Learning Outcomes

Upon completing this lesson, you will be able to:

• Define mechanical energy and state the conditions under which it is conserved.
• State the universal Law of Conservation of Energy.
• Differentiate between conservative and non-conservative (dissipative) forces.
• Apply the principle of energy conservation to solve problems involving dissipative forces like friction.

Topics Covered

• Mechanical Energy and Its Conservation
• The Law of Conservation of Energy
• Energy Conservation with Dissipative Forces

I. Mechanical Energy and Its Conservation

In the previous lesson, we introduced kinetic energy $(KE)$ and potential energy $(PE)$. The sum of these two is called the total mechanical energy $(E)$ of a system.

$$ E = KE + PE $$

The principle of Conservation of Mechanical Energy is a powerful rule that applies under specific conditions. To understand it, we must first define conservative forces.

• Conservative Force: A force is conservative if the work it does on an object moving between two points is independent of the path taken. The work done depends only on the initial and final positions. Gravity and the elastic force of a spring are the most common examples.

The Principle of Conservation of Mechanical Energy: If the work done on an object in a system is done only by conservative forces, then the total mechanical energy of the system remains constant.

In other words, energy can be converted between kinetic and potential forms, but the total amount $(E)$ does not change.

$$ E_{\text{initial}} = E_{\text{final}}\\KE_i + PE_i = KE_f + PE_f $$

II. The Law of Conservation of Energy

While the conservation of mechanical energy is a conditional rule, it is a specific case of one of the most fundamental and universal laws of nature: The Law of Conservation of Energy.

This law states: Energy cannot be created or destroyed; it can only be transformed from one form into another.

The total energy of an isolated system is always constant. This principle applies to all forms of energy, not just mechanical.

• Forms of Energy: Mechanical, Thermal (Heat), Chemical, Electrical, Radiant (Light), Nuclear, etc.

When a ball falls and hits the ground, its mechanical energy seems to disappear. However, it has been converted into other forms: thermal energy (the ball and ground get slightly warmer) and sound energy. The total energy is still conserved.

III. Energy Conservation with Dissipative Forces

What happens when non-conservative forces are present?

• Non-Conservative Force (or Dissipative Force): A force is non-conservative if the work it does depends on the path taken. These forces "dissipate" or remove mechanical energy from a system, usually by converting it into heat. Friction and air resistance are the primary examples.

When dissipative forces are involved, the total mechanical energy is not conserved; it decreases. However, the universal Law of Conservation of Energy still holds. The "lost" mechanical energy has simply been transformed into thermal energy.

We can modify our conservation equation to account for the work done by non-conservative forces $(W_{nc})$.

$$ E_i + W_{nc} = E_f\\KE_i + PE_i + W_{nc} = KE_f + PE_f $$