Potential energy
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</div>Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. This form of energy has the potential to change the state of other objects around it, for example, the configuration or motion.
Various forms of energy can be grouped as potential energy. Each of these forms is associated with a particular kind of force acting in conjunction with some physical property of matter (such as mass, charge, elasticity, temperature etc). For example, gravitational potential energy is associated with the gravitational force acting on object's mass; elastic potential energy with the elastic force (ultimately electromagnetic force) acting on the elasticity of a deformed object; electrical potential energy with the coulombic force; strong nuclear force or weak nuclear force acting on the electric charge on the object; chemical potential energy, with the chemical potential of a particular atomic or molecular configuration acting on the atomic/molecular structure of the chemical substance that constitutes the object; thermal potential energy with the electromagnetic force in conjunction with the temperature of the object.
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[edit] Gravitational potential energy
Gravitational potential energy is the potential energy that an object possesses by virtue of its mass and the gravitational force acting on it.
In everyday experience, gravitational potential energy arises most familiarly when an object is raised in the Earth's gravitational field. The object's increase in gravitational potential energy is equal to the amount of energy required to raise it, or, equivalently, the amount of energy that would be released if it were allowed to fall back to its original level.
For example, consider a book placed on top of a table. To raise the book from the floor to the table, work must be done, and energy supplied. (If the book is lifted by a person then this is provided by the chemical energy obtained from that person's food and then stored in the chemicals of the body.) Assuming perfect efficiency (no energy losses), the energy supplied to lift the book is exactly the same as the increase in the book's gravitational potential energy. The book's potential energy can be released by knocking it off the table. As the book falls, its potential energy is converted to kinetic energy. When the book hits the floor this kinetic energy is converted into heat and sound by the impact.
The factors that affect an object's gravitational potential energy are the height to which it is raised, its mass, and the strength of the gravitational field in which it is raised. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a tall cupboard, and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height on Earth because the Moon's gravity is weaker. (The gravitational force between any two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, according to the Newton's law of gravitation.
[edit] Calculation of gravitational potential energy
Assuming that the opposing gravitational force is constant, the work done in raising an object is equal to the force applied multiplied by the distance through which the object is raised. The gravitational force that must be overcome is equal to the object's mass multiplied by the acceleration due to gravity, so the object's gravitational potential energy, Ug, is given by
- <math>U_g = m g h \,</math>
where
- m is the mass of the object
- g is the acceleration due to gravity (approximately 9.8 m/s2 at the earth's surface)
- h is the height to which the object is raised, relative to a given reference level (such as the earth's surface).
When applying this equation it is essential to use consistent units. Most scientific work is now done in SI units, in which case mass is measured in kilograms (kg), acceleration in metres per second squared (m/s2), and distance (here height) in metres (m). The resulting energy is expressed in joules (kg m2/s2).
The equation shows that gravitational potential energy is proportional to both mass and height. For example, raising two similar objects, or raising the same object twice as far, doubles the potential energy.
The "mgh" formula works well provided that the acceleration due to gravity, g, is very nearly constant over the distance h. On or close to the surface of the earth this assumption is reasonable, but over the much larger distances applying, for example, to spacecraft and astronomical bodies, it is not.
To calculate gravitational potential energy with varying g it is necessary to sum all the individual increments of potential energy as the masses are separated, taking account of the varying value of g as we go. In the limit, as the increments become "infinitely small", the sum becomes an integral.
To simplify the evaluation of the integral we can make the assumption that the gravitational forces act as if the objects' masses were concentrated at their respective centres of mass. This assumption is mathematically exactly correct for a spherically symmetrical object (such as, to a reasonable approximation, a planet). It is not generally correct in other cases, though if the dimensions of an object are very small compared to the distance of separation then it is reasonable to consider it as a point mass and ignore the details of its shape.
With this simplifying assumption, integrating force over distance leads to the following general expression for the gravitational potential energy, Ug, of a system of two masses:
| <math>U_g\,</math> | <math>= \int_{h_1}^{h_2} {G m_1 m_2 \over r^2} dr</math> | |
| <math>= G m_1 m_2 \left ( \frac{1}{h_1} - \frac{1}{h_2} \right )</math> |
where
- <math>m_1</math> and <math>m_2</math> are the masses of the two objects
- <math>G</math> is the gravitational constant (not to be confused with the g used earlier)
- <math>h_1</math> is the reference level (the separation at which potential energy is considered to be zero)
- <math>h_2</math> is the actual distance between the objects.
Subject to the caveats mentioned above, the distances <math>h_1</math> and <math>h_2</math> are measured between the objects' centres of mass.
For example, in the case of a small object above the surface of the earth, with reference level at the surface, <math>m_1</math> and <math>m_2</math> are respectively the masses of the earth and the object, <math>h_1</math> is the distance from the earth's centre to the earth's surface, and <math>h_2</math> is the distance from the earth's centre to the object.
