Conservation of Energy in Systems
Explore energy transformations in complex systems, analyse efficiency using Sankey diagrams, and understand how the law of conservation of energy governs all physical processes.
Pax says: "Energy can never be created or destroyed -- only transformed. Let's trace where the energy goes in real systems and learn why nothing is perfectly efficient!"
Energy Transformations in Systems
The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another or transferred between objects. In any closed system, the total energy remains constant. When we observe energy "losses," the energy has simply been converted into forms that are less useful for the intended purpose -- most commonly thermal energy (heat) and sound.
Energy Transformation Chain: Coal Power Station
Chemical Energy
Stored in coal (100%)
Thermal Energy
Combustion heats water to steam
Kinetic Energy
Steam spins turbine blades
Electrical Energy
Generator produces electricity (~35%)
Key Equations
Conservation of energy: Etotal = Ekinetic + Epotential + Ethermal + Eother = constant
Work-energy theorem: Wnet = ΔEk
Power: P = W / t = E / t (measured in watts, W)
Important: While energy is always conserved, useful energy decreases with each transformation. The second law of thermodynamics tells us that energy tends to spread out and become less concentrated, reducing its ability to do work.
Efficiency and Sankey Diagrams
Efficiency measures how well a system converts input energy into useful output energy. No real system is 100% efficient because some energy is always dissipated as heat, sound, or other non-useful forms. Sankey diagrams are visual tools that show how energy flows through a system, with arrow widths proportional to the amount of energy.
Efficiency Formula
Efficiency (%) = (Useful energy output / Total energy input) × 100
Sankey Diagram: Incandescent Light Bulb
Input
100 J electrical
Light: 5 J (5%)
Heat: 95 J (95%)
Efficiency = 5/100 × 100 = 5% -- most energy is wasted as heat
Typical Efficiencies
- LED bulb: ~40%
- Electric motor: ~85-95%
- Solar panel: ~15-22%
- Coal power station: ~33-40%
- Internal combustion engine: ~25-30%
Reading Sankey Diagrams
- Arrow widths are proportional to energy
- Input arrow enters from the left
- Useful output continues right
- Wasted energy branches off (usually downward)
- Total output = total input (conservation)
Energy in Mechanical Systems
In mechanical systems, energy continuously transforms between kinetic energy (Ek = ½mv2) and gravitational potential energy (Ep = mgh). In an ideal (frictionless) system, the total mechanical energy remains constant. In real systems, friction and air resistance convert some mechanical energy into thermal energy.
Roller Coaster Energy Profile
Top
Mid
Bottom
Potential Energy Kinetic Energy
Note: bars are shorter at bottom due to friction losses (thermal energy)
Conservation with Friction
When friction is present: Ek,initial + Ep,initial = Ek,final + Ep,final + Ethermal. The total energy is still conserved, but some has been converted to thermal energy and is no longer available as useful mechanical energy.
Key Vocabulary
Conservation of Energy
The fundamental principle that energy cannot be created or destroyed, only transformed between different forms or transferred between objects in a system.
Efficiency
The ratio of useful energy output to total energy input, expressed as a percentage. No real system achieves 100% efficiency.
Sankey Diagram
A flow diagram where arrow widths are proportional to energy quantities, visually showing how input energy is distributed between useful output and waste.
Dissipation
The spreading out of energy into the surroundings as thermal energy, making it less useful for doing work. This is an irreversible process.
Worked Examples
A motor uses 2000 J of electrical energy to lift a 15 kg mass through 10 m. Calculate the efficiency of the motor. (g = 9.8 m s-2)
Step 1: Useful energy output = mgh = 15 × 9.8 × 10 = 1470 J
Step 2: Efficiency = (useful output / total input) × 100 = (1470 / 2000) × 100
Answer: Efficiency = 73.5%. The remaining 530 J (26.5%) was dissipated as heat and sound.
A ball of mass 0.5 kg is dropped from a height of 8 m. It bounces back to a height of 5 m. How much energy was lost to thermal energy?
Step 1: Initial Ep = mgh1 = 0.5 × 9.8 × 8 = 39.2 J
Step 2: Final Ep = mgh2 = 0.5 × 9.8 × 5 = 24.5 J
Step 3: Energy lost = 39.2 - 24.5 = 14.7 J dissipated as thermal energy and sound during the bounce.
A coal power station burns coal releasing 1000 MJ of chemical energy. The station is 35% efficient. How much electrical energy is generated, and how much is wasted?
Step 1: Useful electrical energy = 0.35 × 1000 = 350 MJ
Step 2: Wasted energy = 1000 - 350 = 650 MJ
Answer: 350 MJ of electrical energy is produced and 650 MJ is dissipated as thermal energy (mainly heating cooling water and exhaust gases).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
According to the law of conservation of energy, when a ball rolls down a hill and comes to rest at the bottom due to friction:
Question 2
In a Sankey diagram, the width of an arrow represents:
Question 3
A machine uses 500 J of input energy and produces 125 J of useful output. Its efficiency is:
Question 4
A pendulum swings back and forth but gradually comes to rest. The energy has been:
Question 5
An LED bulb and an incandescent bulb both produce the same amount of light. Compared to the incandescent bulb, the LED bulb:
Key Concepts Summary
- ●Energy is always conserved -- it cannot be created or destroyed, only transformed between forms or transferred between objects.
- ●Efficiency = (useful output / total input) × 100%. No real system achieves 100% efficiency due to energy dissipation.
- ●Sankey diagrams visually represent energy flow through systems, with arrow widths proportional to energy quantities.
- ●In mechanical systems, energy transforms between kinetic and potential forms, with friction converting some to thermal energy.
- ●Energy dissipation is irreversible -- thermal energy spreads into the environment and cannot be fully recovered for useful work.