# Efficiency ## What is Efficiency? - **Efficiency** measures how effectively a system converts input energy into useful output energy - It quantifies the success of an energy transfer process - Systems with **high efficiency** transfer most of their input energy into useful forms - Systems with **low efficiency** waste most of their input energy

Efficiency is the ratio of useful power output from a system to its total power input

### Typical Efficiency Values - Different devices have vastly different efficiencies depending on their energy conversion mechanisms - Understanding typical values helps calibrate expectations for real-world systems

- Modern devices like LED lamps achieve very high efficiencies (around 90%) - Older technologies like incandescent bulbs are highly inefficient (around 5%) - Thermal systems (car engines, power stations) often have moderate efficiencies due to fundamental thermodynamic limitations ## Useful vs Wasted Energy - Whether energy is "useful" or "wasted" depends entirely on the purpose of the device - Examples where this distinction matters: ### Lightbulb - **Useful energy**: Light energy (illumination is the goal) - **Wasted energy**: Thermal energy (unwanted heating of surroundings) ### Electric Heater - **Useful energy**: Thermal energy (heating is the goal) - **Wasted energy**: Sound energy (unwanted noise from the element) ### Hair Dryer - **Useful energy**: Thermal energy and kinetic energy of air - **Wasted energy**: Sound energy (motor noise) ## Efficiency Equation - Efficiency can be expressed as a **ratio** (between 0 and 1) or as a **percentage** (between 0% and 100%) ### Power-Based Formula $$\text{Efficiency} = \frac{\text{Useful power output}}{\text{Total power input}} \times 100\%$$ - This form is used when power values are known - Remember that power is the rate of energy transfer: $P = \frac{E}{t}$ ### Energy-Based Formula $$\text{Efficiency} = \frac{\text{Useful energy output}}{\text{Total energy input}} \times 100\%$$ - This form is used when total energy values are known - The energy can be in any form (e.g. kinetic, gravitational potential, elastic, thermal) - Both formulas are equivalent because the time factor cancels out

Efficiency has no units - it is a dimensionless ratio

## Calculating with Efficiency ### Finding Efficiency from Given Values 1. Identify which energy or power is **useful output** 2. Identify the **total input** energy or power 3. Divide output by input and multiply by 100 for percentage ### Finding Input or Output from Efficiency - If efficiency and output are known: $\text{Input} = \frac{\text{Output} \times 100}{\text{Efficiency}}$ - If efficiency and input are known: $\text{Output} = \frac{\text{Efficiency} \times \text{Input}}{100}$ ## Worked Example: Electric Motor **Problem**: An electric motor has an efficiency of 35%. It lifts a 7.2 kg load through a height of 5 m in 3 s. Calculate the power of the motor. **Solution**: **Step 1**: Write the efficiency equation in terms of power $$\text{Efficiency} = \frac{\text{Power output}}{\text{Power input}} \times 100$$ **Step 2**: Rearrange for power input $$\text{Power input} = \frac{\text{Power output} \times 100}{\text{Efficiency}}$$ **Step 3**: Calculate the useful power output - The motor converts electrical energy into gravitational potential energy - Gravitational potential energy gained: $E_p = mgh = 7.2 \times 9.81 \times 5 = 353.16 \text{ J}$ - Time taken: $t = 3 \text{ s}$ - Power output: $P = \frac{E}{t} = \frac{353.16}{3} = 117.72 \text{ W}$ **Step 4**: Substitute values into the power input equation $$\text{Power input} = \frac{117.72 \times 100}{35} = 336 \text{ W}$$

When efficiency is given as a percentage (e.g. 35%), use this value directly in the formula with the $\times 100$ included

## Efficiency Formats ### Ratio Format (0 to 1) - Express efficiency as a decimal: e.g. 0.35 for 35% efficiency - Used in calculations where the equation doesn't include $\times 100$ - Formula becomes: $\text{Efficiency} = \frac{\text{Output}}{\text{Input}}$ ### Percentage Format (0% to 100%) - Express efficiency with the % symbol: e.g. 35% - Used when explicitly asked for percentage - To convert ratio to percentage: multiply by 100 (e.g. $0.35 \times 100 = 35\%$) - To convert percentage to ratio: divide by 100 (e.g. $35\% \div 100 = 0.35$)

Check what format the question requires - if it asks for efficiency "as a ratio", give a decimal or fraction; if it asks for "percentage", include the % symbol

## Key Points to Remember - **No system is 100% efficient** - some energy is always dissipated to the surroundings - Efficiency is always less than or equal to 1 (or 100%) - Higher efficiency means less wasted energy, which is important for: - Reducing energy costs - Minimizing environmental impact - Conserving resources - Common wasted energy forms include thermal energy, sound energy, and light energy (depending on the device) - Always identify what counts as "useful" based on the device's intended purpose