A.3.3 Power and efficiency

Power: The Rate of Doing Work

Mathematically, power ($P$) is defined as the rate of doing work ($W$) over a time interval ($\mathrm{\Delta }t$):

$$P=\frac{W}{\mathrm{\Delta }t}$$

Power Delivered by a Force

1. For a constant force ($F$) moving an object at velocity ($v$), power is given by: $$P=Fv$$
2. It applies when the force and velocity are in the same direction, measuring power in watts ($\text{W}$).
3. If the force and velocity are not aligned, the equation generalizes to $$P=Fv\cos \theta$$ where $\theta$ is the angle between the force and velocity.

This formula highlights the relationship between force, velocity, and power.

Efficiency: Measuring Energy Transfer Effectiveness

It is expressed as a ratio or percentage:

$$\eta =\frac{E_{\text{output}}}{E_{\text{input}}}\times 100$$

Energy Density

It can be expressed in two ways:

• Volumetric Energy Density: Energy per unit volume (${\mathrm{J}/\mathrm{m}}^3$)
• Gravimetric Energy Density: Energy per unit mass ($\mathrm{J}/\mathrm{kg}$)

1. High energy-density fuels can store large amounts of energy in a small amount of space, making them efficient for transportation and storage.
2. The energy density can be expressed using these equations:
   • For volumetric energy density: $$e_v=\frac{E}{V}$$
   • For gravimetric energy density: $$e_m=\frac{E}{m}$$

where:

• $e_v$ is volumetric energy density (${\mathrm{J}/\mathrm{m}}^3$)
• $e_m$ is gravimetric energy density ($\mathrm{J}/\mathrm{kg}$)
• $E$ is energy ($\mathrm{J}$)
• $V$ is volume (${\mathrm{m}}^3$)
• $m$ is mass ($\mathrm{kg}$)

Applications of Power in Real-World Systems

Engines

1. Engines convert fuel energy into mechanical work.
2. Their power output determines performance, such as acceleration and towing capacity.

Electric Appliances

1. Power ratings on appliances indicate energy consumption.
2. A 100 W light bulb uses 100 J of energy per second, while a 1,500 W heater consumes 1,500 J per second.

Reflection and Broader Implications