How To Draw A Potential Energy Diagram

Free energy held by an object because of its position relative to other objects

Potential free energy
In the case of a bow and arrow, when the archer does work on the bow, cartoon the string back, some of the chemical energy of the archer's body is transformed into rubberband potential energy in the aptitude limb of the bow. When the string is released, the force between the string and the arrow does work on the arrow. The potential free energy in the bow limbs is transformed into the kinetic free energy of the arrow as it takes flight.
Mutual symbols	PE, U, or Five
SI unit	joule (J)
Derivations from other quantities	U = m ⋅ yard ⋅ h (gravitational) U = 1⁄2 ⋅ thou ⋅ x 2 (elastic) U = 1⁄two ⋅ C ⋅ V 2 (electric) U = −m ⋅ B (magnetic) U = ${\textstyle \int F(r)\,dr}$

In physics, potential energy is the free energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.^{[1] [2]}

Common types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential free energy of an extended spring, and the electrical potential energy of an electric charge in an electric field. The unit for free energy in the International System of Units (SI) is the joule, which has the symbol J.

The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine,^{[3] [4]} although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a manner that the total piece of work washed by these forces on the body depends only on the initial and concluding positions of the body in space. These forces, that are chosen conservative forces, tin can be represented at every signal in space past vectors expressed as gradients of a certain scalar function called potential.

Since the work of potential forces acting on a torso that moves from a get-go to an end position is adamant only by these 2 positions, and does not depend on the trajectory of the body, in that location is a function known every bit potential that tin be evaluated at the two positions to determine this work.

Overview

There are various types of potential energy, each associated with a particular type of force. For example, the piece of work of an elastic strength is called elastic potential energy; piece of work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy; work of the strong nuclear force or weak nuclear force acting on the baryon charge is chosen nuclear potential energy; work of intermolecular forces is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the piece of work of the Coulomb force during rearrangement of configurations of electrons and nuclei in atoms and molecules. Thermal energy normally has two components: the kinetic energy of random motions of particles and the potential energy of their configuration.

Forces derivable from a potential are too called conservative forces. The work done by a bourgeois force is

$${\displaystyle West=-\Delta U}$$

where ${\displaystyle \Delta U}$ 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 PE, U, V, and Eastward_p .

Potential energy is the free energy by virtue of an object's position relative to other objects.^[5] Potential free 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 piece of work is stored in the forcefulness field, which is said to be stored as potential energy. If the external strength is removed the force field acts on the body to perform the work as it moves the body dorsum 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 top is h . The acceleration g of costless autumn is approximately constant, and then the weight force of the ball mg is constant. The product of force and deportation gives the work done, which is equal to the gravitational potential energy, thus

$${\displaystyle U_{g}=mgh}$$

The more formal definition is that potential free energy is the energy difference between the energy of an object in a given position and its energy at a reference position.

Piece of work and potential energy

Potential energy is closely linked with forces. If the piece of work washed past a force on a trunk that moves from A to B does non depend on the path between these points (if the piece of work is done by a conservative force), so the work of this strength measured from A assigns a scalar value to every other bespeak in space and defines a scalar potential field. In this example, the force can be divers as the negative of the vector gradient of the potential field.

If the work for an practical force is independent of the path, and then the work washed past the force is evaluated from the start to the end of the trajectory of the point of application. This means that at that place is a function U(10), called a "potential," that can be evaluated at the two points ten _A and 10 _B 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

$${\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {ten} _{A})-U(\mathbf {ten} _{B})}$$

where C is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is truthful for any trajectory, C, from A to B.

