Gravitational potential is a fundamental concept in the study of gravity, offering a unique perspective on how mass influences the space around it. In physics, gravitational potential is a scalar field that represents the amount of gravitational potential energy per unit mass at a specific point in space. The gravitational potential at a point in space is a measure of the work that gravity would do in bringing a mass from an infinite distance to that point, per unit mass. As one moves closer to a massive object, the gravitational potential increases, signifying a higher potential energy for an object placed in that field.
Gravitational Potential
Gravitational Potential due to a point mass
Definition: Gravitational potential at a point in space is the work done by an external force in bringing a unit mass from infinity to that point, without changing its kinetic energy.
Formula:
The gravitational potential (V) at a distance r from a mass M can be expressed using the following formula;
\(\displaystyle V=-\frac{{GM}}{r}\)
where:
- V is the gravitational potential,
- G is the gravitational constant (6.67430×10−11 m3 kg−1 s−2),
- M is the mass creating the gravitational field,
- r is the distance from the center of the mass to the point where the potential is being measured.
The negative sign in the formula indicates that gravitational potential is a decrease in energy as you move away from the mass. It is a scalar quantity and is measured in joules per kilogram (J/kg).
Consider a point in space at a distance r from a massive object. The formula \(\displaystyle V=-\frac{{GM}}{r}\) tells us how much potential energy a unit mass would have at that point. As r increases, potential energy decreases, and as r decreases, potential energy increases. Gravitational potential is crucial in understanding orbits, celestial mechanics, and how gravitational forces influence the motion of objects in space.
Gravitational Potential due to a shell
The gravitational potential due to a spherical shell of matter behaves differently depending on whether you are outside the shell, on its surface, or inside the shell. Let’s explore each case:
Outside the Shell: Let’s consider the gravitational potential due to a thin, spherical shell of mass M at a distance r outside the shell. The gravitational potential (V) at this point is given by:
\(\displaystyle V=-\frac{{GM}}{r}(r>R)\)
where (G) is the gravitational constant, (M) is the total mass of the shell, and (r) is the distance from the center of the shell to the point outside it.
The potential behaves as if all the mass of the shell were concentrated at its center, and it decreases with distance (r) from the center.
On the surface of the Shell: Consider a point exactly on the surface of the shell. This point is at a distance R from the center of the shell. The formula for gravitational potential (V) at a point on the surface of the shell is given by
\(\displaystyle V=-\frac{{GM}}{R}\)
The negative sign indicates that the gravitational potential is negative, and it signifies that the potential energy of a unit mass at this point is lower compared to an infinite distance away. It’s a convention to consider potential energy to be zero at an infinite distance. On the surface of the spherical shell, the gravitational potential is constant, and it is the same as the potential at any other point on the surface. This is a unique property of a spherically symmetric mass distribution, and it is a consequence of the cancellation of gravitational forces from all parts of the shell at its surface. The potential is constant on the surface and is the same as the potential at any point outside the shell.
Inside the Shell: Inside a spherical shell, the gravitational potential is constant throughout the interior of the shell, regardless of the distance from the center of the shell. This unique property arises from the spherical symmetry of the mass distribution.
\(\displaystyle V=-\frac{{GM}}{R}\)
Interestingly, the potential is constant and independent of the distance from the center to any point within the shell. Inside a spherically symmetric distribution of mass (like a shell), the gravitational field is zero. This means that the gravitational potential does not change as you move within the shell.
The constancy of gravitational potential inside the shell is a consequence of the cancellation of gravitational forces from all parts of the shell, resulting in a net gravitational field of zero. This is one of the unique features of gravitational fields created by spherical shells. It’s important to note that these results are specific to thin, spherical shells with uniform mass distribution.
Gravitational Potential due to a uniform solid sphere
The gravitational potential (V) due to a uniform solid sphere can be derived and expressed in terms of the radial distance (r) from the center of the sphere. For a uniform solid sphere, the mass distribution is symmetrical, and the potential varies both outside and inside the sphere.
Gravitational Potential at an external point:
Gravitational Potential due to a uniform solid sphere at an external point is the same as that due to a single particle of the same mass placed at its center. Thus
\(\displaystyle V=-\frac{{GM}}{r}(r\ge R)\)
Gravitational Potential at the surface: Here, r = R, and therefore \(\displaystyle V=-\frac{{GM}}{R}\)
Gravitational Potential at an internal point
For a uniform solid sphere, the gravitational potential (V) at an internal point within the sphere (at a distance r from the center where r is less than or equal to the radius R of the sphere) is given by:
\(\displaystyle V=-\frac{{GM}}{{{{R}^{3}}}}\left( {\frac{{{{R}^{2}}}}{2}-\frac{{{{r}^{2}}}}{6}} \right)\)
This equation describes the potential energy of a unit mass at a point inside a sphere of mass M and radius R, due to the gravitational attraction of the sphere. The point is at a distance r from the center of the sphere.
Gravitational Potential due to a uniform ring at a point on its axis
Consider a uniform ring with mass M and radius R. The gravitational potential at a point on its axis, situated at a distance z from the center of the ring, becomes a focal point of investigation.
This formula can be derived by considering the ring as composed of many small elements of mass dM, and adding up the contributions of each element to the potential at the point of interest.
\(\displaystyle V(r)=-\frac{{GM}}{{\sqrt{{{{R}^{2}}+{{r}^{2}}}}}};0\le r\le \infty \)
The potential due to each element is given by: \(\displaystyle dV=-\frac{{GdM}}{{\sqrt{{{{R}^{2}}+{{r}^{2}}}}}}\)
Where the distance from the element to the point is \(\displaystyle \sqrt{{{{R}^{2}}+{{r}^{2}}}}\) by the Pythagorean theorem. The total potential is obtained by integrating the mass of the ring:
\(\displaystyle V=\int{{dV=-\frac{G}{{\sqrt{{{{R}^{2}}+{{r}^{2}}}}}}\int{{dM}}}}=-\frac{{GM}}{{\sqrt{{{{R}^{2}}+{{r}^{2}}}}}}\)
At r = 0, \(\displaystyle V=-\frac{{GM}}{R}\) i.e at the centre of the ring gravitational potential as \(\displaystyle V=-\frac{{GM}}{R}\).