nucleus
6520:
6517:
6497:
6494:
6491:
6486:
6482:
6478:
6474:
6469:
6466:
6461:
6458:
6455:
6446:in a field is
6424:
6420:
6416:
6412:
6408:
6405:
6402:
6372:
6356:
6353:
6329:
6325:
6322:
6316:
6312:
6305:
6300:
6296:
6292:
6289:
6286:
6281:
6276:
6271:
6268:
6264:
6260:
6213:
6187:
6184:
6161:
6155:
6152:
6142:
6138:
6133:
6130:
6119:
6115:
6111:
6108:
6104:
6099:
6096:
6091:
6086:
6064:
6061:
6037:Main article:
6034:
6031:
6016:photosynthesis
6004:chemical bonds
5996:Main article:
5993:
5990:
5981:kinetic energy
5958:counterweights
5933:
5930:
5905:
5902:
5899:
5863:
5860:
5857:
5837:
5834:
5831:
5811:
5808:
5805:
5777:
5774:
5771:
5751:
5748:
5745:
5725:
5722:
5719:
5710:becomes zero:
5696:
5693:
5681:
5676:
5673:
5668:
5665:
5662:
5659:
5656:
5653:
5632:
5624:
5620:
5614:
5610:
5604:
5601:
5594:
5590:
5584:
5580:
5574:
5570:
5566:
5563:
5560:
5557:
5527:
5519:
5515:
5509:
5505:
5499:
5496:
5489:
5485:
5479:
5475:
5469:
5465:
5461:
5458:
5455:
5452:
5423:
5419:
5416:
5413:
5410:
5407:
5404:
5347:
5344:
5341:
5336:
5330:
5326:
5320:
5316:
5309:
5306:
5303:
5300:
5274:at a distance
5270:
5261:
5229:
5226:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5148:
5145:
5142:
5139:
5136:
5075:
5070:
5066:
5062:
5059:
5056:
5051:
5047:
5008:
5005:
4997:kinetic energy
4959:
4956:
4939:
4936:
4924:
4919:
4915:
4912:
4901:
4897:
4893:
4890:
4886:
4881:
4878:
4875:
4872:
4869:
4834:
4808:
4805:
4782:
4776:
4773:
4763:
4759:
4754:
4751:
4740:
4736:
4732:
4729:
4725:
4720:
4716:
4688:
4685:
4673:
4666:
4662:
4656:
4653:
4646:
4643:
4637:
4634:
4626:
4623:
4619:
4615:
4612:
4607:
4604:
4600:
4593:
4590:
4586:
4562:
4556:
4551:
4547:
4543:
4540:
4535:
4532:
4529:
4523:
4517:
4512:
4508:
4504:
4501:
4496:
4493:
4490:
4484:
4481:
4478:
4472:
4469:
4463:
4456:
4452:
4447:
4444:
4441:
4431:
4427:
4419:
4415:
4410:
4406:
4403:
4400:
4397:
4393:
4388:
4383:
4375:
4372:
4366:
4363:
4358:
4353:
4345:
4342:
4336:
4333:
4330:
4325:
4320:
4315:
4312:
4305:
4301:
4296:
4293:
4290:
4280:
4276:
4268:
4264:
4259:
4255:
4252:
4249:
4227:
4218:
4203:
4198:
4193:
4185:
4182:
4176:
4173:
4168:
4163:
4155:
4152:
4146:
4142:
4137:
4132:
4127:
4122:
4119:
4115:
4090:
4087:
4084:
4079:
4075:
4071:
4063:
4059:
4054:
4051:
4048:
4038:
4034:
4026:
4022:
4017:
4013:
4010:
4006:
4002:
3999:
3995:
3987:
3983:
3978:
3975:
3972:
3964:
3959:
3955:
3951:
3947:
3941:
3936:
3932:
3928:
3924:
3919:
3915:
3912:
3909:
3895:
3881:
3843:pointing from
3825:
3822:
3799:
3793:
3790:
3780:
3776:
3771:
3768:
3765:
3759:
3756:
3752:
3724:
3721:
3706:
3701:
3697:
3694:
3691:
3685:
3682:
3679:
3663:
3660:
3656:kinetic energy
3624:
3619:
3615:
3611:
3606:
3603:
3598:
3595:
3592:
3589:
3586:
3577:The function
3566:
3535:
3531:
3527:
3522:
3519:
3514:
3511:
3508:
3504:
3501:
3496:
3493:
3490:
3487:
3482:
3477:
3473:
3469:
3466:
3462:
3458:
3455:
3452:
3446:
3443:
3438:
3435:
3429:
3426:
3421:
3416:
3412:
3408:
3405:
3402:
3399:
3393:
3389:
3385:
3382:
3377:
3372:
3368:
3364:
3361:
3358:
3355:
3350:
3346:
3342:
3336:
3331:
3327:
3323:
3320:
3306:
3299:
3292:
3202:Main article:
3199:
3196:
3170:
3167:
3164:
3159:
3155:
3151:
3148:
3145:
3139:
3135:
3129:
3125:
3117:
3113:
3105:
3101:
3096:
3092:
3089:
3086:
3081:
3077:
3073:
3065:
3061:
3053:
3049:
3044:
3040:
3037:
3023:
3016:
3009:
2955:
2915:
2912:
2909:
2906:
2903:
2898:
2894:
2865:
2862:
2847:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2816:
2812:
2809:
2806:
2803:
2800:
2797:
2767:
2764:
2759:
2754:
2749:
2746:
2743:
2740:
2735:
2730:
2725:
2722:
2719:
2716:
2713:
2709:
2706:
2703:
2700:
2696:
2692:
2689:
2683:
2680:
2676:
2669:
2664:
2660:
2656:
2653:
2650:
2648:
2646:
2643:
2640:
2637:
2632:
2628:
2624:
2618:
2613:
2609:
2605:
2602:
2600:
2597:
2593:
2590:
2586:
2580:
2576:
2572:
2571:
2520:
2516:
2511:
2506:
2501:
2497:
2494:
2490:
2485:
2480:
2475:
2471:
2468:
2465:
2462:
2459:
2456:
2452:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2394:
2390:
2387:
2381:
2378:
2374:
2367:
2362:
2358:
2354:
2351:
2349:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2325:
2321:
2316:
2313:
2310:
2307:
2304:
2300:
2296:
2293:
2290:
2285:
2280:
2276:
2272:
2269:
2267:
2264:
2260:
2257:
2254:
2250:
2246:
2243:
2240:
2235:
2231:
2227:
2226:
2155:
2152:
2122:and the force
2111:
2107:
2103:
2099:
2092:
2089:
2084:
2081:
2075:
2069:
2066:
2061:
2058:
2052:
2046:
2043:
2038:
2035:
2028:
2024:
2021:
2017:
2014:
2010:
2007:
2003:
2000:
1973:
1970:
1960:
1955:
1952:
1949:
1946:
1936:
1931:
1928:
1925:
1922:
1879:
1876:
1866:
1861:
1857:
1854:
1850:
1847:
1837:
1832:
1828:
1825:
1821:
1818:
1794:
1790:
1786:
1783:
1779:
1776:
1772:
1767:
1763:
1759:
1755:
1751:
1748:
1744:
1738:
1734:
1730:
1727:
1703:
1699:
1692:
1689:
1683:
1680:
1676:
1670:
1664:
1661:
1655:
1652:
1648:
1642:
1636:
1633:
1627:
1624:
1620:
1613:
1609:
1604:
1601:
1597:
1593:
1589:
1556:
1553:
1521:
1511:
1506:
1503:
1500:
1497:
1487:
1482:
1479:
1476:
1472:
1468:
1465:
1461:
1455:
1451:
1447:
1444:
1432:
1425:
1382:
1379:
1342:
1339:
1324:
1321:
1318:
1315:
1312:
1307:
1303:
1247:
1220:
1217:
1197:
1194:
1191:
1188:
1185:
1154:acting on the
1146:; work of the
1131:
1128:
1088:electric field
1061:'s concept of
1033:
1032:
1030:
1029:
1022:
1015:
1007:
1004:
1003:
1002:
1001:
988:
971:
970:
966:
965:
960:
955:
950:
945:
940:
935:
930:
925:
920:
915:
910:
905:
900:
895:
890:
885:
880:
875:
870:
865:
860:
855:
850:
844:
843:
840:
839:
836:
835:
832:
831:
812:
811:
806:
801:
796:
794:Coriolis force
791:
790:
789:
779:
774:
769:
763:
762:
757:
756:
753:
752:
748:
747:
742:
737:
736:
735:
730:
720:
715:
710:
703:
692:
691:
690:
685:
672:
671:
666:
661:
656:
649:
644:
639:
633:
632:
629:
628:
625:
624:
621:
620:
617:
616:
611:
606:
601:
596:
591:
585:
579:
577:
570:
567:
564:
563:
560:
559:
555:
554:
549:
544:
539:
534:
529:
524:
519:
513:
508:
502:
497:
486:
481:
476:
471:
466:
465:
464:
459:
449:
444:
439:
434:
428:
427:
424:
423:
420:
419:
415:
414:
409:
404:
399:
394:
389:
384:
379:
373:
372:
369:
368:
365:
364:
362:
361:
356:
351:
345:
342:
341:
336:
322:
319:
313:
309:
303:
285:
284:
278:
277:
269:
268:
257:
254:
250:
247:
244:
241:
238:
113:
108:
105:
104:
98:
92:
91:
78:
76:Common symbols
75:
72:
71:
67:kinetic energy
52:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8968:
8957:
8954:
8952:
8949:
8947:
8944:
8942:
8939:
8938:
8936:
8921:
8920:
8911:
8909:
8908:
8903:
8899:
8897:
8896:
8887:
8885:
8884:
8875:
8874:
8871:
8865:
8862:
8860:
8857:
8855:
8852:
8851:
8849:
8845:
8839:
8838:United States
8836:
8834:
8833:South America
8831:
8829:
8826:
8824:
8821:
8819:
8816:
8814:
8811:
8809:
8806:
8804:
8801:
8799:
8796:
8794:
8791:
8789:
8786:
8784:
8781:
8779:
8776:
8772:
8769:
8768:
8767:
8766:Energy policy
8764:
8762:
8759:
8757:
8754:
8750:
8747:
8745:
8742:
8740:
8737:
8736:
8735:
8732:
8731:
8729:
8727:
8721:
8713:
8710:
8708:
8705:
8704:
8703:
8700:
8696:
8693:
8691:
8690:Solar furnace
8688:
8687:
8686:
8683:
8679:
8676:
8674:
8671:
8670:
8669:
8666:
8664:
8661:
8657:
8654:
8652:
8649:
8648:
8647:
8646:Nuclear power
8644:
8640:
8637:
8635:
8632:
8630:
8627:
8626:
8625:
8622:
8620:
8617:
8613:
8610:
8608:
8605:
8604:
8603:
8600:
8598:
8595:
8593:
8590:
8588:
8585:
8583:
8580:
8579:
8577:
8573:
8572:Energy system
8569:
8563:
8560:
8558:
8555:
8553:
8550:
8546:
8543:
8542:
8541:
8538:
8536:
8533:
8531:
8528:
8526:
8525:Gravitational
8523:
8521:
8518:
8514:
8511:
8509:
8506:
8504:
8501:
8500:
8499:
8496:
8494:
8491:
8490:
8488:
8486:
8482:
8476:
8473:
8471:
8468:
8466:
8463:
8461:
8458:
8454:
8453:Hydrogen fuel
8451:
8450:
8449:
8446:
8442:
8439:
8438:
8437:
8434:
8430:
8427:
8425:
8422:
8421:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8396:
8394:
8392:
8388:
8382:
8379:
8377:
8374:
8372:
8369:
8367:
8364:
8362:
8359:
8357:
8354:
8352:
8349:
8347:
8344:
8342:
8339:
8337:
8334:
8332:
8329:
8327:
8324:
8322:
8319:
8317:
8314:
8312:
8309:
8307:
8304:
8302:
8299:
8297:
8294:
8292:
8289:
8287:
8284:
8280:
8277:
8276:
8275:
8274:Gravitational
8272:
8270:
8267:
8265:
8262:
8260:
8257:
8255:
8252:
8250:
8247:
8243:
8240:
8239:
8238:
8235:
8234:
8232:
8228:
8220:
8217:
8215:
8212:
8210:
8207:
8205:
8202:
8200:
8197:
8195:
8192:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8170:
8167:
8165:
8162:
8160:
8157:
8155:
8152:
8150:
8147:
8145:
8144:Heat transfer
8142:
8140:
8139:Heat capacity
8137:
8135:
8132:
8130:
8127:
8125:
8122:
8120:
8117:
8115:
8112:
8111:
8110:
8107:
8105:
8102:
8098:
8095:
8093:
8092:Negative mass
8090:
8089:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8072:Energy system
8070:
8068:
8065:
8063:
8060:
8056:
8053:
8052:
8051:
8048:
8046:
8043:
8041:
8038:
8037:
8035:
8029:
8023:
8020:
8018:
8015:
8013:
8010:
8009:
8006:
8002:
7995:
7990:
7988:
7983:
7981:
7976:
7975:
7972:
7966:
7963:
7962:
7958:
7952:
7950:0-7167-0809-4
7946:
7942:
7937:
7933:
7927:
7923:
7918:
7917:
7913:
7904:
7897:
7894:
7889:
7885:
7878:
7875:
7870:
7868:0-471-43132-X
7864:
7860:
7859:
7854:
7848:
7845:
7840:
7838:0-19-851791-2
7834:
7830:
7829:
7821:
7818:
7814:
7810:
7807:
7801:
7798:
7794:
7790:
7784:
7781:
7777:
7776:
7775:The Economist
7771:
7766:
7763:
7759:
7758:
7754:
7750:
7747:
7742:
7739:
7735:
7734:
7730:
7726:
7723:
7717:
7714:
7710:
7709:
7704:
7700:
7697:
7691:
7688:
7684:
7681:
7677:
7674:
7668:
7665:
7660:
7659:
7658:The Economist
7654:
7648:
7645:
7640:
7638:0-201-14942-7
7634:
7630:
7623:
7620:
7615:
7609:
7606:
7603:
7598:
7595:
7590:
7584:
7580:
7579:
7571:
7568:
7563:
7557:
7553:
7552:
7544:
7541:
7536:
7534:0-13-109686-9
7530:
7526:
7522:
7521:
7513:
7510:
7505:
7503:0-226-76420-6
7499:
7495:
7488:
7486:
7482:
7470:
7466:
7462:
7458:
7454:
7450:
7446:
7439:
7437:
7433:
7429:
7425:
7421:
7417:
7413:
7410:
7406:
7400:
7398:
7394:
7389:
7383:
7379:
7375:
7374:
7369:
7362:
7359:
7354:
7348:
7344:
7340:
7333:
7330:
7323:
7321:
7319:
7315:
7311:
7307:
7303:
7299:
7294:
7292:
7276:
7256:
7247:
7233:
7230:
7227:
7224:
7218:
7215:
7212:
7206:
7200:
7197:
7194:
7188:
7183:
7177:
7173:
7152:
7132:
7112:
7109:
7106:
7086:
7083:
7080:
7060:
7057:
7051:
7048:
7045:
7039:
7033:
7030:
7027:
7021:
7016:
7010:
7004:
7000:
6976:
6973:
6970:
6944:
6941:
6938:
6915:
6912:
6909:
6889:
6886:
6883:
6875:
6870:
6868:
6852:
6830:
6827:
6822:
6816:
6813:
6808:
6805:
6802:
6796:
6793:
6788:
6781:
6778:
6775:
6769:
6763:
6760:
6756:
6750:
6746:
6743:
6737:
6732:
6728:
6725:
6718:
6714:
6711:
6708:
6700:
6684:
6679:
6675:
6672:
6669:
6663:
6660:
6657:
6649:
6645:
6641:
6625:
6605:
6597:
6593:
6588:
6586:
6582:
6578:
6574:
6570:
6566:
6558:
6556:
6554:
6550:
6544:
6542:
6538:
6534:
6530:
6526:
6518:
6516:
6512:
6495:
6492:
6489:
6480:
6472:
6467:
6464:
6459:
6456:
6453:
6444:
6440:
6439:magnetization
6435:
6422:
6414:
6406:
6403:
6400:
6391:
6387:
6362:
6354:
6352:
6350:
6346:
6345:
6323:
6320:
6303:
6298:
6294:
6290:
6287:
6269:
6266:
6262:
6250:
6247:to any point
6246:
6242:
6238:
6233:
6230:
6226:
6221:
6219:
6212:
6208:
6204:
6159:
6140:
6136:
6131:
6128:
6117:
6113:
6109:
6106:
6102:
6097:
6094:
6089:
6074:
6070:
6062:
6057:
6053:
6049:
6046:
6040:
6032:
6030:
6028:
6023:
6021:
6017:
6013:
6009:
6005:
5999:
5991:
5989:
5987:
5982:
5976:
5973:
5969:
5967:
5963:
5959:
5954:
5951:
5949:
5945:
5939:
5931:
5929:
5927:
5921:
5919:
5903:
5900:
5897:
5888:
5883:
5878:
5861:
5858:
5855:
5835:
5832:
5829:
5809:
5806:
5803:
5795:
5790:
5775:
5772:
5769:
5746:
5743:
5723:
5720:
5717:
5708:
5703:
5694:
5692:
5679:
5674:
5671:
5666:
5663:
5660:
5657:
5654:
5651:
5630:
5622:
5618:
5612:
5608:
5602:
5599:
5592:
5588:
5582:
5578:
5572:
5568:
5564:
5561:
5558:
5555:
5547:
5545:
5525:
5517:
5513:
5507:
5503:
5497:
5494:
5487:
5483:
5477:
5473:
5467:
5463:
5459:
5456:
5453:
5450:
5441:
5437:
5421:
5414:
5411:
5408:
5402:
5386:
5380:
5377:
5370:
5364:
5358:
5345:
5342:
5339:
5334:
5328:
5324:
5318:
5314:
5307:
5304:
5301:
5298:
5290:
5287:
5283:
5278:
5269:
5260:
5254:
5249:
5245:
5241:
5236:
5227:
5225:
5212:
5209:
5203:
5200:
5197:
5194:
5182:
5180:
5175:
5169:
5163:
5146:
5143:
5140:
5137:
5134:
5126:
5122:
5118:
5113:
5103:
5097:
5093:
5086:
5073:
5068:
5064:
5057:
5054:
5049:
5045:
5036:
5033:equation for
5031:
5027:
5022:
5016:
5006:
5004:
5000:
4998:
4990:
4985:
4981:
4979:
4973:
4972:Gravity field
4969:
4965:
4957:
4955:
4953:
4948:
4946:
4937:
4935:
4922:
4917:
4913:
4910:
4899:
4895:
4891:
4888:
4884:
4879:
4873:
4867:
4859:
4856:to any point
4855:
4851:
4847:
4842:
4840:
4833:
4829:
4825:
4780:
4761:
4757:
4752:
4749:
4738:
4734:
4730:
4727:
4723:
4718:
4706:
4705:Coulomb's Law
4702:
4698:
4694:
4686:
4684:
4671:
4664:
4660:
4654:
4651:
4644:
4641:
4635:
4632:
4624:
4621:
4617:
4613:
4610:
4605:
4602:
4598:
4591:
4588:
4584:
4573:
4560:
4549:
4545:
4538:
4533:
4530:
4527:
4521:
4510:
4506:
4499:
4494:
4491:
4488:
4482:
4479:
4476:
4470:
4467:
4461:
4454:
4450:
4445:
4442:
4439:
4429:
4425:
4417:
4413:
4408:
4404:
4401:
4398:
4395:
4386:
4373:
4370:
4364:
4361:
4356:
4343:
4340:
4331:
4323:
4313:
4303:
4299:
4294:
4291:
4288:
4278:
4274:
4266:
4262:
4257:
4253:
4250:
4247:
4239:
4235:
4230:
4226:
4221:
4217:
4201:
4196:
4183:
4180:
4174:
4171:
4166:
4153:
4150:
4144:
4135:
4130:
4120:
4117:
4105:are given by
4104:
4088:
4085:
4082:
4073:
4061:
4057:
4052:
4049:
4046:
4036:
4032:
4024:
4020:
4015:
4011:
4008:
4000:
3997:
3985:
3981:
3976:
3973:
3970:
3957:
3953:
3934:
3930:
3917:
3913:
3910:
3907:
3894:
3890:
3880:
3876:
3870:
3865:
3862:Let the mass
3860:
3858:
3854:
3850:
3846:
3842:
3797:
3778:
3774:
3769:
3766:
3763:
3757:
3754:
3742:
3738:
3734:
3730:
3722:
3720:
3717:
3704:
3699:
3695:
3692:
3689:
3683:
3680:
3677:
3669:
3661:
3659:
3657:
3653:
3649:
3645:
3641:
3636:
3622:
3617:
3613:
3609:
3604:
3601:
3596:
3590:
3584:
3575:
3573:
3569:
3562:
3558:
3552:
3533:
3529:
3525:
3520:
3517:
3512:
3509:
3506:
3502:
3499:
3491:
3485:
3475:
3471:
3464:
3460:
3456:
3453:
3450:
3444:
3441:
3436:
3433:
3427:
3424:
3419:
3414:
3410:
3406:
3403:
3400:
3397:
3391:
3387:
3383:
3380:
3375:
3370:
3366:
3362:
3359:
3356:
3353:
3344:
3334:
3329:
3325:
3321:
3318:
3305:
3298:
3291:
3287:
3280:
3276:
3272:
3268:
3264:
3260:
3256:
3252:
3247:
3241:
3237:
3228:
3224:
3218:
3214:
3210:
3205:
3197:
3195:
3191:
3187:
3186:
3168:
3165:
3157:
3153:
3149:
3146:
3143:
3137:
3133:
3127:
3123:
3115:
3111:
3103:
3099:
3094:
3090:
3087:
3084:
3075:
3063:
3059:
3051:
3047:
3042:
3038:
3035:
3022:
3015:
3008:
3004:
3003:
2995:
2991:
2987:
2983:
2979:
2975:
2971:
2967:
2966:
2958:
2951:
2950:
2943:
2941:
2937:
2933:
2929:
2913:
2910:
2907:
2904:
2901:
2896:
2892:
2879:
2878:counterweight
2875:
2870:
2863:
2861:
2858:
2845:
2837:
2829:
2821:
2817:
2810:
2807:
2801:
2795:
2787:
2782:
2765:
2757:
2744:
2741:
2733:
2720:
2717:
2714:
2711:
2701:
2687:
2681:
2678:
2674:
2667:
2662:
2658:
2654:
2651:
2649:
2641:
2638:
2635:
2626:
2616:
2611:
2607:
2603:
2601:
2591:
2588:
2578:
2574:
2561:
2556:
2552:
2549:
2545:
2540:
2535:
2518:
2514:
2509:
2499:
2492:
2488:
2483:
2473:
2466:
2457:
2440:
2431:
2414:
2411:
2408:
2399:
2379:
2376:
2372:
2365:
2360:
2356:
2352:
2350:
2342:
2339:
2336:
2330:
2323:
2314:
2305:
2283:
2278:
2274:
2270:
2268:
2258:
2255:
2233:
2229:
2215:
2211:
2207:
2201:
2197:
2193:
2186:
2182:
2178:
2174:
2169:
2165:
2161:
2153:
2151:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2109:
2101:
2097:
2090:
2082:
2073:
2067:
2059:
2050:
2044:
2036:
2026:
2022:
2019:
2015:
2008:
2005:
2001:
1989:
1984:
1971:
1950:
1947:
1926:
1923:
1920:
1910:
1906:
1902:
1896:
1893:
1877:
1855:
1852:
1848:
1826:
1823:
1819:
1816:
1808:
1792:
1784:
1781:
1777:
1774:
1765:
1761:
1757:
1749:
1746:
1736:
1732:
1728:
1725:
1717:
1701:
1697:
1690:
1681:
1678:
1668:
1662:
1653:
1650:
1640:
1634:
1625:
1622:
1611:
1607:
1602:
1599:
1591:
1578:
1574:
1570:
1566:
1562:
1561:line integral
1554:
1552:
1550:
1546:
1543:The function
1541:
1539:
1535:
1501:
1498:
1477:
1474:
1466:
1463:
1453:
1449:
1445:
1442:
1431:
1424:
1420:
1416:
1410:
1408:
1404:
1400:
1396:
1392:
1388:
1380:
1378:
1360:
1356:
1352:
1348:
1340:
1338:
1335:
1322:
1319:
1316:
1313:
1310:
1305:
1301:
1291:
1285:
1279:
1273:
1267:
1265:
1261:
1257:
1252:
1250:
1243:
1239:
1235:
1218:
1195:
1189:
1186:
1183:
1175:
1170:
1168:
1164:
1160:
1157:
1153:
1149:
1145:
1141:
1140:Coulomb force
1137:
1129:
1127:
1125:
1124:
1119:
1118:
1113:
1112:
1107:
1106:
1099:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1068:
1066:
1065:
1060:
1056:
1052:
1048:
1044:
1040:
1028:
1023:
1021:
1016:
1014:
1009:
1008:
1006:
1005:
999:
989:
986:
981:
975:
974:
973:
972:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
845:
838:
837:
830:
826:
822:
818:
815:
814:
810:
807:
805:
802:
800:
797:
795:
792:
788:
785:
784:
783:
780:
778:
775:
773:
770:
768:
765:
764:
760:
755:
754:
746:
743:
741:
738:
734:
731:
729:
726:
725:
724:
721:
719:
716:
714:
711:
709:
704:
701:
697:
694:
693:
689:
686:
683:
679:
675:
674:
670:
667:
665:
662:
660:
657:
655:
650:
648:
645:
643:
640:
638:
635:
634:
627:
626:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
586:
584:
583:
578:
575:
574:
569:
568:
562:
561:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
523:
520:
518:
514:
512:
509:
507:
503:
501:
498:
495:
491:
487:
485:
482:
480:
477:
475:
472:
470:
467:
463:
460:
458:
455:
454:
453:
450:
448:
445:
443:
440:
438:
435:
433:
430:
429:
422:
421:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
375:
374:
367:
366:
360:
357:
355:
352:
350:
347:
346:
343:
339:
320:
317:
307:
301:
287:
286:
283:
279:
275:
274:
255:
252:
245:
239:
236:
228:
225:
223:
219:
218:
213:
212:
207:
202:
198:
197:
192:
191:
176:
171:
167:
166:
161:
160:
145:
139:
138:gravitational
135:
134:
129:
128:
123:
122:
117:
114:
106:
102:
99:
97:
93:
90:
86:
82:
79:
73:
68:
64:
60:
56:
55:bow and arrow
50:
45:
40:
37:
33:
19:
8917:
8905:
8893:
8881:
8663:Oil refinery
8607:Cogeneration
8540:Nuclear fuel
8346:Quintessence
8325:
8134:Free entropy
8067:Energy level
8031:Fundamental
7940:
7921:
7902:
7896:
7883:
7877:
7857:
7847:
7827:
7820:
7800:
7783:
7773:
7765:
7755:
7741:
7731:
7716:
7706:
7690:
7682:
7667:
7656:
7647:
7628:
7622:
7608:
7597:
7577:
7570:
7550:
7543:
7519:
7512:
7493:
7472:. Retrieved
7452:
7448:
7423:
7419:
7411:
7408:
7404:
7372:
7361:
7342:
7332:
7310:contractible
7295:
7248:
6871:
6867:reduced mass
6647:
6643:
6639:
6589:
6576:
6572:
6568:
6562:
6545:
6522:
6510:
6442:
6436:
6389:
6358:
6348:
6342:
6248:
6244:
6240:
6236:
6234:
6222:
6210:
6206:
6202:
6072:
6068:
6066:
6042:
6024:
6012:solar energy
6001:
5977:
5970:
5964:, crane, or
5955:
5952:
5941:
5922:
5918:gravity well
5886:
5876:
5791:
5706:
5701:
5698:
5548:
5541:
5384:
5381:
5375:
5368:
5362:
5359:
5291:
5285:
5276:
5267:
5258:
5252:
5234:
5231:
5183:
5178:
5173:
5167:
5161:
5111:
5101:
5095:
5091:
5087:
5029:
5025:
5014:
5010:
5001:
4993:
4975:
4951:
4949:
4941:
4857:
4853:
4849:
4845:
4843:
4831:
4827:
4823:
4703:is given by
4700:
4696:
4692:
4690:
4574:
4237:
4233:
4228:
4224:
4219:
4215:
4102:
3900:is given by
3892:
3888:
3878:
3874:
3868:
3863:
3861:
3852:
3848:
3844:
3739:is given by
3736:
3732:
3728:
3726:
3718:
3665:
3637:
3576:
3571:
3564:
3560:
3556:
3550:
3311:, to obtain
3303:
3296:
3289:
3285:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3250:
3245:
3239:
3235:
3232:
3189:
3184:
3183:
3028:, to obtain
3020:
3013:
3006:
3001:
3000:
2993:
2989:
2985:
2981:
2977:
2973:
2969:
2964:
2963:
2953:
2948:
2947:
2944:
2939:
2935:
2931:
2927:
2883:
2859:
2785:
2783:
2554:
2550:
2547:
2543:
2538:
2536:
2213:
2209:
2205:
2199:
2195:
2191:
2184:
2180:
2176:
2172:
2163:
2159:
2157:
2143:
2139:
2135:
2127:
2123:
1988:del operator
1985:
1908:
1904:
1900:
1897:
1891:
1715:
1576:
1572:
1568:
1564:
1558:
1548:
1544:
1542:
1537:
1533:
1429:
1422:
1418:
1414:
1411:
1398:
1394:
1390:
1384:
1344:
1336:
1289:
1283:
1277:
1271:
1268:
1253:
1245:
1241:
1237:
1233:
1171:
1167:fossil fuels
1133:
1121:
1117:vector field
1115:
1110:
1104:
1100:
1098:(symbol J).
1094:(SI) is the
1069:
1064:potentiality
1063:
1050:
1042:
1036:
827: /
823: /
821:displacement
819: /
680: /
642:Displacement
580:
571:
565:Formulations
552:Virtual work
492: /
461:
432:Acceleration
425:Fundamentals
226:
215:
209:
205:
194:
188:
174:
163:
157:
143:
142:
131:
125:
119:
115:
96:SI unit
88:
84:
80:
36:
8919:WikiProject
8739:Agriculture
8668:Solar power
8634:Tidal power
8508:Natural gas
8498:Fossil fuel
8441:Latent heat
8409:Electricity
7474:15 February
7428:pp. 203–208
7416:pp. 106–117
7302:closed form
7246:as before.
6022:reactions.
