If the mass of the earth is Me, show that the gravitational...

If the mass of the earth is $M_{e},$ show that the gravitational potential energy of a body of mass $m$ located a distance $r$ from the center of the earth is $V_{g}=-G M_{e} m / r$ Recall that the gravitational force acting between the earth and the body is $F=G\left(M_{e} m / r^{2}\right),$ Eq. $13-1 .$ For the calculation, locate the datum at $r \rightarrow \infty$. Also, prove that $F$ is a conservative force.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. It is defined relative to a chosen reference point, usually where the potential energy is zero. The energy is obtained by integrating the gravitational force over the displacement from the reference point to the object's position, leading to an expression that is inversely proportional to the distance from the source of the gravitational field.

Gravitational Force

Gravitational force is the attractive force that exists between any two bodies that have mass. According to Newton's law of universal gravitation, this force is directly proportional to the product of the masses involved and inversely proportional to the square of the distance between their centers. It is the underlying force responsible for the characteristic inverse-square behavior seen in gravitational interactions.

Conservative Force

A conservative force is one for which the work done in moving an object between two points is independent of the path taken. This property implies that a potential energy function can be defined for the force. In the context of gravity, because the work done depends only on the initial and final positions in the gravitational field, the gravitational force is classified as conservative.

Integration of Force to Determine Potential Energy

Determining the potential energy from a force field involves integrating the force with respect to displacement. For gravitational fields, integrating the gravitational force from the reference point (typically at infinity where the potential energy is assumed zero) to a point at a finite distance yields the gravitational potential energy. This process highlights the mathematical connection between force and potential energy in conservative fields.

Reference Datum in Potential Energy Calculations

The choice of a reference datum is crucial in potential energy calculations because potential energy is defined up to an arbitrary constant. In gravitational field problems, infinity is often selected as the reference point where the potential energy is set to zero. This standard convention simplifies the resulting expression and ensures consistency across different problems in gravitational physics.

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