Gravitational Potential of a Point Mass

Gravitational potential at a particular location is equal to the work per unit mass required to move an object to that location from a fixed location. In this case, the mass plays the role of the charge, and the gravitational potential is analogous to the electric potential. The location where the potential is zero is at a far distance from the mass, resulting in a negative potential at any finite distance of the mass. 

Mathematically, the gravitational potential of a point mass is also known as the Newtonian potential, which is fundamental in studying potential theory. Besides other uses, it can be used to solve the electrostatic and magnetostatic fields, which are generated by uniformly charged bodies. In other words, the gravitational potential, meaning at a point mass, is the work done in bringing the unit mass from infinity to the point being considered without acceleration. 

It can be formulated as follows:

VG = W/m 

Where, V = A vector quantity is the one that has the magnitude and direction distinctively defined

G = gravitational potential

W = Weight

M = Mass

The gravitation potential of a point mass is a vector quantity. A vector quantity is the one that has the magnitude and direction distinctively defined—denoted with V, the SI unit of gravitational potential J kg-1. Also, the c.g.s unit is erg g-1. 

Its dimensional formula is 

[V] = [W]/[m] = [M1L2T-2]/[M1] 

[M1L2T-2]/[M1] = [M0L2T-2]

Its dimensions are = [M0L2T-2]

Expression and Derivation of the Gravitational Potential

Let R be the radius of the earth, M be the mass of the earth, and P be the point at the distance r. r is greater than the Radius of the object (R). Let the centre of the earth be O, and the distance from it be y. A is the point on y. Assume that a unit mass is at A. This is how we find the mass acting on it –

F = GMm/x2 = GM(1)/x2 = GM/x2

No acceleration will be taken into consideration in the further steps. The mass M is moved from points A to B through a small distance = dx. Note that no acceleration has taken place. The work done is shown as

dW = F.dx = GM/x2.dx 

Integration is to be used in the expression to obtain the total work. 

The result obtained looks like this:

W = -GM[1/R – 0]

Therefore, W = -GM/R

This work obtained is the gravitational potential at the point given by the expression. 

V = -GM/r

G in this equation is the gravitational constant, and M stands for the mass. Gravitational potential is universal and stays the same irrespective of the changes the body or the medium may undergo. 

Features of the Gravitational Potential of a Point Mass

• The gravitational potential of a point mass is maximum at infinity. 

• As the gravitational field intensity is zero at infinity, we can say that it goes on decreasing as we move the test mass closer to the attracting body. A conclusion can thus be drawn that the gravitational potential of a point mass is a negative quantity. That is because it makes the body move from a higher to a lower potential. 

• If it is equally spread on a surface, or we can say that the value of the gravitational potential is equal and same for all the points on the surface, the surface is called an equipotential surface. 

• We can keep the example of electric potential in mind while studying the gravitational potential of a point mass. This is because, like an electric potential moving from a higher to a lower potential, the gravitational potential moves from a higher potential to a lower potential. 

• It is a scalar quantity with the dimensional formula [M0L2T-2].

Exceptional Cases

• If we consider the mass of uniform solid spheres at the centre of their uniform sphere along with other uniform spheres is at the centre potential at the outside points of those spheres can be deduced. 

• It is witnessed that the potential inside the solid sphere at all points remains constant. The potential inside is the same as the potential on the sphere’s surface. If we assume the mass of that sphere to be M and the radius to be r, then the potential inside the sphere and at any point is equal to

V = -GM/r

The gravitational potential is often used to factor out the mass. Thus, we can calculate the function that depends just on the sources. We can then apply it to any test mass. Thus, another equation for the gravitational potential of a mass can be formulated as follows:

 Φ(r)=−rGM

Gravity is a superposable force. We can say that the gravitational force is exerted on some test mass by a collection of test masses. The force is the cumulation of the forces exerted by all of them in isolation and individually. 

Conclusion 

Gravitational potential energy is the energy of an object with a mass and in relation to the other object with a nonzero mass. Besides many other important things, it is used in power clocks. In the power clocks, the falling weights run the mechanism. It is also used by the elevators, lifts, etc., with the mechanism of counterweights. It can be used to calculate the energy stored in water towers.