Show that the gravitational field is conservative. Disregard the curvature of the earth and assume that \(g\) is a constant. What if one takes into account the curvature of the earth and the fact that \(g\). decreases with altitude? Is the gravitational field still conservative?

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. Think of it as the energy stored in an object ready to be converted into motion. For example, a book on a shelf has GPE relative to the floor, and this energy can be released if the book falls.

The formula to calculate GPE is \( GPE = m \times g \times h \) where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point. When GPE changes, it tells us that work is done by or against the force of gravity. If you lift the book higher, you increase its GPE, doing work against gravity. If you let the book fall, gravity does work on the book, and the GPE decreases, typically converting into kinetic energy or motion.

Work is a concept that tells us how much force has acted over a distance. In the case of gravity, the work done by gravity is the force of gravity on an object times the vertical distance the object moves. If an object doesn't move vertically, even if gravity is acting on it, gravity does no work. An easy formula to remember for the work done by gravity is \( W = m \times g \times (h_A - h_B) \) where \( h_A \) and \( h_B \) are the heights at the start and the end of the motion, respectively.

This calculation is simplified by assuming \( g \) is constant, which is nearly accurate for small changes in altitude on Earth. However, for large changes in altitude or on a cosmic scale, \( g \) does vary. Despite this variability, gravity still does the same amount of work as long as the starting and ending points remain the same.

Gravitational acceleration, usually denoted as \( g \), is the acceleration of an object due to the force of gravity alone. Near the surface of the Earth, \( g \) is approximately \( 9.8 \, m/s^2 \). This value can be thought of as the speed an object gains each second when falling freely under the influence of Earth's gravity.

Variation with Altitude

While we often approximate \( g \) as constant near Earth's surface, it actually decreases with altitude because the gravitational force weakens with distance from the center of the Earth. This is why astronauts experience weightlessness in orbit; they are in a state of continuous free fall around the Earth, with \( g \) effectively being zero relative to their spacecraft.

A conservative force is a type of force with a very special property: when it does work on an object and moves it from one point to another, the total work done is independent of the path taken. Gravity is a classic example of a conservative force; no matter how you move an object in a gravitational field, the work done by gravity depends only on the vertical displacement.

This means that if you were to lift a ball straight up or carry it up a winding staircase, the work done by gravity would be the same, as long as the start and end points are at the same height. This concept is vital because it allows us to use potential energy as a simple way to describe the workings of gravity. In a gravitational field, potential energy can transform into kinetic energy and vice versa without any loss of total energy, which is a hallmark of a conservative field.