Power: The Rate of Doing Work
Power
Power is the rate at which energy is transferred or converted.
Mathematically, power ($P$) is defined as the rate of doing work ($W$) over a time interval ($\Delta t$):
$$
P = \frac{W}{\Delta t}
$$
The unit of power is the watt (W), where 1 watt equals 1 joule per second (1 W = 1 J/s).
ExampleIf a motor does 500 J of work in 10 seconds, its power is:
$$
P = \frac{500 \, \text{J}}{10 \, \text{s}} = 50 \, \text{W}
$$
Power Delivered by a Force
- For a constant force ($F$) moving an object at velocity ($v$), power is given by: $$P = Fv$$
- It applies when the force and velocity are in the same direction, measuring power in watts ($\text{W}$).
- 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.Example
A car engine exerts a force of 1,000 N to maintain a speed of $20 \ \text{m s}^{-1}$. The power output is: $$
P = Fv = 1,000 \, \text{N} \times 20 \, \text{m s}^{-1}$$
$$ = 20,000 \, \text{W} \, (20 \, \text{kW})
$$
The formula $P = Fv$ is especially useful for systems with constant velocity, such as vehicles cruising at steady speeds.
Efficiency: Measuring Energy Transfer Effectiveness
Efficiency
Efficiency quantifies how effectively a system converts input energy into useful output energy.
It is expressed as a ratio or percentage:
$$
\eta = \frac{E_{\text{output}}}{E_{\text{input}}} \times 100%
$$
Efficiency is always less than 100% due to energy losses, often as heat or sound.
Example- An electric motor receives 500 J of electrical energy and outputs 350 J of mechanical work.
- Its efficiency is:
$$
\eta = \frac{350 \, \text{J}}{500 \, \text{J}} \times 100\% = 70\%
$$
Efficiency is a dimensionless quantity, often expressed as a percentage.
TipEfficiency can also be calculated as a ratio of powers instead of a ratio of energies.
Energy Density
Energy density
Energy density is defined as the amount of energy stored per unit volume or mass of a substance.
- It can be expressed in two ways:
