Introduction

Gravitational potential energy plays a crucial role in understanding the behavior of objects under the influence of gravity. The equation for gravitational potential energy, U -frac{GMm}{R}, provides a mathematical representation of this energy. In this article, we will delve into the derivation of this equation from the perspective of work done against the gravitational force.

Understanding Gravitational Force

The gravitational force F between two masses M and m, separated by a distance R, is described by Newton's law of gravitation:

F frac{GMm}{R^2}

Here,

(G) is the gravitational constant, (M) is the mass of the larger body, (m) is the mass of the smaller body, and (R) is the distance between the centers of the two masses.

Work Done Against Gravitational Force

To derive the gravitational potential energy, we need to calculate the work done to move an object (mass (m)) from a reference point (typically at infinity) to a distance (R) from the larger mass ((M)). The work done W in moving the object is given by the integral of the gravitational force:

W -int_{infty}^{R} F dr

The negative sign indicates that work is done against the gravitational force, which is attractive.

Substituting the Gravitational Force

Substituting the expression for F into the work integral, we get:

W -int_{infty}^{R} frac{GMm}{r^2} dr

Evaluating the Integral

Now, we can evaluate the integral:

W -GMm int_{infty}^{R} frac{1}{r^2} dr

The integral (int frac{1}{r^2} dr) evaluates to (-frac{1}{r}). Thus, we get:

W -GMm left[-frac{1}{r}right]_{infty}^{R}

Calculating this gives:

W -GMm left(-frac{1}{R} - 0right) -frac{GMm}{R}

Conclusion

Therefore, the gravitational potential energy U at a distance (R) from the mass (M) is given by:

U -frac{GMm}{R}

This equation reflects the fact that work must be done to separate two masses against their attractive gravitational force. The potential energy approaches zero as the distance approaches infinity, which is a common reference point in gravitational problems.