If we try to calculate an "absolute" potential energy by setting the reference level at zero then the formula "blows up" with division by zero. In other words, we can only actually use this formula to measure the difference in potential energy between one non-zero separation and another.
In practice it is often convenient to take the reference level at infinite separation (i.e. <math>h_1 = \infty</math>), in which case the formula becomes:
- <math>U_g = \frac{-G m_1 m_2}{r}</math>
where r is now the distance between the centres of mass of the two objects (again noting the earlier caveats). For a small object above the surface of the earth, r is the distance from the object to the earth's centre (and similarly for other spherical bodies).
Using this convention, potential energy is zero when r is infinitely large, and negative at any finite r. However, the difference in potential energy at different values of r – the quantity we are actually interested in – takes the expected sign.
Ug as calculated above measures the potential energy of the whole system. This can be visualised as if two bodies in space were released from rest and allowed to come together under the force of gravity. The sum of the kinetic energy gained by the two objects is exactly equal to the decrease in the potential energy of the system. The ratio of the objects' individual kinetic energy gains is equal to the reciprocal of the ratio of their masses. So, in the case of a relatively light object falling towards a very massive object (such as the earth), the contribution from the massive object is insignificant. In some sense, therefore, we can say that almost all the potential energy of the system is embodied in the light object, and almost none in the very massive object.
See also two-body problem and gravitational binding energy.
[edit] Gravitational potential
Gravitational potential is the potential energy per unit mass of an object due to its position in a gravitational field. The gravitational potential due to a point mass:
- <math>E(r) = \frac{-Gm}{r} \ </math>
where:
- <math>G \ </math> is the universal gravitational constant,
- <math>r \ </math> is the distance to the center of mass of the object,
- <math>m \ </math> is the mass of the point object.
In astrodynamics the gravitational potential function has to account for the non-spherical and non-homogeneous nature of typical sources of gravitational potential. In this case a gravitational potential may depend on polar <math>\phi\!\,</math> and azimuth <math>\lambda\!\,</math> direction of vector <math>r\!\,</math>.
The most widely used form of the gravitational potential function depends on <math>\phi\!\,</math> (latitude) and potential coefficients, Jn, called the zonal coefficients:
- <math> E(r,\phi) = \frac{GM}{r} \left [1 - \sum_{n=2}^N J_{n} \left (\frac{R}{r} \right)^2 P_n (\sin \phi) \right ] </math>
[edit] Elastic potential energy
Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (often termed under the word stress by physicists). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object.
[edit] Calculation of elastic potential energy
In the case of a spring of natural length l and modulus of elasticity λ under an extension of x, elastic potential energy can be calculated using the formula:
- <math>E = \frac{\lambda x^2}{2l}</math>
This formula is obtained from the integral of Hooke's Law:
- <math>U_e = \int {k x}\, dx = \frac {1} {2} k x^2</math>
The equation is often used in calculations of positions of mechanical equilibrium.
In the general case, elastic energy is given by the Helmholtz potential per unit of volume f as a function of the strain tensor components εij:
- <math> f(\epsilon_{ij}) = \lambda \left ( \sum_{i=1}^{3} \epsilon_{ii}\right)^2+2\mu \sum_{i=1}^{3} \sum_{j=1}^{3} \epsilon_{ij}^2</math>
Where λ and μ are the Lamé elastical coefficients. The connection between stress tensor components and strain tensor components is:
- <math> \sigma_{ij} = \left ( \frac{\partial f}{\partial \epsilon_{ij}} \right)_S </math>
For a material of Young's modulus, Y (same as modulus of elasticity λ), cross sectional area, A0, initial length, l0, which is stretched by a length, <math>\Delta l</math>:
- <math>U_e = \int {\frac{Y A_0 \Delta l} {l_0}}\, dl = \frac {Y A_0 {\Delta l}^2} {2 l_0}</math>
- where Ue is the elastic potential energy.
The elastic potential energy per unit volume is given by:
- <math>\frac{U_e} {A_0 l_0} = \frac {Y {\Delta l}^2} {2 l_0^2} = \frac {1} {2} Y {\varepsilon}^2</math>
- where <math>\varepsilon = \frac {\Delta l} {l_0}</math> is the strain in the material.
[edit] Chemical energy
Chemical energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction. For 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 solar energy to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical reactions.
The similar term chemical potential is used by chemists to indicate the potential of a substance to undergo a chemical reaction.
[edit] Electrical potential energy
An object can also have potential energy by virtue of its electric charge and several forces related to their presence. There are three main kinds of this kind of potential energy; electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy)and nuclear potential energy.
[edit] Electrostatic potential energy
In case the electric charge of an object can be assumed to be it rest, it has potential energy due to its position relative to other charged objects.
