Power

1. Overview

Power is a measure of how quickly work is being done or how rapidly energy is being transferred. In everyday terms, two machines might perform the exact same task (like lifting a crate), but the one that completes the task faster is considered more "powerful." Understanding power is essential for evaluating the performance of engines, motors, and electrical appliances.

Key Definitions

• Power: The rate at which work is done or the rate at which energy is transferred.
• Watt (W): The SI unit of power. One Watt is equal to one Joule of energy transferred per second ($1\text{ W} = 1\text{ J/s}$).

Core Content

Power is always about the relationship between energy and time. If you do work faster, your power output is higher.

Understanding the Equations There are two ways to express the same concept:

1. Work-based: $P = \frac{W}{t}$ (Used when a force moves an object).
2. Energy-based: $P = \frac{\Delta E}{t}$ (Used when energy changes from one form to another).

Power in Action

• If two motors lift the same weight to the same height, they do the same amount of work.
• However, if Motor A does it in 5 seconds and Motor B does it in 10 seconds, Motor A has twice the power of Motor B.

Worked Example: Lifting a Load A crane lifts a $200\text{ kg}$ crate to a height of $10\text{ m}$ in $4\text{ seconds}$. Calculate the power of the crane. (Take $g = 10\text{ m/s}^2$)

• Step 1: Calculate the Force (Weight) $F = m \times g = 200\text{ kg} \times 10\text{ m/s}^2 = 2000\text{ N}$
• Step 2: Calculate the Work Done $W = F \times d = 2000\text{ N} \times 10\text{ m} = 20,000\text{ J}$
• Step 3: Calculate Power $P = \frac{W}{t} = \frac{20,000\text{ J}}{4\text{ s}} = 5000\text{ W}$ (or $5\text{ kW}$)

Extended Content (Extended Only)

The IGCSE syllabus classifies the definition and calculation of Power under the Core curriculum. There is no additional calculation or theory required specifically for Extended students for this sub-topic.

Key Equations

Equation	Symbols	Units
$P = \frac{W}{t}$	$P$ = Power, $W$ = Work done, $t$ = time	$P$ (Watts), $W$ (Joules), $t$ (seconds)
$P = \frac{\Delta E}{t}$	$P$ = Power, $\Delta E$ = Energy transferred, $t$ = time	$P$ (Watts), $E$ (Joules), $t$ (seconds)

Common Mistakes to Avoid

• ❌ Wrong: Using minutes or hours for time in your calculation.
  • ✓ Right: Always convert time into seconds (e.g., $2\text{ mins} = 120\text{ s}$) before using the formula.
• ❌ Wrong: Using mass ($kg$) directly in the work equation ($W = F \times d$).
  • ✓ Right: You must convert mass to weight (Force) by multiplying by $g$ ($10\text{ m/s}^2$ or $9.8\text{ m/s}^2$).
• ❌ Wrong: Thinking the most powerful motor is the one that lifts the heaviest mass OR the highest distance regardless of time.
  • ✓ Right: Power is a combined effect. You must calculate $(m \times g \times h) / t$ to compare them fairly.
• ❌ Wrong: Rearranging the formula to $P = W \times t$ or $P = W / t^2$.
  • ✓ Right: Power is simply the rate: $P = \frac{\text{Work}}{\text{Time}}$. If you multiply Work by Time, the units no longer represent Power.

Exam Tips

1. Check your units first: If a question gives you power in kilowatts (kW), multiply it by $1000$ to get Watts before you start. If time is in minutes, multiply by $60$.
2. Two-step problems: Many exam questions won't give you "Work." They will give you mass, height, and time. You must remember to calculate Work ($m \times g \times h$) first, then divide by time to find Power.
3. The "Watt" Definition: If asked to define a Watt, remember it is "one Joule per second." This helps you remember the formula ($J / s$).