Understanding Gravitational Potential: Definition, Expression, and Proof

Gravitational potential is a fundamental concept in physics that helps us understand the interaction between masses in a gravitational field. It is defined as the work done per unit mass in bringing a small test mass from infinity to a specific point against the gravitational force. This article delves into the definition, expression, and proof of gravitational potential, providing a comprehensive understanding of this crucial physical quantity.

Definition of Gravitational Potential

Gravitational potential at a point in a gravitational field is fundamentally defined as the work done per unit mass to bring a small test mass from infinity to that point. It is a scalar quantity denoted by V.

Expression for Gravitational Potential

The expression for gravitational potential, V, at a distance r from a point mass M, is given by the formula:

V -frac{GM}{r}

where:

G is the universal gravitational constant, approximately 6.674 times 10^{-11} text{N m}^2/text{kg}^2. M is the mass creating the gravitational field. r is the distance from the mass to the point where the potential is being measured.

Proof of the Expression

The proof of the expression for gravitational potential involves the concept of work done against gravitational force. The gravitational force F acting on a mass m at a distance r from a mass M is given by Newton's law of gravitation:

F frac{GMm}{r^2}

To bring a mass m from a distance r_1 to r_2 (where r_2 > r_1), the work done W against the gravitational force is calculated as the integral of the force over distance:

W int_{r_1}^{r_2} F , dr int_{r_1}^{r_2} frac{GMm}{r^2} , dr

Integrating the Force

Evaluating the integral:

W GMm int_{r_1}^{r_2} frac{1}{r^2} , dr

The integral of frac{1}{r^2} is -frac{1}{r}, thus:

W GMm left[-frac{1}{r}right]_{r_1}^{r_2} GMm left(-frac{1}{r_2} frac{1}{r_1}right)

Simplifying gives:

W GMm left(frac{1}{r_1} - frac{1}{r_2}right)

Gravitational Potential

The gravitational potential V at a point r is defined as the work done per unit mass:

V frac{W}{m} frac{GM}{r} - frac{GM}{r_1}

As r_1 approaches infinity, V approaches zero:

V_r -frac{GM}{r}

This expression highlights that the gravitational potential is negative, indicating that work must be done against the field to move a mass from a point in the field to infinity.