5966:sash window
5882:real number
5794:singularity
5644:therefore,
3563:-velocity,
2558:, then the
2148:force field
1111:force field
963:von Neumann
630:Core topics
8951:Potentials
8935:Categories
8702:Wind power
8624:Hydropower
8575:components
8530:Hydropower
8520:Geothermal
8470:Sound wave
8381:Zero-point
8311:Mechanical
8296:Ionization
8269:Electrical
8164:Negentropy
8045:Energetics
7914:References
7318:exact form
6541:beta decay
6527:inside an
5280:using the
3723:Derivation
2130:must be a
1809:to obtain
1579:) so that
1258:such as a
1142:is called
898:d'Alembert
878:Maupertuis
841:Scientists
723:Rigid body
397:Kinematics
8813:Australia
8749:Transport
8744:Computing
8712:Wind farm
8639:Wave farm
8513:Petroleum
8493:Bioenergy
8465:Radiation
8404:Capacitor
8326:Potential
7469:250895349
7277:ϕ
7231:−
7207:−
7181:→
7049:−
7031:−
7014:→
7008:→
6974:−
6942:−
6853:μ
6831:μ
6814:μ
6797:−
6764:−
6715:−
6709:ϕ
6664:−
6606:ϕ
6525:particles
6481:⋅
6473:∫
6460:−
6415:⋅
6411:μ
6407:−
6371:μ
6321:⋅
6295:∫
6291:−
6259:Δ
6235:The work
6186:^
6154:^
6114:ε
6110:π
6098:−
5750:∞
5664:∑
5658:−
5562:−
5457:−
5412:−
5305:−
5244:integrate
5207:Δ
5192:Δ
5061:Δ
5058:−
5017:= 9.8 m/s
4896:ε
4892:π
4844:The work
4807:^
4775:^
4735:ε
4731:π
4655:˙
4645:−
4636:˙
4622:−
4614:−
4603:−
4522:−
4471:˙
4409:∫
4405:−
4374:˙
4371:θ
4344:˙
4332:⋅
4258:∫
4254:−
4184:˙
4181:θ
4154:˙
4074:⋅
4016:∫
4012:−
3998:⋅
3918:∫
3914:−
3824:^
3792:^
3758:−
3684:−
3461:∫
3411:∫
3407:−
3367:∫
3363:−
3345:⋅
3326:∫
3163:Δ
3095:∫
3076:⋅
3043:∫
2952:= (0, 0,
2874:trebuchet
2838:⋅
2822:⋅
2815:∇
2811:−
2742:−
2659:∫
2655:−
2627:⋅
2608:∫
2589:⋅
2579:γ
2575:∫
2496:Φ
2493:−
2470:Φ
2444:Φ
2441:−
2418:Φ
2386:Φ
2357:∫
2315:⋅
2292:Φ
2289:∇
2275:∫
2256:⋅
2242:Φ
2239:∇
2234:γ
2230:∫
2088:∂
2080:∂
2065:∂
2057:∂
2042:∂
2034:∂
2023:−
2013:∇
2009:−
1999:∇
1948:−
1849:−
1782:⋅
1771:∇
1762:∫
1747:⋅
1733:∫
1688:∂
1675:∂
1660:∂
1647:∂
1632:∂
1619:∂
1596:∇
1499:−
1464:⋅
1450:∫
1355:Aristotle
1216:Δ
1193:Δ
1190:−
1059:Aristotle
943:Liouville
825:frequency
745:Vibration
462:potential
387:Continuum
382:Celestial
359:Textbooks
237:∫
8883:Category
8448:Hydrogen
8414:Enthalpy
8316:Negative
8306:Magnetic
8291:Internal
8249:Chemical
8114:Enthalpy
8033:concepts
7855:(1975).
7809:Archived
7749:Archived
7725:Archived
7705:page 6,
7699:Archived
7676:Archived
7125:, where
5988:(ARES).
5962:elevator
5948:Dinorwig
5240:calculus
5125:altitude
4952:relative
3648:stressed
3559:and the
2562:yields,
2324:′
1856:′
1827:′
1778:′
1682:′
1654:′
1626:′
1603:′
1130:Overview
998:Category
923:Hamilton
908:Lagrange
903:Clairaut
868:Horrocks
829:velocity
799:Pendulum
787:reactive
759:Rotation
728:dynamics
678:Inertial
664:Friction
547:Velocity
522:Momentum
402:Kinetics
392:Dynamics
370:Branches
354:Timeline
222:magnetic
201:electric
8895:Commons
8723:Use and
8582:Biomass
8552:Radiant
8399:Battery
8371:Thermal
8366:Surface
8351:Radiant
8321:Phantom
8301:Kinetic
8279:Binding
8259:Elastic
8242:Nuclear
8237:Binding
8124:Entropy
8022:Outline
8012:History
6865:is the
6216:is the
5121:gravity
4837:is the
3855:is the
3640:elastic
3242:, 0, 0)
3227:Archery
3213:Springs
1375:
1363:
1341:History
1264:gravity
1136:elastic
1045:is the
1039:physics
958:Koopman
918:Poisson
913:Laplace
858:Huygens
853:Galileo
698: (
637:Damping
490:Inertia
484:Impulse
457:kinetic
407:Statics
377:Applied
349:History
182:⁄
170:elastic
151:⁄
8907:Portal
8828:Mexico
8823:Europe
8818:Canada
8803:Africa
8726:supply
8535:Marine
8424:Fossil
8376:Vacuum
8129:Exergy
8050:Energy
8001:Energy
7947:
7928:
7865:
7835:
7683:Stucky
7635:
7585:
7558:
7531:
7500:
7467:
7384:
7349:
7312:, its
6845:where
6565:forces
6172:where
6056:Plasma
5360:where
5159:where
5123:, and
4970:, and
4793:where
4214:where
3810:where
3670:, is:
2926:where
2541:, let
1532:where
1387:forces
1260:spring
1256:forces
1244:, and
1208:where
1159:charge
1156:baryon
1086:in an
1082:of an
1047:energy
996:
948:Appell
933:Cauchy
928:Jacobi
873:Halley
863:Newton
848:Kepler
700:linear
696:Motion
542:Torque
517:Moment
452:Energy
442:Couple
8847:Misc.
8557:Solar
8361:Sound
8230:Types
8104:Power
8055:Units
8017:Index
7465:S2CID
7324:Notes
7304:. As
7300:as a
7291:Earth
6243:from
4852:from
3570:, is
3257:) = (
2972:) = (
2189:from
1096:joule
953:Gibbs
938:Routh
893:Euler
532:Speed
527:Space
469:Force
101:joule
87:, or
8808:Asia
8562:Wind
8503:Coal
8475:Work
8436:Heat
8419:Fuel
8356:Rest
8254:Dark
8219:Work
8087:Mass
7945:ISBN
7926:ISBN
7863:ISBN
7833:ISBN
7633:ISBN
7583:ISBN
7556:ISBN
7529:ISBN
7498:ISBN
7476:2023
7382:ISBN
7347:ISBN
7269:and
6642:and
6437:The
6229:work
6209:and
5932:Uses
5848:for
5792:The
5736:and
5265:and
5117:mass
5035:work
4830:and
4223:and
3886:to
3851:and
3731:and
3574:/2.