The function U(x) is called the potential energy associated with the practical forcefulness. Examples of forces that accept 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 bend C takes a special grade if the strength F is related to a scalar field Φ(x) and so that

$${\displaystyle \mathbf {F} ={\nabla \Phi }=\left({\frac {\partial \Phi }{\fractional x}},{\frac {\partial \Phi }{\partial y}},{\frac {\partial \Phi }{\fractional z}}\right).}$$

In this case, work along the curve is given by

$${\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{C}\nabla \Phi \cdot d\mathbf {x} ,}$$

which can be evaluated using the gradient theorem to obtain

$${\displaystyle W=\Phi (\mathbf {x} _{B})-\Phi (\mathbf {10} _{A}).}$$

This shows that when forces are derivable from a scalar field, the work of those forces forth a bend C is computed past evaluating the scalar field at the commencement indicate 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 exist independent of the path.

Potential energy U = −Φ(ten) is traditionally defined as the negative of this scalar field then that piece of work past the strength field decreases potential energy, that is

$${\displaystyle W=U(\mathbf {ten} _{A})-U(\mathbf {x} _{B}).}$$

In this case, the application of the del operator to the work function yields,

$${\displaystyle {\nabla West}=-{\nabla U}=-\left({\frac {\fractional U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}\correct)=\mathbf {F} ,}$$

and the force F is said to be "derivable from a potential."^[6] This too necessarily implies that F must be a conservative vector field. The potential U defines a force F at every point x in infinite, then the set of forces is chosen a forcefulness field.

Computing potential free energy

Given a forcefulness field F(ten), evaluation of the work integral using the slope theorem tin can be used to find the scalar function associated with potential energy. This is washed by introducing a parameterized curve γ(t) = r(t) from γ(a) = A to γ(b) = B , and computing,

$${\displaystyle {\begin{aligned}\int _{\gamma }\nabla \Phi (\mathbf {r} )\cdot d\mathbf {r} &=\int _{a}^{b}\nabla \Phi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt,\\&=\int _{a}^{b}{\frac {d}{dt}}\Phi (\mathbf {r} (t))dt=\Phi (\mathbf {r} (b))-\Phi (\mathbf {r} (a))=\Phi \left(\mathbf {10} _{B}\correct)-\Phi \left(\mathbf {x} _{A}\right).\stop{aligned}}}$$

For the strength field F, let v = d r/dt , and so the slope theorem yields,

$${\displaystyle {\begin{aligned}\int _{\gamma }\mathbf {F} \cdot d\mathbf {r} &=\int _{a}^{b}\mathbf {F} \cdot \mathbf {5} \,dt,\\&=-\int _{a}^{b}{\frac {d}{dt}}U(\mathbf {r} (t))\,dt=U(\mathbf {x} _{A})-U(\mathbf {x} _{B}).\finish{aligned}}}$$

The power practical 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 signal of application, that is

$${\displaystyle P(t)=-{\nabla U}\cdot \mathbf {v} =\mathbf {F} \cdot \mathbf {v} .}$$

Examples of work that can be computed from potential functions are gravity and spring forces.^[7]

Potential energy for near Earth gravity

A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over two hundred meters

For small height changes, gravitational potential energy can be computed using

$${\displaystyle U_{thou}=mgh,}$$

where m is the mass in kg, k is the local gravitational field (9.8 metres per second squared on world), h is the height to a higher place a reference level in metres, and U is the energy in joules.

In classical physics, gravity exerts a abiding downward force F = (0, 0, F_z ) on the heart of mass of a body moving almost the surface of the Earth. The work of gravity on a body moving along a trajectory r (t) = (ten(t), y(t), z(t)), such as the track of a roller coaster is calculated using its velocity, v = (5 _x, 5 _y, five _z), to obtain

$${\displaystyle W=\int _{t_{ane}}^{t_{two}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\,dt=\int _{t_{i}}^{t_{2}}F_{z}v_{z}\,dt=F_{z}\Delta z.}$$

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 r (t).

Potential energy for a linear spring

Archery is one of humankind's oldest applications of elastic potential energy

A horizontal jump exerts a force F = (−kx, 0, 0) that is proportional to its deformation in the axial or 10 direction. The piece of work of this bound on a body moving along the space bend s(t) = (x(t), y(t), z(t)), is calculated using its velocity, v = (v _x, v _y, v _z), to obtain

$${\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} \,dt=-\int _{0}^{t}kxv_{x}\,dt=-\int _{0}^{t}kx{\frac {dx}{dt}}dt=\int _{x(t_{0})}^{x(t)}kx\,dx={\frac {i}{2}}kx^{two}}$$

For convenience, consider contact with the leap occurs at t = 0, and so the integral of the product of the altitude ten and the x-velocity, xv_ten , is x ²/2.

The function

$${\displaystyle U(x)={\frac {1}{2}}kx^{2},}$$

is called the potential free energy of a linear spring.

Rubberband potential free energy is the potential energy of an elastic object (for instance a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises equally a consequence of a forcefulness that tries to restore the object to its original shape, which is about often the electromagnetic forcefulness between the atoms and molecules that constitute the object. If the stretch is released, the free energy is transformed into kinetic energy.

Potential energy for gravitational forces between two bodies

The gravitational potential role, also known as gravitational potential free energy, is:

$${\displaystyle U=-{\frac {GMm}{r}},}$$

The negative sign follows the convention that work is gained from a loss of potential free energy.

Derivation

The gravitational force between two bodies of mass Yard and m separated past a altitude r is given by Newton's police

$${\displaystyle \mathbf {F} =-{\frac {GMm}{r^{ii}}}\mathbf {\chapeau {r}} ,}$$

where ${\displaystyle \mathbf {\hat {r}} }$ is a vector of length 1 pointing from 1000 to thousand and G is the gravitational abiding.

Let the mass m move at the velocity v and then the work of gravity on this mass as it moves from position r(t ₁) to r(t ₂) is given by

$${\displaystyle W=-\int _{\mathbf {r} (t_{one})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.}$$

The position and velocity of the mass g are given by

$${\displaystyle \mathbf {r} =r\mathbf {due east} _{r},\qquad \mathbf {v} ={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},}$$

where due east _r and e _t are the radial and tangential unit vectors directed relative to the vector from M to m. Apply this to simplify the formula for work of gravity to,

$${\displaystyle Westward=-\int _{t_{one}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {due east} _{r})\cdot ({\dot {r}}\mathbf {eastward} _{r}+r{\dot {\theta }}\mathbf {eastward} _{t})\,dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{iii}}}r{\dot {r}}dt={\frac {GMm}{r(t_{ii})}}-{\frac {GMm}{r(t_{1})}}.}$$

This calculation uses the fact that

$${\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{ii}}}.}$$

Potential energy for electrostatic forces between two bodies

The electrostatic force exerted by a accuse Q on another charge q separated by a distance r is given past Coulomb'due south Law

$${\displaystyle \mathbf {F} ={\frac {1}{four\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}$$

where ${\displaystyle \mathbf {\hat {r}} }$ is a vector of length ane pointing from Q to q and ε ₀ is the vacuum permittivity. This may also be written using Coulomb constant k _e = 1 ⁄ 4πε ₀ .

The piece of work Westward required to motion q from A to whatsoever betoken B in the electrostatic force field is given by the potential part

$${\displaystyle U(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r}}.}$$

Reference level

The potential energy is a part of the state a system is in, and is divers relative to that for a particular land. This reference state is not always a real land; it may as well be a limit, such every bit with the distances betwixt all bodies tending to infinity, provided that the free energy involved in tending to that limit is finite, such as in the instance of changed-foursquare law forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience.

Typically the potential free energy of a system depends on the relative positions of its components just, and so the reference state can also be expressed in terms of relative positions.

Gravitational potential free energy

Gravitational energy is the potential free energy associated with gravitational forcefulness, as work is required to elevate objects against World's gravity. The potential energy due to elevated positions is chosen gravitational potential energy, and is evidenced past water in an elevated reservoir or kept behind a dam. If an object falls from one indicate to some other betoken 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.

