Efficiency

• The Law of Conservation of Energy states that energy cannot be created or destroyed, only transformed from one form to another.

• In any energy transfer, the total input energy must equal the sum of the useful output energy and the wasted energy: $E_{input} = E_{useful} + E_{wasted}$

• Dissipation is the process where energy spreads out into the surroundings, usually as thermal energy (heat), making it less useful for doing work.

• Because some energy is always dissipated due to factors like friction or electrical resistance, the useful output is always less than the total input in real systems.

Calculating Efficiency

• To calculate efficiency using energy values, use the formula:

$\eta = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} \times 100\%$

• To calculate efficiency using power values, use the formula:

$\eta = \frac{\text{Useful Power Output}}{\text{Total Power Input}} \times 100\%$

Step-by-Step Analysis

1. Identify the Input: Determine the total energy or power supplied to the device (e.g., electrical energy from a socket).
2. Identify the Useful Output: Isolate the energy form that performs the intended task (e.g., the work done lifting a load).
3. Calculate Wasted Energy: If not given, find it by subtracting useful output from total input.
4. Apply the Formula: Divide the useful value by the total value and convert to a percentage.

• The 100% Rule: Always check your final answer; if the efficiency is greater than 100%, you have likely swapped the input and output values in your calculation.

• Unit Consistency: Ensure both input and output are in the same units (e.g., both in Joules or both in kiloJoules) before dividing.

• Decimal vs. Percentage: Read the question carefully to see if it asks for a ratio (0.45) or a percentage (45%).

• Sankey Diagram Interpretation: The width of the arrows in a Sankey diagram is proportional to the amount of energy; use a ruler to compare widths if numerical values are missing.

• Energy 'Disappearance': Students often think wasted energy is 'lost' or 'gone'. It still exists but has been transferred to a non-useful store (usually the thermal store of the surroundings).

• Confusing Efficiency with Speed: A machine being 'efficient' does not necessarily mean it is 'fast'. Efficiency is about waste reduction, not velocity.

• Input/Output Swap: A common error is dividing the larger number by the smaller number. Remember: Input is always the denominator because it is the total available.