Gravitational Potential Energy

In these revision notes for Gravitational Potential Energy, we cover the following key points:

• What is Potential Energy?
• What are conservative forces?
• What is gravitational potential energy and how it relates with potential energy in general?
• What does the law of mechanical energy conservation says?
• How can we calculate the final velocity of a falling object when the initial falling height is given, without using the kinematic formulae?
• What does the negative sign before the potential energy implies?

Gravitational Potential Energy Revision Notes

Potential Energy represents the energy stored in a system when no change of system's structure is considered. This stored energy can be converted into other forms of energy in certain conditions.

Since Potential Energy is a stored energy, it depends of the parameters of the system.

In general, the parameters that affect the value of potential energy of a certain system and all physical quantities involved in their interaction, are:

1. The force F⃗ of interaction between the objects, and
2. The linear distance r⃗ between them

Therefore, the general formula of potential energy is

In other words, "Potential energy PE represents the work done by one of the objects in the system (usually the largest) to bring the other object from the position r⃗ to zero, i.e. to bring it at the place where the first object is."

This means Potential Energy is the energy possessed by an object due to its position relative to other objects.

The negative sign provides the convention that work done against a force field increases the potential energy, while work done by the force field decreases the potential energy.

Gravitational potential energy is the energy of the object due to its position in relation to the Earth.

Since all forces generated by a field (distant forces) are all conservative, i.e. their work does not depend on the path followed, we can conclude that gravitational force is a conservative force.

Since W = - ΔGPE, the change in GPE when an object moves from the ground to a certain height or vice versa is

The sign in the above equation depends on the direction of process. When the object moves up, its GPE increases, so the sign of ΔGPE is positive (and therefore, work is negative). This means we are doing work against gravity. When the object moves down, its GPE decreases, so the sign of ΔGPE is negative (and therefore, work is positive). This means we are doing work in the direction of gravity.

Mechanical Energy (ME) represents the sum of Kinetic and Potential Energy of a system. In the specific case, when this potential energy is produced by the gravitational field of the Earth, we write:

If no external factors such as friction with the ground, air drag etc., are relevant, we obtain the Law of Mechanical Energy Conservation, which says:

"In absence of external factors, the mechanical energy of a system is conserved."

This means that

For two situations (1) and (2) [or initial and final], we can write mathematically the above law as: