Definition

Power is the rate of doing work:

$P = \frac{dW}{dt} = \frac{W}{t}\ (\text{average}) \quad [\text{M}^1\text{L}^2\text{T}^{-3}]\ (\text{W})$

$P_{\text{inst}} = \vec{F} \cdot \vec{v} = Fv\cos\theta$

Unit Conversions

• 1 W = 1 J/s
• 1 kW = 1000 W
• 1 horsepower (hp) = 746 W ≈ 0.746 kW
• 1 kWh = $3.6 \times 10^{6}$ J (commercial unit of energy)

Efficiency

$\eta = \frac{P_{\text{output}}}{P_{\text{input}}} \times 100\% \leq 100\%$

Real machines always have η < 100% due to friction and heat losses.

Key Applications

• Vehicle engine: F = P/v (higher speed ⟹ lower driving force at same power)
• Pump: P = ρgQh where Q = volume flow rate
• Motor lifting: P = mgv (at constant lifting speed)