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Encyclopedia > Gravitational potential energy

Potential energy (U, or Ep), a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. For example a mass released above the Earth has energy resulting from the gravitational attraction of the Earth which is transferred in to kinetic energy. The concept of a scalar is used in mathematics, physics, and computing. ... In physics, a potential is a scalar quantity that can be used to analyze the effects of complicated vectorial forces and similar quantities by means of simple conservation laws. ... Matter is anything that has mass and occupies space. ... Earth, also known as the Earth or Terra, is the third planet outward from the Sun. ... Gravitation is the tendency of masses to move toward each other. ... Kinetic energy (also called vis viva, or living force) is energy possessed by a body by virtue of its motion. ...

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### Gravitational potential energy

This energy is stored as a result of the elevated position of an object such as a rock on top of a hill or water behind a dam. It is written as Scrivener Dam, Canberra Australia, was engineered to withstand a once-in-5000-years flood event A dam (a common Teutonic word, compare to Dutch dam, Swedish and German damm, and the Gothic verb faurdammjan, to block up) is a barrier across flowing water that obstructs, directs or retards the flow...

where m is the mass of the object, g the acceleration due to gravity and h the height above a chosen reference level (typical units would be kilograms for m, metres/second2 for g, and metres for h).

In relation to spacecraft and astronomy g is not constant and the formula becomes an integral. In the case of a sphere of uniform mass (such as a planet), with h measured above the surface, the integral takes the form: Ariane 5 lifts off with the Rosetta probe on 2nd of March, 2004. ... Astronomy (Greek: αστρονομία = άστρον + νόμος, literally, law of the stars) is the science involving the observation and explanation of events occurring beyond the Earth and its atmosphere. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

Where h0 is the radius of the sphere, M is the mass of the sphere, and G is the gravitational constant. According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

If h is instead taken to be the distance from the center of the sphere, then outside the sphere the potential energy relative to that at the center has two terms:

which evaluates to:

[We may also want to link to an explanation of that second term (the gravitational forces created by hollow spherical shells)]

A frequently adopted convention is that an object infinitely far away from an attracting source has zero potential energy. Relative to this, an object at a finite distance r from a source of gravitation has negative potential energy. If the source is approximated as a point mass, the potential energy simplifies to: A point mass is in physics an idealisation of a body whose dimensions can be neglected compared to the distances of its movement. ...

See also Gravitational binding energy. The gravitational binding energy of an object is the amount of energy required to accelerate every component of that object to the escape velocity of every other component. ...

### Elastic potential energy

This energy is stored as the result of a deformed solid such as a stretched spring. As a result of Hooke's law, it is given by: In physics, Hookes law of elasticity states that if a force (F) is applied to an elastic spring or prismatic rod (with length L and cross section A), its extension is linearly proportional to its tensile stress σ and modulus of elasticity (E): ΔL = 1/E × F × L/A...

where k is the spring constant (a measure of the stiffness of the spring), expressed in N/m, and x is the displacement from the equilibrium position, expressed in metres (see Main Article: Elastic potential energy). The elastic potential energy stored in an elastic string or spring of natural length l and modulus of elasticity λ under an extension of x is given by: This equation is often used in calculations of positions of mechanical equilibrium. ...

### Chemical energy

Chemical energy is a form of potential energy related to the breaking and forming of chemical bonds. In chemistry, a chemical bond is the force, which holds together atoms in molecules or crystals. ...

### Rest mass energy

Albert Einstein's famous equation, derived in his special theory of relativity, can be written: Portrait of Albert Einstein taken by Yousuf Karsh on February 11, 1948 Albert Einstein (March 14, 1879 – April 18, 1955) was a theoretical physicist who is widely regarded as the greatest scientist of the 20th century. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ...

where E0 is the rest mass energy, m is 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.) Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Cherenkov effect in a swimming pool nuclear reactor. ... The article on the vacuum cleaner is located elsewhere. ...

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 petajoule/kg ≈ 21 megatons/kg) This article describes the SI prefix peta. ... A megaton or megatonne is a unit of mass equal to 1,000,000 metric tons, i. ...

### Electrical potential energy

The electrical potential energy per unit charge is called electrical potential. It is expressed in volts. The fact that a potential is always relative to a reference point is often made explicit by using the term potential difference. The term voltage is also common. Electrical potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential or the electric potential, typically measured in volts. ... The volt is the SI derived unit for electric potential and voltage (derived from the ampere and watt). ... In the physical sciences, potential difference is the difference in potential between two points in a conservative vector field. ...

The electrical potential energy between two charges q1 and q2 is:

The electric potential generated by charges q1 (denoted V1) and q2 (denoted V2) is:

## Relation between potential energy and force

Potential energy is closely linked with forces. If the work done going around a loop 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. In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ...

For example, gravity is a conservative force. The work done by a unit mass going from point A with U = a to point B with U = b by gravity is (ba) and the work done going back the other way is (ab) so that the total work done from A conservative force is a force which is path-independent. ...

The nice thing about potential energy is that you can add any number to all points in space and it doesn't affect the physics. If we redefine the potential at A to be a + c and the potential at B to be b + c [where c can be any number, positive or negative, but it must be the same number for all points] then the work done going from

as before.

In practical terms, this means that you can set the zero of U 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. Earth, also known as the Earth or Terra, is the third planet outward from the Sun. ...

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. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... Properties For alternative meanings see atom (disambiguation). ...

A conservative force can be expressed in the language of differential geometry as an exact form. Because Euclidean space is contractible, its de Rham cohomology vanishes, so every exact form is closed, 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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

## 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 a mass or charge, etc. being attracted.

E.g. a mass, being an area of attraction, may be called a gravitational well. See also potential well. The Whole Earth Lectronic Link (or The WELL) is one of the oldest virtual communities still online. ... A potential well is the region surrounding a local potential energy minimum. ...

Results from FactBites:

 Gravitational Potential Energy (348 words) Gravitational potential energy is energy an object possesses because of its position in a gravitational field. The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
 PowerPedia:Potential energy - PESWiki (2552 words) Potential energy is energy that is "captured" in an object, with the potential to be released. Gravitational potential energy is the energy that would be released if an object in a gravitational field (such as the earth's gravitational field) were allowed to fall from its current position to a given reference level (such as the surface of the earth). Gravitational potential is the potential energy per unit mass of an object due to its position in a gravitational field.
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