Unit – III C |

## The concept of Gravitational Potential:

- The gravitational potential at a point is defined as the work done in bringing the unit mass from infinity to that point without acceleration.

- Gravitational potential is a scalar quantity. It is denoted by V. Its S.I. unit is J kg
^{-1}. Its c.g.s. unit is erg g^{-1}.

- Dimensions of gravitational potential are [M
^{0}L^{2}T^{-2}].

### Expression for Gravitational Potential:

- Let M = Mass of the Earth and R = Radius of the Erath

- Let P be the point at a distance of ‘r’ (r > R) at which gravitational potential is to be calculated. Let A be the point at a distance of ‘x’ from the centre of the Earth O. Let us consider a unit mass at A. The Force acting on the unit mass is given by

- Let the mass be moved from A to B without acceleration through very small distance ‘dx’. The work done is given by

- Total work done can be obtained as

- This work is gravitational potential at that point. Thus the gravitational potential at a point at a distance of r from the centre of the earth is given by

Where G is universal gravitation constant and M is the mass of the earth.

### Characteristics of Gravitational Potential:

- As gravitational field intensity is zero at infinity and it goes on decreasing as the test mass approach the attracting body. Thus the body moves from higher potential to lower potential, hence gravitational potential is a negative quantity.
- It is maximum at infinity (zero).
- Gravitational potential is a scalar quantity.
- Its dimensions are [M
^{0}L^{2}T^{-2}]. - Its S.I. unit is J kg
^{-1}. Its c.g.s. unit is erg g^{-1}. - If the gravitational potential at all the points on the surface is the same, then the surface is called as the equipotential surface.
- Gravitational potential can be studied in analogy with an electrical potential. As the current moves from higher electrical potential to the lower electrical potential, similarly mass moves from higher gravitational potential to the lower gravitational potential.

### Gravitational Potential due to the Earth at Different Points:

- At infinity gravitational potential is negligible. It increases as we approach the earth. It further increases inside the earth. It is maximu at the centre of the earth.
- This distribution is the same for a solid sphere.

### Gravitational Potential due to a Spherical Shell (Hollow Sphere) at Different Points:

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