Potential energy

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Potential energy is stored energy. The energy is stored by doing work against a force such as gravity or the spring in a clockwork motor.

Contents

Examples

A book on a table has greater gravitational potential energy than the same book on the floor. However the same book has less gravitational potential energy than if it were even higher. In raising the book from the floor to the table, work was done by someone which is now stored as potential energy. (This energy was provided by the chemical energy stored in food). The presence of this potential energy could be demonstrated by sliding the book off the table. The book would gain kinetic energy from its velocity until it reached the floor. The kinetic energy would then be converted into heat and sound by the impact.

In another example in Wales at Dinorwig there are two lakes, one higher than the other. At times when surplus electricity is not required (and so is cheap), water is pumped to the higher lake. At times of peak demand for electricity, the water flows through turbines and generates electricity once more (see also pumped storage). (The process is not completely efficient and much of the original energy from the surplus electricity is wasted by friction). In this example the potential energy is stored by doing work against the force of gravity.

The factors that affect the amount of gravitational potential energy that is created are: the mass of the object, the distance that it is raised and the gravitational field strength. Raising the same object to the same height on the Moon would require less energy than on earth because the force of gravity on the Moon's surface is less.

Simple calculation

The work done in raising an object is the force overcome multiplied by the distance that it was raised. Thus raising two similar objects or raising the object twice as far would produce twice as much potential energy. The gravitational force that must be overcome is the book's mass multiplied by the force of gravity.

The potential energy of a body = m g h \, 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 per second squared for g, and metres for h).

An even simpler way of saying this is weight times height.

Types

Gravitational potential energy

The work of constant gravitational force F=mg on, say, rock of mass m (when rock is lowered from elevation h) is equal W = Fh = mgh - thus this value is called gravitational potential energy of rock.

It is written as

U_g = m g h \,

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 per second squared 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:

U_g = \int_{h_0}^{h + h_0} {GmM \over r^2} dr

Where h0 is the radius of the sphere, M is the mass of the sphere, and G is the gravitational constant.

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:

U_g = \int_{h_0}^h {GmM \over r^2} dr + \int_0^{h_0} {GmM \over h_0^2} {r \over h_0} dr

which evaluates to:

U_g = GmM \left[{1 \over h_0} - {1 \over h}\right] + {1 \over 2} {GmM \over h_0} = GmM \left[{3 \over 2h_0} - {1 \over h}\right]


Conventional reference state for a system of interacting parts is the state at which parts are infinitely separated (thus not interacting). 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:

U_g = - {GmM \over r}

When an object is lifted, the object and the Earth are moved apart such that each is moved a distance inversely proportional to its mass. In both moves the force is of the same magnitude, so the energy involved in moving the Earth is much smaller. Similarly when the object is dropped the velocities are inversely proportional to the masses, so the kinetic energy also. See also two-body problem and gravitational binding energy.

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:

U_e = {1\over2}kx^2

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).

Chemical energy

Chemical energy is a form of potential energy related to the breaking and forming of chemical bonds.

Rest mass energy

Albert Einstein's famous equation, derived in his special theory of relativity, can be written:

E_0 = m c^2 \,

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)

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.

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

U = \frac{q_1 q_2} {4 \pi \epsilon_o r}

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

V_1 = \frac{q_1} {4 \pi \epsilon_o r}
V_2 = \frac{q_2} {4 \pi \epsilon_o r}

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.

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 (b − a) and the work done going back the other way is (a − b) so that the total work done from

U_{A \to B \to A} = (b - a) + (a - b) = 0 \,

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

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

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.

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 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.

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.

See also

References

  • Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.), Brooks/Cole. ISBN 0534408427.
  • Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.), W. H. Freeman. ISBN 0716708094.cs:Potenciální energie

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