A derivation of Poisson’s equation for gravitational potential Dr. Christian Salas November 3, 2009 1 Introduction A distribution of matter of density ˆ= ˆ(x;y;z) gives rise to a gravitational potential ˚which satis es Poisson’s equation r2˚= 4ˇGˆ at points inside the distribution, where the Laplacian operator r2 … U = mgh applies only for a uniform field, so it does not apply here where the field goes as 1/r 2. 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. Gravitational potential energy (GPE) is an important physical concept that describes the energy something possesses due to its position in a gravitational field. This total initial energy is then compared with the sum of the potential and kinetic energies at an infinite separation, in order to determine the escape velocity … The second term is a correction factor. I was shown this derivation for the gravitational potential energy, and I'm not very happy about it assuming that $\frac{1}{\infty} = 0$. For ordinary heights, this term is essentially one. Let's confirm this using a really high height — the top of the spire on the Burj Khalifa in the United Arab Emirates (818 m). Definition: The True Equation for Gravitational Potential Energy = − This equation is universal, regardless of how long or short the distance is. https://www.physicseasytips.com/gravitational-potential-energy Gravitational fields are conservative, which means that the energy used to move between two points within the field is independent of the path taken. The difference in gravitational potential energy of an object (in the Earth-object system) between two rungs of a ladder will be the same for the first two rungs as for the last two rungs. Thank you Maheshwar Gundelli for A2A Referring to the figure below Source: SciPhy – Physics and More Let us have a solid sphere of radius $a$ of mass $M$ having constant density. We know that the magnitude of the gravitational force is given by: F = -GmM/r 2. Use the connection between force and potential energy to determine the general form of gravitational potential energy. Is there a better derivation, either using a completely different method, or one similar that avoids $\frac{1}{\infty}$? Gravitational Potential Energy. The GPE formula GPE = mgh shows that it depends on the mass of the object, the acceleration due to … However, it is still often more convenient to use the simpler equation, as it requires neither the distance from the object to the core of the planet involved nor the mass of the planet involved. The kinetic energy of a particle with mass m and velocity v is defined as KE = mv 2 /2. Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. The derivation starts with the initial gravitational potential energy at the given altitude and the initial kinetic energy of the object. Converting Between Potential Energy and Kinetic Energy. Thus, the total energy of the system is the sum of the kinetic energy and gravitational potential energy: • The gravitational potential energy due to a single object looks like: • If a particle begins a distance R from the object, the initial speed needed for escape is: 0 • If a particle has total energy > 0, the kinetic energy is positive for all values of r – Such a particle can “escape” from the gravitational … The first part of this expression is our old friend, the original equation for gravitational potential energy. Gravitational Potential Energy.