Gravitational force keeps the planets in orbit effectually the Sun

Consider a book placed on top of a table. As the volume is raised from the floor to the table, some external force works confronting the gravitational force. If the book falls back to the floor, the "falling" energy the volume receives is provided by the gravitational strength. 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 flooring this kinetic free energy is converted into heat, deformation, and sound past the impact.

The factors that affect an object'due south gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field information technology is in. Thus, a book lying on a tabular array has less gravitational potential free energy than the same book on top of a taller cupboard and less gravitational potential energy than a heavier book lying on the aforementioned tabular array. An object at a certain meridian above the Moon'due south surface has less gravitational potential energy than at the aforementioned acme higher up 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 non assumed to be a abiding. The post-obit sections provide more than detail.

Local approximation

The strength of a gravitational field varies with location. However, when the change of altitude is small in relation to the distances from the centre 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 abiding. Near the surface of the Earth, for example, we assume that the dispatch due to gravity is a abiding m = ix.8 m/s² ("standard gravity"). In this case, a simple expression for gravitational potential energy can exist derived using the W = Fd equation for piece of work, and the equation

$${\displaystyle W_{F}=-\Delta U_{F}.}$$

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 Westward = Fd ). The upward forcefulness 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 production mgh . Thus, when accounting only for mass, gravity, and distance, the equation is:^[eight]

$${\displaystyle U=mgh}$$

where U is the potential energy of the object relative to its being on the Earth's surface, g is the mass of the object, k is the acceleration due to gravity, and h is the altitude of the object.^[9] If m is expressed in kilograms, g in yard/s² and h in metres so U will be calculated in joules.

Hence, the potential difference is

$${\displaystyle \Delta U=mg\Delta h.}$$

General formula

Nonetheless, over big variations in altitude, the approximation that g is abiding is no longer valid, and nosotros have to use calculus and the general mathematical definition of work to make up one's mind gravitational potential energy. For the computation of the potential free energy, we tin can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation, with respect to the distance r between the 2 bodies. Using that definition, the gravitational potential energy of a system of masses yard ₁ and 1000 ₂ at a distance r using gravitational abiding G is

$${\displaystyle U=-Thou{\frac {m_{ane}M_{two}}{r}}+Thou,}$$

where K is an arbitrary abiding dependent on the choice of datum from which potential is measured. Choosing the convention that Thou = 0 (i.e. in relation to a signal at infinity) makes calculations simpler, albeit at the cost of making U negative; for why this is physically reasonable, run across below.

Given this formula for U , the total potential energy of a system of n bodies is found by summing, for all ${\textstyle {\frac {n(n-one)}{2}}}$ pairs of two bodies, the potential energy of the system of those two bodies.

Gravitational potential summation ${\displaystyle U=-m\left(K{\frac {M_{1}}{r_{1}}}+M{\frac {M_{2}}{r_{2}}}\right)}$

Considering the system of bodies as the combined set of pocket-sized particles the bodies consist of, and applying the previous on the particle level we go the negative gravitational bounden free energy. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since information technology too includes the negative gravitational binding energy of each torso. The potential energy of the organisation of bodies as such is the negative of the energy needed to split the bodies from each other to infinity, while the gravitational binding energy is the energy needed to split all particles from each other to infinity.

$${\displaystyle U=-m\left(G{\frac {M_{1}}{r_{one}}}+G{\frac {M_{2}}{r_{2}}}\right)}$$

therefore,

$${\displaystyle U=-chiliad\sum One thousand{\frac {Grand}{r}},}$$

Negative gravitational free energy

As with all potential energies, only differences in gravitational potential energy affair for well-nigh concrete purposes, and the pick of zero point is capricious. Given that there is no reasonable criterion for preferring one particular finite r over some other, at that place seem to be merely two reasonable choices for the distance at which U becomes cypher: ${\displaystyle r=0}$ and ${\displaystyle r=\infty }$ . The choice of ${\displaystyle U=0}$ at infinity may seem peculiar, and the effect that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, admitting negative.

The singularity at ${\displaystyle r=0}$ in the formula for gravitational potential energy means that the merely other apparently reasonable alternative selection of convention, with ${\displaystyle U=0}$ for ${\displaystyle r=0}$ , would result in potential energy beingness positive, but infinitely large for all nonzero values of r , and would brand calculations involving sums or differences of potential energies across what is possible with the real number system. Since physicists abominate infinities in their calculations, and r is ever non-goose egg in practice, the choice of ${\displaystyle U=0}$ at infinity is by far the more preferable choice, even if the thought of negative free energy in a gravity well appears to be peculiar at commencement.

The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where the full free energy of the universe can meaningfully be considered; see inflation theory for more than on this.^[10]

Uses

Gravitational potential energy has a number of practical uses, notably the generation of pumped-storage hydroelectricity. For example, in Dinorwig, Wales, in that location are ii lakes, i at a higher elevation than the other. At times when surplus electricity is not required (and then 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 need for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic free energy and so back into electricity. The process is non completely efficient and some of the original free energy from the surplus electricity is in fact lost to friction.^{[11] [12] [13] [14] [15]}

Gravitational potential energy is also used to power clocks in which falling weights operate the machinery.

Information technology'due south also used by counterweights for lifting up an lift, crane, or sash window.

Roller coasters are an entertaining way to utilize potential energy – chains are used to move a automobile up an incline (edifice up gravitational potential energy), to so have that energy converted into kinetic energy every bit it falls.

Another applied use is utilizing gravitational potential energy to descend (peradventure coast) downhill in transportation such equally the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline. In some cases the kinetic energy obtained from the potential free 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 free energy (in the course of rail cars raised to higher elevations) that is then converted to electric energy when needed past an electric grid, is being undertaken in the United States in a system called Advanced Rail Energy Storage (ARES).^{[16] [17] [xviii]}

Chemical potential free energy

Chemical potential energy is a course of potential energy related to the structural organisation of atoms or molecules. This organisation may exist the result of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance tin exist transformed to other forms of energy by a chemic reaction. As an example, when a fuel is burned the chemic energy is converted to oestrus, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemic energy through the process known equally photosynthesis, and electric energy can be converted to chemic energy through electrochemical reactions.

The like term chemical potential is used to indicate the potential of a substance to undergo a modify of configuration, be it in the grade of a chemical reaction, spatial ship, particle exchange with a reservoir, etc.

Electric potential free energy

An object tin have potential energy by virtue of its electrical charge and several forces related to their presence. There are two main types of this kind of potential free energy: electrostatic potential free energy, electrodynamic potential energy (also sometimes called magnetic potential free energy).

Plasma formed inside a gas filled sphere

Electrostatic potential energy

Electrostatic potential energy between two bodies in space is obtained from the strength exerted by a charge Q on another accuse q which is given past

$${\displaystyle \mathbf {F} _{e}=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}$$

where ${\displaystyle \mathbf {\hat {r}} }$ is a vector of length ane pointing from Q to q and ε ₀ is the vacuum permittivity. This may also be written using Coulomb'due south constant k _e = one ⁄ 4πε ₀ .

If the electric charge of an object tin exist assumed to be at rest, then it has potential free energy due to its position relative to other charged objects. The electrostatic potential free energy is the free energy of an electrically charged particle (at rest) in an electrical field. It is defined as the work that must be washed to move it from an infinite altitude away to its present location, adapted for not-electrical forces on the object. This energy will generally be not-cypher if there is another electrically charged object nearby.

The work W required to move q from A to any indicate B in the electrostatic strength field is given past

$${\displaystyle \Delta U_{AB}({\mathbf {r} })=-\int _{A}^{B}\mathbf {F_{e}} \cdot d\mathbf {r} }$$

typically given in J for Joules. A related quantity chosen electric potential (commonly denoted with a Five for voltage) is equal to the electrical potential energy per unit accuse.

