Energy

Energy is a fundamental quantity that every physical system possesses. Energy of physical system in a certain given state is defined as the amount of work W needed to change the state of the system from some initial state (called reference state or reference level) to the given state.

Types of energy

Kinetic energy

Main article: Kinetic energy.

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Kinetic energy is the portion of energy related to the motion of a body.

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:E_k = int mathbf{v} cdot mathrm{d}mathbf{p}

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The equation above says that the kinetic energy (E_k) is equal to the integral of the dot product of the velocity (mathbf{v}) of a body and the infinitesimal of the body's momentum (mathbf{p}).

Related Topics:
Dot product - Velocity - Infinitesimal - Momentum

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For non-relativistic velocities, that is velocities much smaller than the speed of light, we can use the Newtonian approximation

Related Topics:
Relativistic - Speed of light

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:E_k = egin{matrix} rac{1}{2} end{matrix} mv^2

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where

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Ek is kinetic energy

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m is mass of the body

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v is velocity of the body

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At near-light velocities, we use the correct relativistic formula:

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:E_k = m c^2 (gamma - 1) = gamma m c^2 - m c^2 ;!

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:gamma = rac{1}{sqrt{1 - (v/c)^2}}

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where

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v is the velocity of the body

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m is its rest mass

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c is the speed of light in a vacuum, which is approximately 300,000 kilometers per second

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gamma m c^2 , is the total energy of the body

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m c^2 , is again the rest mass energy.

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In the form of a Taylor series, the relativistic formula for can be written as:

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:E_k = rac{1}{2} mv^2 - rac{3}{8} rac{mv^4} {c^2} + cdots

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Hence, the second and higher terms in the series correspond with the "inaccuracy" of the Newtonian approximation for kinetic energy in relation to the relativistic formula.

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Potential energy

Main article: Potential energy.

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In contrast to kinetic energy, which is the energy of a system due to its motion, or the internal motion of its particles, the potential energy of a system is the energy associated with the spatial configuration of its components and their interaction with each other. Any number of particles which exert forces on each other automatically constitute a system with potential energy. Such forces, for example, may arise from electrostatic interaction (see Coulomb's law), or gravity.

Related Topics:
Kinetic energy - System - Potential energy - Coulomb's law - Gravity

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In an isolated system consisting of two stationary objects that exert a force f(x) on each other and lay on the x-axis, their potential energy is most generally defined as

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:E_p = -int f(x) , dx

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where the force between the objects varies only with distance x and is integrated along the line connecting the two objects.

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To further illustrate the relationship between force and potential energy, consider the same system of two objects situated along the x-axis. If the potential energy due to one of the objects at any point x is U(x), then the force on the that object x is

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:f(x) = -rac{dU(x)}{dx}

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This mathematical relationship demonstrates the direct connection between force and potential energy: the force between two objects is in the direction of decreasing potential energy, and the magnitude of the force is proportional to the extent to which potential energy decreases. A large force is associated with a large decrease in potential energy, while a small force is associated with a small decrease in potential energy. Notice how, in this case, the force on an object depends entirely on its potential energy.

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These two relationships – the definition of potential energy based on force, and the dependence of force on potential energy – show how the concepts of force and potential energy are intimately linked: if two objects do not exert forces on each other, there is no potential energy between them. If two objects do exert forces on each other, then potential energy naturally arises in the system as part of the system's total energy. Since potential energy arises from forces, any change in the system's spatial configuration will either increase or decrease the system's potential energy as the objects are repositioned.

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When a system moves to a lower potential energy state, energy is either released in some form or converted into another form of energy, such as kinetic energy. The potential energy can be "stored" as gravitational energy, elastic energy, chemical energy, rest mass energy or electrical energy, but arises in all cases from the spatial positioning and interaction of objects within a system. Unlike kinetic energy, which exists in any moving body, potential energy exists in any body which is interacting with another object.

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For example a mass released above the Earth initially has potential energy resulting from the gravitational attraction of the Earth, which is transferred to kinetic energy as the gravitational force acts on the object and its potential energy is decreased as it falls.

Related Topics:
Earth - Gravitational attraction

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Equation:

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:E_p = mgh ;

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where m is the mass, h is the height and g is the value of acceleration due to gravity at the Earth's surface (see gee).

Related Topics:
Acceleration - Gee

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Internal energy

Main article: Internal energy.

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Internal energy is the kinetic energy associated with the motion of molecules, and the potential energy associated with the rotational, vibrational and electric energy of atoms within molecules. Internal energy, like energy, is a quantifiable state function of a system.

Related Topics:
Kinetic energy - Molecule - Potential energy - Rotational - Vibrational - Electric - Atom - Internal energy - State function

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