Mechanics: Conservation of Energy

Key Concepts

The law of conservation of energy states that the total energy of an isolated system remains constant—it is said to be conserved over time.
Energy can be neither created nor destroyed, but it can change from one form to another.
Mechanical energy consists of kinetic and potential energy. In a closed system, the total energy (mechanical plus non-mechanical) remains constant.
Kinetic energy is the energy an object possesses due to its motion. It is calculated as 1/2mv², where m is mass and v is the velocity.
Potential energy is the stored energy an object has due to its position or state. Gravitational potential energy is given by mgh, where m is mass, g is acceleration due to gravity, and h is height above the ground.
Work done is the amount of energy transferred by a force. It is given by the product of the force (F) and the distance (d) moved in the direction of the force (cosθ) - W = Fdcosθ.
Energy can also be stored or transferred via vibrations (like sound waves) - this energy is called vibration energy or sound energy.

Roller coasters are an excellent example of the principle of conservation of energy in action. The coaster car gains potential energy as it climbs to the top of a hill. As the car races down the hill, the potential energy gets converted back into kinetic energy.
The principle is also evident in pendulums, where the kinetic energy transforms into potential and back, causing the pendulum to swing back and forth, before friction (energy dissipation as heat) eventually brings it to a stop.

The principle of the conservation of energy is one of the most fundamental principles in nature and is a cornerstone of physics.
This principle is applicable in a wide range of physical phenomena and is a powerful tool for solving problems in mechanics.
You need to understand the concepts of potential energy, kinetic energy and work done to apply the law of conservation of energy effectively.
Remember that energy is always conserved in some form—it might be converted into sound, thermal energy, light, or other forms, but the total amount of energy in a closed system remains constant.