Power, energy consumption and efficiency
Your family's electric bill arrives and you wonder why the space heater costs so much more to run than a phone charger left plugged in all week. A 1,500 W electric kettle and a 150 W immersion heater both heat water, yet the kettle boils a liter in a few minutes while the small heater takes about ten times as long; and two motors can draw the same electrical energy while one wastes far more of it as heat. The work done may be the same, and yet one machine is quicker, or cheaper to run, than the other. Power and efficiency are the two quantities that tell apart "how much work is done" from "how well and how fast it is done".
Power: how quickly energy is transferred
Two cranes lift a pallet of $500\ \text{kg}$ to a height of $10\ \text{m}$: both do exactly the same work of $49{,}000\ \text{J}$, but one takes $10\ \text{s}$ and the other $40\ \text{s}$. Which quantity measures how quickly that same energy is transferred?
Rearranging $P = F \cdot v$ gives $F = P/v$: with the same engine power, a truck moving slowly can exert a much larger force. In everyday life in the US you will also see the horsepower: $1\ \text{hp} \approx 746\ \text{W}$, a unit used for car and lawn mower engines. Rearranging $P = W/t$ gives $t = W/P$, the time a job takes. Some problems also use the metric horsepower, written CV (from the French cheval vapeur), a historical unit: $1\ \text{CV} = 736\ \text{W}$, slightly less than the US horsepower. James Watt originally defined horsepower as the power needed to lift $75\ \text{kg}$ by $1\ \text{m}$ every second, which gives about $735\ \text{W}$; to convert, multiply the number of CV by 736 to get watts.
The motor of a garage door does $6000\ \text{J}$ of work in $30\ \text{s}$ when lifting it. The average power is calculated as $P = W/t$. a) Calculate the power of the…
The motor of a garage door does $6000\ \text{J}$ of work in $30\ \text{s}$ when lifting it. The average power is calculated as $P = W/t$.
- a) Calculate the power of the motor, in watts.
- b) Express it in kilowatts.
In the technical specifications of a car, a power rating of $100\ \text{CV}$ is listed. The metric horsepower is a historical unit of power:…
In the technical specifications of a car, a power rating of $100\ \text{CV}$ is listed. The metric horsepower is a historical unit of power: $1\ \text{CV} = 736\ \text{W}$.
- a) Express the car's power in watts.
- b) Express it in kilowatts, the unit used by industry today.
A construction tower crane lifts a $400\ \text{kg}$ load of blocks to a height of $20\ \text{m}$, at constant speed, in $40\ \text{s}$ (take $g = 10\ \text{m/s}^2$). a)…
A construction tower crane lifts a $400\ \text{kg}$ load of blocks to a height of $20\ \text{m}$, at constant speed, in $40\ \text{s}$ (take $g = 10\ \text{m/s}^2$).
- a) Calculate the work done by the crane against weight.
- b) Calculate the useful power it develops, in watts and in kilowatts.
Energy consumed: $E = P \cdot t$ and the kWh
An electric heater has a power rating of $2000\ \text{W}$ and operates for $3\ \text{h}$. The energy consumed is calculated as $E = P \times t$. a) Calculate the energy…
An electric heater has a power rating of $2000\ \text{W}$ and operates for $3\ \text{h}$. The energy consumed is calculated as $E = P \times t$.
- a) Calculate the energy consumed, in $\text{kWh}$ (use power in $\text{kW}$ and time in hours).
- b) Given that $1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$, express it in joules.
Order these three appliances from highest to lowest power: a $350\ \text{W}$ blender, a $2000\ \text{W}$ hair dryer, and a $9\ \text{W}$ LED light bulb. Then, calculate…
Order these three appliances from highest to lowest power: a $350\ \text{W}$ blender, a $2000\ \text{W}$ hair dryer, and a $9\ \text{W}$ LED light bulb. Then, calculate the energy consumed by the most powerful appliance operating for $10\ \text{min}$, expressed in joules (remember to convert time to seconds).
Efficiency: useful power compared with total power
When $100\ \text{J}$ of electrical energy is supplied to a traditional light bulb, only $5\ \text{J}$ is given off as visible light and the remaining $95\ \text{J}$ heats the glass. How do we measure quantitatively the fraction of energy that a device actually makes use of compared with the fraction it wastes?
