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# Power, energy loss and efficiency

## Power as work done over time taken

Power $(P)$ is the energy generated per unit time.

Because energy is the ability to do work, power can also be defined as the work done per unit time.

A motor which uses a high power can do more work (or use more energy) in a fixed amount of time than a motor with a low power.

The power is given by: $$\Tblue{\text{power}}=\frac{\Tred{\text{work done}}}{\Torange{\text{time taken}}}=\frac{\Tred{\text{energy generated}}}{\Torange{\text{time taken}}}$$

The SI unit of power is the watt $(\text{W})$. One watt is equivalent to one joule per second (i.e. $1\text{ W}=1\text{ J s}^{-1}$).

A light bulb that does $\Tred{500 \text{ J}}$ of work in $\Torange{5 \us}$ has a power of $\Tblue{100 \text{ W}}.$

A microwave with a power of $\Tblue{0.8 \text{ kW}}$ delivers $\Tred{16 \text{ kJ}}$ of energy during a $\Torange{20 \us}$ period.

Coal is burnt in this power station to create electrical power for homes.

## Power as force times velocity

Power $(P)$ is the work done (or energy generated) per unit time.

The work done is equal to the product of the force and the displacement in the direction of the force.

Remember also that the velocity is equal to the displacement per unit time.

Power is therefore: $$\Tblue{\text{power}} = \dfrac{\Tred{\text{force}} \times \Torange{\text{displacement}}}{\Torange{\text{time taken}}} = \Tred{\text{force}} \times \Torange{\text{velocity}}$$

A car travelling at $\Torange{30 \umps}$ faces a resistive force of $\Tred{ 100 \text{ N}}$. The power generated by the car's engine is $$\Tblue{P} = \Tred{100 \text{ N}} \times \Torange{30 \umps} = \Tblue{3000 \text{ W}}$$

Cyclists generates large amounts of power with their legs to travel at high velocities.

## Energy loss and efficiency

Energy can be transferred in and out of a closed system. In most real systems, not all of the energy input into the system gets output into useful energy/work as some energy is lost to the surroundings as heat.

Some of the energy obtained by burning fuel in a car engine is lost as heat to the surroundings instead of being converted into kinetic energy to drive the car forward.

This means that most real systems are not $100\%$ efficient. The efficiency of a system is the percentage of input energy/work (or power) that gets transformed into useful output energy/work (or power). This is given by: \begin{align*} \text{Efficiency}&=\frac{E_{\text{out}}}{E_{\text{in}}}\times 100\%\\ &=\frac{W_{\text{out}}}{W_{\text{in}}}\times 100\%\\ &=\frac{P_{\text{out}}}{P_{\text{in}}}\times 100\% \end{align*} $W_{\text{in}}$, $E_{\text{in}}$, $P_{\text{in}}$ and $W_{\text{out}}$, $E_{\text{out}}$, $P_{\text{out}}$ are the input and output work, energy and power respectively.