3238:= (−
2212:) =
2198:) =
2179:) =
1428:and
1347:work
537:Time
500:Mass
59:work
8429:Oil
7525:168
7457:doi
7420:(2)
7409:(1)
7308:is
6928:is
6650:is
6618:or
6571:to
6549:Sun
6205:to
5796:at
5371:= 0
5289:is
5112:mgh
4989:Sun
4826:to
4236:to
3847:to
3839:is
3644:bow
3553:= 0
3288:= (
3273:),
3265:),
3005:= (
2988:),
2980:),
2203:to
1903:= −
1393:to
1150:or
1037:In
208:= −
146:=
103:(J)
8937::
7886:.
7791:,
7772:,
7655:.
7527:.
7484:^
7463:.
7453:24
7451:.
7447:.
7435:^
7396:^
7380:.
7378:74
7370:.
7341:.
7061:0.
6869:.
6543:.
6535:.
6220:.
5968:.
5119:,
5102:mg
5096:Fd
5094:=
5030:Fd
5028:=
4966:,
4841:.
3859:.
3658:.
3565:xv
3302:,
3295:,
3281:))
3240:kx
3194:.
3019:,
3012:,
2996:))
2872:A
2555:dt
2546:=
2150:.
1907:′(
1575:′(
1290:mg
1251:.
1240:,
1236:,
1234:PE
1067:.
1041:,
229:=
224:)
214:⋅
193:⋅
187:⋅
177:=
162:⋅
156:⋅
130:⋅
124:⋅
118:=
83:,
81:PE
7993:e
7986:t
7979:v
7953:.
7934:.
7871:.
7841:.
7641:.
7616:.
7591:.
7564:.
7537:.
7506:.
7478:.
7459::
7430:.
7390:.
7355:.
7257:U
7234:a
7228:b
7225:=
7222:)
7219:c
7216:+
7213:a
7210:(
7204:)
7201:c
7198:+
7195:b
7192:(
7189:=
7184:B
7178:A
7174:U
7153:c
7133:c
7113:c
7110:+
7107:b
7087:c
7084:+
7081:a
7058:=
7055:)
7052:b
7046:a
7043:(
7040:+
7037:)
7034:a
7028:b
7025:(
7022:=
7017:A
7011:B
7005:A
7001:U
6980:)
6977:b
6971:a
6968:(
6948:)
6945:a
6939:b
6936:(
6916:b
6913:=
6910:U
6890:a
6887:=
6884:U
6828:U
6823:=
6817:r
6809:m
6806:M
6803:G
6794:=
6789:r
6785:)
6782:m
6779:+
6776:M
6773:(
6770:G
6761:=
6757:)
6751:r
6747:m
6744:G
6738:+
6733:r
6729:M
6726:G
6719:(
6712:=
6685:.
6680:r
6676:m
6673:M
6670:G
6661:=
6658:U
6648:r
6644:m
6640:M
6626:V
6577:A
6573:B
6569:A
6511:M
6496:,
6493:V
6490:d
6485:B
6477:M
6468:2
6465:1
6457:=
6454:U
6443:M
6423:.
6419:B
6404:=
6401:U
6390:B
6349:V
6328:r
6324:d
6315:e
6311:F
6304:B
6299:A
6288:=
6285:)
6280:r
6275:(
6270:B
6267:A
6263:U
6249:B
6245:A
6241:q
6237:W
6214:0
6211:ε
6207:q
6203:Q
6183:r
6160:,
6151:r
6141:2
6137:r
6132:q
6129:Q
6118:0
6107:4
6103:1
6095:=
6090:e
6085:F
6073:q
6069:Q
5904:0
5901:=
5898:U
5887:r
5877:r
5862:0
5859:=
5856:r
5836:0
5833:=
5830:U
5810:0
5807:=
5804:r
5776:0
5773:=
5770:U
5747:=
5744:r
5724:0
5721:=
5718:r
5707:U
5702:r
5680:,
5675:r
5672:M
5667:G
5661:m
5655:=
5652:U
5631:)
5623:2
5619:r
5613:2
5609:M
5603:G
5600:+
5593:1
5589:r
5583:1
5579:M
5573:G
5569:(
5565:m
5559:=
5556:U
5526:)
5518:2
5514:r
5508:2
5504:M
5498:G
5495:+
5488:1
5484:r
5478:1
5474:M
5468:G
5464:(
5460:m
5454:=
5451:U
5422:2
5418:)
5415:1
5409:n
5406:(
5403:n
5390:n
5385:U
5376:U
5369:K
5363:K
5346:,
5343:K
5340:+
5335:r
5329:2
5325:M
5319:1
5315:m
5308:G
5302:=
5299:U
5286:G
5277:r
5271:2
5268:M
5262:1
5259:m
5253:r
5235:g
5213:.
5210:h
5204:g
5201:m
5198:=
5195:U
5179:h
5174:g
5168:m
5162:U
5147:h
5144:g
5141:m
5138:=
5135:U
5107:h
5092:W
5074:.
5069:F
5065:U
5055:=
5050:F
5046:W
5026:W
5019:(
5015:g
4923:.
4918:r
4914:q
4911:Q
4900:0
4889:4
4885:1
4880:=
4877:)
4874:r
4871:(
4868:U
4858:B
4854:A
4850:q
4846:W
4835:0
4832:ε
4828:q
4824:Q
4804:r
4781:,
4772:r
4762:2
4758:r
4753:q
4750:Q
4739:0
4728:4
4724:1
4719:=
4715:F
4701:r
4697:q
4693:Q
4672:.
4665:2
4661:r
4652:r
4642:=
4633:r
4625:2
4618:r
4611:=
4606:1
4599:r
4592:t
4589:d
4585:d
4561:.
4555:)
4550:1
4546:t
4542:(
4539:r
4534:m
4531:M
4528:G
4516:)
4511:2
4507:t
4503:(
4500:r
4495:m
4492:M
4489:G
4483:=
4480:t
4477:d
4468:r
4462:r
4455:3
4451:r
4446:M
4443:m
4440:G
4430:2
4426:t
4418:1
4414:t
4402:=
4399:t
4396:d
4392:)
4387:t
4382:e
4365:r
4362:+
4357:r
4352:e
4341:r
4335:(
4329:)
4324:r
4319:e
4314:r
4311:(
4304:3
4300:r
4295:M
4292:m
4289:G
4279:2
4275:t
4267:1
4263:t
4251:=
4248:W
4238:m
4234:M
4229:t
4225:e
4220:r
4216:e
4202:,
4197:t
4192:e
4175:r
4172:+
4167:r
4162:e
4151:r
4145:=
4141:v
4136:,
4131:r
4126:e
4121:r
4118:=
4114:r
4103:m
4089:.