The motor of a Ferris wheel consumes $5000\ \text{J}$ of electrical energy and delivers $4000\ \text{J}$ of useful work in raising the cabins. Efficiency is defined as…
The motor of a Ferris wheel consumes $5000\ \text{J}$ of electrical energy and delivers $4000\ \text{J}$ of useful work in raising the cabins. Efficiency is defined as $\eta = \frac{W_{util}}{W_{total}}$. The diagram summarizes the energy balance of the motor:
- a) Calculate the efficiency of the motor and express it as a percentage.
- b) How much energy was lost? What do you think it has been transformed into?
An industrial electric motor consumes $8\ \text{kWh}$ of electrical energy during its shift and delivers $6\ \text{kWh}$ of useful work (…
An industrial electric motor consumes $8\ \text{kWh}$ of electrical energy during its shift and delivers $6\ \text{kWh}$ of useful work ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).
- a) Calculate its efficiency as a percentage.
- b) Calculate the energy lost, in $\text{kWh}$ and in joules.
The motor of a workshop compressor draws a power of $5\ \text{kW}$ from the power line and has an efficiency of $80\ \%$. Efficiency can also be expressed using power:…
The motor of a workshop compressor draws a power of $5\ \text{kW}$ from the power line and has an efficiency of $80\ \%$. Efficiency can also be expressed using power: $\eta = \frac{P_{util}}{P_{total}}$.
- a) Calculate the useful power of the motor.
- b) Calculate the power lost.
- c) If the motor operates for $4\ \text{h}$, how much useful energy does it deliver? Express it in $\text{kWh}$ and in joules ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).
A jet ski travels at a constant speed of $10\ \text{m/s}$ and its engine develops a useful power of $25\ \text{kW}$. a) What net resistance force does the engine…
A jet ski travels at a constant speed of $10\ \text{m/s}$ and its engine develops a useful power of $25\ \text{kW}$.
- a) What net resistance force does the engine overcome at that speed? First rearrange for force in the expression $P = F \times v$.
- b) If the efficiency of the engine is $25\ \%$, what total power does it consume?
Efficiency chains: several conversions in a row
In a mini hydroelectric power station, the turbine has an efficiency of $90\ \%$ and the generator coupled to it has an efficiency of $80\ \%$. The diagram shows the…
In a mini hydroelectric power station, the turbine has an efficiency of $90\ \%$ and the generator coupled to it has an efficiency of $80\ \%$. The diagram shows the conversion chain:
- a) Explain why the overall efficiency of the chain is the product of the individual efficiencies and calculate it.
- b) If the water supplies $1\times10^{7}\ \text{J}$ of mechanical energy to the power station, how much electrical energy is obtained?
- c) Express it in $\text{kWh}$ ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).
Energy cost of an installation
A bakery oven has a power rating of $2500\ \text{W}$ and operates for $1.5\ \text{h}$ each day. The price of electricity is $0.2\ \text{\$/kWh}…
A bakery oven has a power rating of $2500\ \text{W}$ and operates for $1.5\ \text{h}$ each day. The price of electricity is $0.2\ \text{\$/kWh}$.
- a) Calculate the energy consumed by the oven in a month of $30$ days, in $\text{kWh}$.
- b) Calculate the monthly electricity cost of the oven.
A solar irrigation pump raises $200\ \text{kg}$ of water (remember: $1\ \text{L}$ of water has a mass of $1\ \text{kg}$) from a tank to a terraced field located…
A solar irrigation pump raises $200\ \text{kg}$ of water (remember: $1\ \text{L}$ of water has a mass of $1\ \text{kg}$) from a tank to a terraced field located $15\ \text{m}$ higher, in $5\ \text{min}$ (take $g = 10\ \text{m/s}^2$).
- a) Calculate the useful work done by the pump on the water (remember to convert time to seconds).
- b) Calculate the useful power, in watts.
- c) If the efficiency of the pump is $50\ \%$, what electrical power does it absorb?
- d) If it operates for $2\ \text{h}$ a day and electricity costs $0.15\ \text{\$/kWh}$, how much does irrigation cost per day?
Back to the start: The space heater costs more to run because its power is much higher, so $E = P \cdot t$ is larger for the same time on, and the utility meter charges for every kWh drawn. A kettle with the same work to do but twice the power finishes in half the time ($P = W/t$), while the motor that makes better use of the energy has a higher efficiency ($\eta = P_{\text{useful}}/P_{\text{total}}$), so less is dissipated as heat.