Potential energy

The potential energy (including potential energy ) describes the energy of a physical system, which is determined by its current configuration in itself (eg compressed spring ) or its position in a force field. A compressed spring has potential energy because of their configuration more relaxed than one. A body can vary depending on its location, for example a gravity field or an electrical field have different potential energy.

In everyday life, is a potential energy, for example, the energy that a body has by its altitude: If a stone falls from height of 20 meters, it has twice the ability to work as at 10 meter height of fall. While if the potential energy is converted into kinetic energy or other forms of energy and reduced. In hydroelectric power plants can convert potential energy of the water of a reservoir into electrical energy.

Just like other forms of energy, the potential energy is a state variable of a physical system (eg, general body, Reservoir, particles). In a closed system ( eg ball in marble run without friction, elastic shock balls ), the potential energy at state changes while increasing or decreasing, such as local displacement of the body, at its height change or excitation of an atom by radiation. Because of the principle of conservation of energy then takes but a different form of energy (such as kinetic energy, electric field energy ) from the same extent or to.

As a symbol for the potential energy E pot or U is most often used in theoretical physics V is widespread. It is often inaccurate spoken of potential when the potential energy is intended.

• 4.1 Example: Potential energy in the electric field
• 4.2 Relationship between potential energy and potential 4.2.1 Example gravitational field

Potential energy in gravitational field

Introduction

As an introduction, we consider a cyclist who is running on a flat track, then a mountain moves up and down as a last resort. Consideration should first be carried out without friction.

On level ground, the cyclist is traveling at a certain speed, which corresponds to a certain kinetic energy. He now travels up the mountain, he must expend more energy to the same speed (and therefore the same kinetic energy) to maintain. Because of the conservation of energy but no energy can be lost and the energy spent by a cyclist in the rise more, you have somewhere to flow out: The more energy used is converted into potential energy. The higher it rises, the more potential energy the cyclist. On the descent, the cyclist would even slow down to maintain its speed and thus keep its kinetic energy constant. Brake it does not, it is faster and has more kinetic energy. The increase its kinetic energy can not be without loss of another form of energy but due to the energy conservation. The increase of the kinetic energy resulting from the loss of potential energy.

Cyclists in detail with friction

The cyclists reached on level ground without much trouble 20 km / h, as it only needs to compete against air resistance and rolling friction. Does he come now to an increasing stretch, he must more effort than before at the same speed. After reaching the summit, it is downhill and the cyclist continues to roll without pedaling, even having to brake so that it is not too fast.

On the driver's wheel, including two forces: the frictional force and the weight force. In the first stretch, the weight force is perpendicular to the road and has therefore the application of the resolution of forces no force component in the direction of movement. Now takes a rise in the decomposition of the gravitational force produces a force component counter to the direction of movement. After the summit exceeded gravity has a component in the direction of movement and against the frictional force.

For a movement against the weight of work must be expended on the body, which is now stored as potential energy in it. With a movement which contains a component in the direction the weight of the body is doing work, overcomes some friction and its potential energy is decreased. The path component in the direction of gravitational force is called height and together with the force we have:

• M - mass
• G - acceleration of gravity, specifically the gravitational field strength of the earth
• H - height above the ground
• Condition: height significantly less than the Earth's radius h « RE

More general description

In general, the gravitational field strength and the weight is location-dependent. Thus, the following applies:

The negative sign arises from the fact that you have to move something in opposition to the force acting to increase the potential energy.

Example: Potential energy at the earth's surface

Substituting constant (which approximately applies to the Earth's surface for small height differences ), we obtain again the equation described in the previous section:

Example: Potential energy on a planetary surface

The potential energy of a particle on a surface of the planet corresponds to the work which must be performed in order to transport this particle to infinity (that is, to remove from the gravitational field, because the gravitational field disappears at infinity ). In addition, the potential energy of a particle on the surface of the planet ( unless it is considered only a system of a planet and a test particle ) maximum. With the agreement, that the origin lies in the planet center and the planet has a radius that is obtained by inserting the Newton 's law of gravitation

As potential energy of the test particle on the planet's surface. In the last step, the new planet was dependent constant

Defined. Here, the mass of the planet, the mass of the test particle and the gravitational constant. If one calculates from the data of the earth, the result is about the so-called acceleration due to gravity.

Potential energy of a stretched spring

Of the spring

Results for the potential energy

Here, k is the spring constant and x is the deflection of the spring from the resting position.

Potential energy and conservation of energy

In a closed system with no energy exchange with the environment and neglecting any friction applies to each of the conservation of energy in classical mechanics:

• U - potential energy
• T - kinetic energy
• E - mechanical energy

In words, the sum of the potential and kinetic energy, including the rotational energy is constant and corresponds to the total energy of the mechanical system.

In a higher formulation of mechanics, the Hamiltonian formalism, we also write

Where H is the Hamiltonian and L is the Lagrangian.

Formal definition

As a conservative force field defines the force on a test body at any place and mathematically is a gradient field, there is a field of force equivalent to the scalar field. This is the potential energy for the respective location. From the inversion of the integral work follows, which requires an energy increase along a path, a force component in the opposite direction of the path. By breaking down the force field in Cartesian components arise depending on the location following partial derivatives:

Generally, this can be expressed by the nabla operator.

The reversal of the discharge leads to the integral, and determines the change in potential energy in the force field as a work integral with a negative sign. This tends to also understand the applicability to different force fields.

In order to increase the potential energy of a body, field work must be done against the forces of a conservative force field. Thus, each afflicted body mass in a gravitational field potential energy. However, this can only be increased or decreased when the body is moved against or in the direction of gravitational force. With a displacement perpendicular to the field lines, the body maintains its potential energy. One such area is called equipotential surface or line and corresponds to a contour line on the map. The field line, however, describes the direction of the slope.

Unless occur no friction losses or other forms of interaction with the environment is for a displacement in conservative force fields, the principle of path independence. This means regardless of the chosen path must be the same amount of field work so that a body be done flows from the starting point to the destination point. This reflects the principle of energy conservation plays again as the work of the energy change corresponds.

The choice of the reference level can be arbitrary, but reduce pragmatic reasons the selection. If in doubt always suitable as zero level is the starting point of the examined body. When gravitational field often forms the earth's surface as the reference point, or generally the lowest point of the area. In addition, the reference point can be moved to an infinitely distant place (). The reverse of which constitutes the maximum potential energy, in which a body is moved from its starting point from the force field out, a central field of force, it is assumed.

When electric charges of opposite sign, this leads to the minimum potential energy.

Example: Potential energy in the electric field

The force on a charge in a given electric field is calculated from:

Substituting into the integral work now shows the relationship between the potential energy of a charge and the Coulomb potential, which also constitutes a scalar field. Both fields differs only charge the proportionality factor:

Is the so-called Coulomb potential.

Relationship between potential energy and potential

The concept of potential energy is closely linked with the term of the potential, which is an equivalent representation of a conservative force field. The potential energy of a physical system is the product of the coupling constant of the particle with respect to the force field, which it is exposed (eg mass in the case of the gravitational field, the charge in the case of electric field ), and the potential of the force field:

The potential is due to the definition of the force field. Based on this definition, the potential energy is defined only for particles in conservative force fields and the zero point of the energy scale defined as desired.

Example gravitational field

The force on a test body of mass m in a given gravitational field is calculated from:

Substituting into the integral work now shows the relationship between the potential energy of a mass and the gravitational potential, which also constitutes a scalar field.

Clearly, the factor m describes the dependence of the specimen and the potential of the field property.