The Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. This is one of the most fundamental principles in physics, governing everything from simple mechanical systems to complex biological processes.
In any isolated system, the total amount of energy remains constant over time. While energy may change forms—from potential to kinetic, from chemical to thermal—the total sum never changes. This principle helps us analyze and predict the behavior of physical systems.
Historical Note
The conservation of energy principle was developed in the mid-19th century by scientists including James Joule, Hermann von Helmholtz, and Julius Robert Mayer. It unified previously separate concepts of heat and mechanical work.
Forms of Energy
Energy exists in many different forms, each with its own characteristics and formulas. Understanding these forms is essential for applying conservation principles.
Kinetic Energy
Kinetic energy is the energy of motion. Any object that is moving possesses kinetic energy proportional to its mass and the square of its velocity.
Notice that kinetic energy depends on velocity squared. This means doubling your speed quadruples your kinetic energy—which is why car crashes at high speeds are so much more dangerous than at low speeds.
Gravitational Potential Energy
Potential energy is stored energy based on position or configuration. Gravitational potential energy depends on an object's height above a reference point.
Other Forms of Energy
- Elastic potential energy: Stored in stretched or compressed springs
- Thermal energy: The kinetic energy of molecular motion (heat)
- Chemical energy: Stored in chemical bonds (fuels, batteries, food)
- Electrical energy: Energy carried by moving charges
- Nuclear energy: Energy stored in atomic nuclei
Energy Transformations
Energy constantly transforms from one form to another. Recognizing these transformations is key to applying conservation principles.
A Falling Ball
When you drop a ball from height h:
- At the top: PE = mgh, KE = 0
- While falling: PE decreases, KE increases
- Just before hitting ground: PE ≈ 0, KE = mgh
Total mechanical energy (PE + KE) remains constant: mgh = ½mv²
A Pendulum Swing
A swinging pendulum continuously converts energy:
- At highest points: Maximum PE, minimum KE (momentarily at rest)
- At lowest point: Minimum PE, maximum KE (fastest speed)
In an ideal frictionless pendulum, this conversion continues forever.
Applying Conservation of Energy
The conservation principle is powerful because it lets us analyze motion without knowing every detail of the path. We only need to compare energy at two points.
Problem-Solving Strategy
- Define your system: What objects and energy types are involved?
- Choose reference points: Where is PE = 0? What are the initial and final states?
- Identify all energy forms: List KE, PE, and other relevant energies at each point
- Apply conservation: Total energy at start = Total energy at end
- Solve for unknowns: Use the equation to find velocity, height, etc.
Example: Roller Coaster Problem
A roller coaster starts at height 20m. What is its speed at height 5m? (Ignore friction)
Solution:
Initial: PE₁ = mgh₁ = m(9.8)(20), KE₁ = 0
Final: PE₂ = mgh₂ = m(9.8)(5), KE₂ = ½mv²
Conservation: mgh₁ = mgh₂ + ½mv²
Solving: v² = 2g(h₁ - h₂) = 2(9.8)(15) = 294
v = 17.1 m/s
Pro Tip
Notice that mass cancels out in many energy problems. This means the speed of a falling object doesn't depend on its mass—a key insight Galileo discovered centuries ago.
Energy and Real-World Systems
In real systems, mechanical energy isn't perfectly conserved because some energy transforms into thermal energy (heat) through friction, air resistance, and other dissipative forces.
The Role of Friction
Friction converts mechanical energy into thermal energy. While total energy is still conserved, useful mechanical energy decreases:
This is why perpetual motion machines are impossible—friction always converts some mechanical energy to heat, requiring continuous energy input to maintain motion.
Efficiency
Efficiency measures how well a system converts input energy to useful output energy:
Energy in Everyday Life
Understanding energy conservation helps explain everyday phenomena:
- Braking a car: Kinetic energy transforms to thermal energy in brake pads
- Eating food: Chemical energy transforms to kinetic energy (movement) and thermal energy (body heat)
- Charging a phone: Electrical energy transforms to chemical energy stored in the battery
- Solar panels: Light energy transforms to electrical energy
Common Misconception
"Energy is used up" is technically incorrect. Energy transforms to less useful forms (often heat), but it's never destroyed. What we "use" is the availability of energy to do work.
Connection to Other Physics Concepts
Conservation of energy connects to many other physics principles:
- Work-energy theorem: Work done equals change in kinetic energy
- Momentum: Also conserved, but independent of energy conservation
- Thermodynamics: Energy conservation is the First Law of Thermodynamics
- Mass-energy equivalence: Einstein showed E = mc², connecting mass and energy