Specific potential energy

Specific potential energy
Specific potential energy
Common symbols	pe, or eu
SI unit	J/kg
In SI base units	m2/s2
Derivations from; other quantities	eu = g h

In classical mechanics and gravitational physics, massic gravitational potential energy (MGPE) is a fundamental concept that relates to the gravitational potential a body possesses due to its position in a gravitational field relative to a reference point. This energy is associated with the gravitational force acting on a body and its displacement within the gravitational field. It depends on the height of a body and the strength of the gravitational field. The term "massic" in this context refers to the gravitational energy per unit mass.^[1]^[2]

Overview

Definition

The massic gravitational potential energy (MGPE) of an object is defined as the gravitational potential at a location which is the gravitational potential energy (GPE) at that location per unit mass (m):

MGPE={GPE \over m}

In some situations, the equations can be simplified by assuming a field that is nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant. In that case, the difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height (∆h):

MGPE\approx g\Delta h.

Massic gravitational potential energy is typically measured in Joules per kilogram (J/kg) in the International System of Units. The dimensional formula for massic gravitational potential energy is [L]²[T]^-2, consistent with energy measurements.

Key Concepts

Frame of reference: The choice of reference point is arbitrary but is crucial for determining the potential energy. Commonly, the Earth's surface is chosen as a reference, and heights are measured vertically above it.
Zero Potential Energy at Infinity: As an object moves infinitely far away from the gravitational source, its potential energy approaches zero. This choice simplifies calculations and is a convention in many physics problems.

Mathematical form

The gravitational potential V at a distance x from a point mass of mass M can be defined as the work W that needs to be done by an external agent to bring a unit mass in from infinity to that point:^[3]^[4]^[5]^[6]

V(\mathbf {x} )={\frac {W}{m}}={\frac {1}{m}}\int _{\infty }^{x}\mathbf {F} \cdot d\mathbf {x} ={\frac {1}{m}}\int _{\infty }^{x}{\frac {GmM}{x^{2}}}dx=-{\frac {GM}{x}},

where G is the gravitational constant, and F is the gravitational force. The product GM is the standard gravitational parameter and is often known to higher precision than G or M separately. The potential has units of energy per mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero.

The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is

\mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},

where x is a vector of length x pointing from the point mass toward the small body and ${\hat {\mathbf {x} }}$ is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law:

|\mathbf {a} |={\frac {GM}{x^{2}}}.

The potential associated with a mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x₁, ..., x_n and have masses m₁, ..., m_n, then the potential of the distribution at the point x is

V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{|\mathbf {x} -\mathbf {x} _{i}|}}.

Points x and r, with r contained in the distributed mass (gray) and differential mass dm(r) located at the point r.

If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R³, then the potential is the convolution of $- G / | r |$ with dm. In good cases this equals the integral

V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {r} |}}\,dm(\mathbf {r} ),

where $| x - r |$ is the distance between the points x and r. If there is a function ρ(r) representing the density of the distribution at r, so that $dm (r) = ρ (r) dv (r)$ , where dv(r) is the Euclidean volume element, then the gravitational potential is the volume integral

V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {r} |}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).

If V is a potential function coming from a continuous mass distribution ρ(r), then ρ can be recovered using the Laplace operator, $Δ$ :

\rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).

This holds pointwise whenever ρ is continuous and is zero outside of a bounded set. In general, the mass measure dm can be recovered in the same way if the Laplace operator is taken in the sense of distributions. As a consequence, the gravitational potential satisfies Poisson's equation. See also Green's function for the three-variable Laplace equation and Newtonian potential.

The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones.^[7] These include the sphere, where the three semi axes are equal; the oblate (see reference ellipsoid) and prolate spheroids, where two semi axes are equal; the degenerate ones where one semi axes is infinite (the elliptical and circular cylinder) and the unbounded sheet where two semi axes are infinite. All these shapes are widely used in the applications of the gravitational potential integral (apart from the constant G, with 𝜌 being a constant charge density) to electromagnetism.

References

↑ Serway, Raymond A., and Jewett, John W. Physics for Scientists and Engineers. Cengage Learning, 2013
↑ Halliday, David, Resnick, Robert, and Walker, Jearl. Fundamentals of Physics. John Wiley & Sons, 2013.
↑ Marion, J.B.; Thornton, S.T. (1995). Classical Dynamics of particles and systems (4th ed.). Harcourt Brace & Company. p. 192. ISBN 0-03-097302-3.
↑ Arfken, George B.; Weber, Hans J. (2005). Mathematical Methods For Physicists International Student Edition (6th ed.). Academic Press. p. 72. ISBN 978-0-08-047069-6.
↑ Sang, David; Jones, Graham; Chadha, Gurinder; Woodside, Richard; Stark, Will; Gill, Aidan (2014). Cambridge International AS and A Level Physics Coursebook (illustrated ed.). Cambridge University Press. p. 276. ISBN 978-1-107-69769-0.
↑ Muncaster, Roger (1993). A-level Physics (illustrated ed.). Nelson Thornes. p. 106. ISBN 978-0-7487-1584-8.
↑ MacMillan, W.D. (1958). The Theory of the Potential. Dover Press.

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[1] Serway, Raymond A., and Jewett, John W. Physics for Scientists and Engineers. Cengage Learning, 2013

[2] Halliday, David, Resnick, Robert, and Walker, Jearl. Fundamentals of Physics. John Wiley & Sons, 2013.

[3] Marion, J.B.; Thornton, S.T. (1995). Classical Dynamics of particles and systems (4th ed.). Harcourt Brace & Company. p. 192. ISBN 0-03-097302-3.

[4] Arfken, George B.; Weber, Hans J. (2005). Mathematical Methods For Physicists International Student Edition (6th ed.). Academic Press. p. 72. ISBN 978-0-08-047069-6.

[5] Sang, David; Jones, Graham; Chadha, Gurinder; Woodside, Richard; Stark, Will; Gill, Aidan (2014). Cambridge International AS and A Level Physics Coursebook (illustrated ed.). Cambridge University Press. p. 276. ISBN 978-1-107-69769-0.

[6] Muncaster, Roger (1993). A-level Physics (illustrated ed.). Nelson Thornes. p. 106. ISBN 978-0-7487-1584-8.

[7] MacMillan, W.D. (1958). The Theory of the Potential. Dover Press.

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