4086:t
4083:d
4078:v
4070:r
4062:3
4058:r
4053:m
4050:M
4047:G
4037:2
4033:t
4025:1
4021:t
4009:=
4005:r
4001:d
3994:r
3986:3
3982:r
3977:m
3974:M
3971:G
3963:)
3958:2
3954:t
3950:(
3946:r
3940:)
3935:1
3931:t
3927:(
3923:r
3911:=
3908:W
3898:)
3896:2
3893:t
3891:(
3889:r
3884:)
3882:1
3879:t
3877:(
3875:r
3869:v
3864:m
3853:G
3849:m
3845:M
3821:r
3798:,
3789:r
3779:2
3775:r
3770:m
3767:M
3764:G
3755:=
3751:F
3737:r
3733:m
3729:M
3705:,
3700:r
3696:m
3693:M
3690:G
3681:=
3678:U
3623:,
3618:2
3614:x
3610:k
3605:2
3602:1
3597:=
3594:)
3591:x
3588:(
3585:U
3572:x
3567:x
3561:x
3557:x
3551:t
3534:2
3530:x
3526:k
3521:2
3518:1
3513:=
3510:x
3507:d
3503:x
3500:k
3495:)
3492:t
3489:(
3486:x
3481:)
3476:0
3472:t
3468:(
3465:x
3457:=
3454:t
3451:d
3445:t
3442:d
3437:x
3434:d
3428:x
3425:k
3420:t
3415:0
3404:=
3401:t
3398:d
3392:x
3388:v
3384:x
3381:k
3376:t
3371:0
3360:=
3357:t
3354:d
3349:v
3341:F
3335:t
3330:0
3322:=
3319:W
3309:)
3307:z
3304:v
3300:y
3297:v
3293:x
3290:v
3286:v
3279:t
3277:(
3275:z
3271:t
3269:(
3267:y
3263:t
3261:(
3259:x
3255:t
3253:(
3251:s
3246:x
3236:F
3192:)
3190:t
3188:(
3185:r
3169:.
3166:z
3158:z
3154:F
3150:=
3147:t
3144:d
3138:z
3134:v
3128:z
3124:F
3116:2
3112:t
3104:1
3100:t
3091:=
3088:t
3085:d
3080:v
3072:F
3064:2
3060:t
3052:1
3048:t
3039:=
3036:W
3026:)
3024:z
3021:v
3017:y
3014:v
3010:x
3007:v
3002:v
2994:t
2992:(
2990:z
2986:t
2984:(
2982:y
2978:t
2976:(
2974:x
2970:t
2968:(
2965:r
2959:)
2956:z
2954:F
2949:F
2940:U
2936:h
2932:g
2928:m
2914:,
2911:h
2908:g
2905:m
2902:=
2897:g
2893:U
2846:.
2842:v
2834:F
2830:=
2826:v
2818:U
2808:=
2805:)
2802:t
2799:(
2796:P
2786:v
2766:.
2763:)
2758:B
2753:x
2748:(
2745:U
2739:)
2734:A
2729:x
2724:(
2721:U
2718:=
2715:t
2712:d
2708:)
2705:)
2702:t
2699:(
2695:r
2691:(
2688:U
2682:t
2679:d
2675:d
2668:b
2663:a
2652:=
2642:,
2639:t
2636:d
2631:v
2623:F
2617:b
2612:a
2604:=
2596:r
2592:d
2585:F
2553:/
2551:r
2548:d
2544:v
2539:F
2519:.
2515:)
2510:A
2505:x
2500:(
2489:)
2484:B
2479:x
2474:(
2467:=
2464:)
2461:)
2458:a
2455:(
2451:r
2447:(
2438:)
2435:)
2432:b
2429:(
2425:r
2421:(
2415:=
2412:t
2409:d
2406:)
2403:)
2400:t
2397:(
2393:r
2389:(
2380:t
2377:d
2373:d
2366:b
2361:a
2353:=
2343:,
2340:t
2337:d
2334:)
2331:t
2328:(
2320:r
2312:)
2309:)
2306:t
2303:(
2299:r
2295:(
2284:b
2279:a
2271:=
2263:r
2259:d
2253:)
2249:r
2245:(
2214:B
2210:b
2208:(
2206:γ
2200:A
2196:a
2194:(
2192:γ
2187:)
2185:t
2183:(
2181:r
2177:t
2175:(
2173:γ
2164:x
2162:(
2160:F
2144:x
2140:F
2136:U
2128:F
2124:F
2110:,
2106:F
2102:=
2098:)
2091:z
2083:U
2074:,
2068:y
2060:U
2051:,
2045:x
2037:U
2027:(
2020:=
2016:U
2006:=
2002:W
1972:.
1969:)
1964:B
1959:x
1954:(
1951:U
1945:)
1940:A
1935:x
1930:(
1927:U
1924:=
1921:W
1911:)
1909:x
1905:U
1901:U
1892:C
1878:.
1875:)
1870:A
1865:x
1860:(
1853:U
1846:)
1841:B
1836:x
1831:(
1824:U
1820:=
1817:W
1793:,
1789:x
1785:d
1775:U
1766:C
1758:=
1754:x
1750:d
1743:F
1737:C
1729:=
1726:W
1716:U
1702:.
1698:)
1691:z
1679:U
1669:,
1663:y
1651:U
1641:,
1635:x
1623:U
1612:(
1608:=
1600:U
1592:=
1588:F
1577:x
1573:U
1569:F
1565:C
1549:x
1547:(
1545:U
1538:C
1534:C
1520:)
1515:B
1510:x
1505:(
1502:U
1496:)
1491:A
1486:x
1481:(
1478:U
1475:=
1471:x
1467:d
1460:F
1454:C
1446:=
1443:W
1433:B
1430:x
1426:A
1423:x
1419:x
1417:(
1415:U
1399:A
1395:B
1391:A
1372:2
1369:/
1366:1
1323:.
1320:h
1317:g
1314:m
1311:=
1306:g
1302:U
1284:g
1278:h
1272:m
1248:p
1246:E
1242:V
1238:U
1219:U
1196:U
1187:=
1184:W
1026:e
1019:t
1012:v
702:)
321:t
318:d
312:p
308:d
302:=
297:F
256:r
253:d
249:)
246:r
243:(
240:F
227:U
220:(
217:B
211:m
206:U
203:)
199:(
196:V
190:C
184:2
180:1
175:U
172:)
168:(
165:x
159:k
153:2
149:1
144:U
140:)
136:(
133:h
127:g
121:m
116:U
89:V
85:U
34:.
20:)
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