Magnetic potential free energy

The energy of a magnetic moment ${\displaystyle {\boldsymbol {\mu }}}$ in an externally produced magnetic B-field B has potential free energy^[19]

$${\displaystyle U=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}$$

The magnetization M in a field is

$${\displaystyle U=-{\frac {ane}{2}}\int \mathbf {M} \cdot \mathbf {B} \,dV,}$$

where the integral tin be over all space or, equivalently, where K is nonzero.^[20] Magnetic potential energy is the form of energy related non simply to the distance between magnetic materials, but likewise to the orientation, or alignment, of those materials within the field. For case, the needle of a compass has the lowest magnetic potential energy when it is aligned with the northward and south poles of the Earth's magnetic field. If the needle is moved by an exterior strength, torque is exerted on the magnetic dipole of the needle by the Globe'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 every bit the Earth's magnetic field. Two magnets volition have potential free energy in relation to each other and the distance between them, but this also depends on their orientation. If the reverse poles are held autonomously, the potential free energy will be higher the further they are autonomously and lower the closer they are. Conversely, similar poles will have the highest potential energy when forced together, and the lowest when they spring apart.^{[21] [22]}

Nuclear potential energy

Nuclear potential energy is the potential energy of the particles within an atomic nucleus. The nuclear particles are jump together by the potent nuclear strength. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta disuse.

Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, merely collections of them tin can have less mass than if they were individually free, in which instance this mass difference can exist liberated equally heat and radiation in nuclear reactions (the heat and radiation accept the missing mass, but it oftentimes escapes from the arrangement, where it is non measured). The free energy from the Sun is an case of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about iv million tonnes of solar matter per second into electromagnetic energy, which is radiated into space.

Forces and potential energy

Potential free energy is closely linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points, and then the work of this force measured from A assigns a scalar value to every other point in infinite and defines a scalar potential field. In this case, the forcefulness can be divers as the negative of the vector slope of the potential field.

For instance, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by ${\displaystyle \phi }$ or ${\displaystyle V}$ , corresponding to the energy per unit mass as a office of position. The gravitational potential energy of two particles of mass Thousand and thousand separated by a altitude r is

$${\displaystyle U=-{\frac {GMm}{r}},}$$

The gravitational potential (specific energy) of the two bodies is

$${\displaystyle \phi =-\left({\frac {GM}{r}}+{\frac {Gm}{r}}\correct)=-{\frac {G(M+m)}{r}}=-{\frac {GMm}{\mu r}}={\frac {U}{\mu }}.}$$

where ${\displaystyle \mu }$ is the reduced mass.

The work done against gravity by moving an infinitesimal mass from point A with ${\displaystyle U=a}$ to point B with ${\displaystyle U=b}$ is ${\displaystyle (b-a)}$ and the piece of work done going back the other mode is ${\displaystyle (a-b)}$ and so that the full work done in moving from A to B and returning to A is

$${\displaystyle U_{A\to B\to A}=(b-a)+(a-b)=0.}$$

If the potential is redefined at A to be ${\displaystyle a+c}$ and the potential at B to be ${\displaystyle b+c}$ , where ${\displaystyle c}$ is a abiding (i.e. ${\displaystyle c}$ can exist any number, positive or negative, but information technology must exist the same at A as it is at B) then the work done going from A to B is

$${\displaystyle U_{A\to B}=(b+c)-(a+c)=b-a}$$

equally earlier.

In practical terms, this ways that one can gear up the zero of ${\displaystyle U}$ and ${\displaystyle \phi }$ anywhere one likes. 1 may set it to be aught at the surface of the Earth, or may find it more than user-friendly to gear up aught at infinity (as in the expressions given earlier in this department).

A conservative strength tin be expressed in the language of differential geometry as a closed form. Equally Euclidean space is contractible, its de Rham cohomology vanishes, and then every closed form is also an exact grade, 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.