The electrostatic potential energy is the energy of an electrically charged particle in an electric field. It is defined as the work that must be done to move it from an infinite distance away to its present location, in the absence of any non-electrical forces on the object. This energy is non-zero if there is another electrically charged object nearby.
The simplest example is the case of two point-like objects A1 and A2 with electrical charges q1 and q2. The work W required to move A1 from an infinite distance to a distance d away from A2 is given by:
- <math>W=k\frac {q_1q_2} d</math>
where k is Coulomb's constant, equal to <math>\frac 1 {4\pi\epsilon_0}</math>.
This equation is obtained by integrating the Coulomb force between the limits of infinity and d.
A related measure called electrical potential is equivalent to electrical potential energy divided by electric charge.
[edit] Electrodynamic potential energy
In case, a charged object or its constituent charged particles, are not at rest, it generates a magnetic field giving rise to yet another form of potential energy, often termed as magnetic potential energy. This kind of potential energy is a result of the phenomenon magnetism, whereby an object that is magnetic, has the potential to move other similar objects. Magnetic objects are said to have some magnetic moment. Magnetic fields and their effects are best studied under electrodynamics.
[edit] Nuclear potential energy
Nuclear potential energy, is the potential of the particles inside an atomic nucleus, some of which are indeed electrically charged. This kind of potential energy is different from the previous two kinds of electrical potential energies because in this case the charged particles are extremally close to each other. The nuclear particles are bound together not because of the coulombic force but due to strong nuclear force that binds nuclear particles more strongly and closely. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta decay.
Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them have less mass than if they were individually free, and this mass difference is liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun, also called solar energy, is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million metric tons of solar matter per second into light, which is radiated into space.
[edit] Thermal potential energy
An object can also change the state of motion of other objects by virtue of its temperature and electromagnetic force, that is through radiations. This kind of potential energy can be termed as thermal potential energy.
[edit] Rest mass energy
Albert Einstein's famous equation, derived in his special theory of relativity, can be written:
- <math>E_0 = m c^2 \,</math>
where E0 is the rest mass energy, m is the rest mass of the body, and c is the speed of light in a vacuum. (The subscript zero is used here to distinguish this form of energy from the others that follow. In most other contexts, the equation is written with no subscript.)
The rest mass energy is the amount of energy inherent in the mass when it is at rest. This equation quantifies the equivalence of mass and energy: A small amount of mass is equivalent to a very large amount of energy. (i.e., 90 petajoules per kilogram ≈ 21 megaton of TNT per kilogram)
[edit] Relation between potential energy and force
Potential energy is closely linked with forces. If the work done moving along a path which starts and ends in the same location is zero, then the force is said to be conservative and it is possible to define a numerical value of potential associated with every point in space. A force field can be re-obtained by taking the vector gradient of the potential field.
For example, gravity is a conservative force. The work done by a unit mass going from point A with <math>U = a</math> to point B with <math>U = b</math> by gravity is <math>(b - a)</math> and the work done going back the other way is <math>(a - b)</math> so that the total work done from
- <math>U_{A \to B \to A} = (b - a) + (a - b) = 0 \,</math>
If we redefine the potential at A to be <math>a + c</math> and the potential at B to be <math>b + c</math> [where <math>c</math> can be any number, positive or negative, but it must be the same number for all points] then the work done going from
- <math>U_{A \to B} = (b + c) - (a + c) = b - a \,</math>
as before.
In practical terms, this means that you can set the zero of <math>U</math> anywhere you like. You might set it to be zero at the surface of the Earth or you might find it more convenient to set it zero at infinity.
A thing to note about conservative forces is that the work done going from A to B does not depend on the route taken. If it did then it would be pointless to define a potential at each point in space. An example of a non-conservative force is friction. With friction, the route you take does affect the amount of work done, and it makes no sense at all to define a potential associated with friction.
All the examples above are actually force field stored energy (sometimes in disguise). For example in elastic potential energy, stretching an elastic material forces the atoms very slightly further apart. Powerful electromagnetic forces try to keep the atoms at their optimal distance and so elastic potential is actually electromagnetic potential. Having said that, scientists rarely talk about forces on an atomic scale. Everything is phrased in terms of energy rather than force. You can think of potential energy as being derived from force or you can think of force as being derived from potential energy.
A conservative force can be expressed in the language of differential geometry as a closed form. Because Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is exact, i.e., is the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.
[edit] Graphical representation
A graph of a 1D or 2D potential function with the function value scale increasing upward is useful to visualize the potential field: a ball rolling to the lowest part corresponds to an object such as a mass or charge being attracted.
When using this type of analogy, a mass, being an area of attraction, is often called a gravitational well, or potential well.
Potential energy is stored energy. Potential can very much be converted into Kinetic energy. This also goes along with the Law of Conversation of Energy. The Law of Conversation of Energy means that energy change it's form but not be destroyed.
[edit] References
- Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
- Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.
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