Notes

1. ^ Jain, Mahesh C. (2009). "Central forces and laws: a brief review". Textbook of Engineering Physics, Part one. PHI Learning Pvt. Ltd. p. 10. ISBN978-81-203-3862-3.
2. ^ McCall, Robert P. (2010). "Energy, Work and Metabolism". Physics of the Human being Torso. JHU Press. p. 74. ISBN978-0-8018-9455-8.
3. ^ William John Macquorn Rankine (1853) "On the general law of the transformation of energy," Proceedings of the Philosophical Gild of Glasgow, vol. 3, no. 5, pages 276–280; reprinted in: (1) Philosophical Magazine, series 4, vol. v, no. 30, pp. 106–117 (February 1853); and (2) W. J. Millar, ed., Miscellaneous Scientific Papers: past W. J. Macquorn Rankine, ... (London, England: Charles Griffin and Co., 1881), part 2, pp. 203–208.
4. ^ Smith, Crosbie (1998). The Science of Energy – a Cultural History of Energy Physics in Victorian Britain. The University of Chicago Press. ISBN0-226-76420-6.
5. ^ Brown, Theodore 50. (2006). Chemistry The Key Science. Upper Saddle River, New Jersey: Pearson Pedagogy, Inc. pp. 168. ISBN0-xiii-109686-9.
6. ^ John Robert Taylor (2005). Classical Mechanics. University Scientific discipline Books. p. 117. ISBN978-ane-891389-22-1.
7. ^ Burton Paul (1979). Kinematics and dynamics of planar machinery. Prentice-Hall. ISBN978-0-13-516062-half-dozen.
8. ^ Feynman, Richard P. (2011). "Work and potential energy". The Feynman Lectures on Physics, Vol. I. Basic Books. p. 13. ISBN978-0-465-02493-iii.
9. ^ "Hyperphysics – Gravitational Potential Energy".
10. ^ Guth, Alan (1997). "Appendix A, Gravitational Free energy". The Inflationary Universe. Perseus Books. pp. 289–293. ISBN0-201-14942-seven.
11. ^ "Energy storage – Packing some power". The Economist. 3 March 2022.
12. ^ Jacob, Thierry.Pumped storage in Switzerland – an outlook beyond 2000 Archived 23 July 2022 at WebCite Stucky. Accessed: 13 February 2022.
13. ^ Levine, Jonah G. Pumped Hydroelectric Energy Storage and Spatial Diversity of Wind Resource equally Methods of Improving Utilization of Renewable Free energy Sources Archived 1 August 2022 at the Wayback Car page half dozen, University of Colorado, December 2007. Accessed: 12 February 2022.
14. ^ Yang, Chi-Jen. Pumped Hydroelectric Storage Archived 5 September 2022 at the Wayback Automobile Duke University. Accessed: 12 February 2022.
15. ^ Energy Storage Archived 7 April 2022 at the Wayback Machine Hawaiian Electric Company. Accessed: 13 February 2022.
16. ^ Packing Some Power: Energy Technology: Meliorate ways of storing free energy are needed if electricity systems are to become cleaner and more efficient, The Economist, 3 March 2022
17. ^ Downing, Louise. Ski Lifts Help Open up $25 Billion Market for Storing Power, Bloomberg News online, 6 September 2022
18. ^ Kernan, Aedan. Storing Free energy on Rail Tracks Archived 12 April 2022 at the Wayback Motorcar, Leonardo-Energy.org website, 30 Oct 2022
19. ^ Aharoni, Amikam (1996). Introduction to the theory of ferromagnetism (Repr. ed.). Oxford: Clarendon Pr. ISBN0-19-851791-2.
20. ^ Jackson, John David (1975). Classical electrodynamics (second ed.). New York: Wiley. ISBN0-471-43132-10.
21. ^ Livingston, James D. (2011). Ascension Force: The Magic of Magnetic Levitation. President and Fellows of Harvard Higher. p. 152.
22. ^ 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. ISBN978-1-4390-4844-three.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). Due west. H. Freeman. ISBN0-7167-0809-four.

External links

• What is potential energy?

Source: https://en.wikipedia.org/wiki/Potential_energy

Posted by: howardextouralke.blogspot.com

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