Power, energy consumption and efficiency

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 $P = W/t$.

1. a) Calculate the power of the motor, in watts.
2. b) Express it in kilowatts.

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 $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done by the crane against weight.
2. b) Calculate the useful power it develops, in watts and in kilowatts.

A passenger car maintains a constant speed of $90\ \text{km/h}$ on a motorway. At that speed, air resistance and friction add up to a force of $2000\ \text{N}$ that the engine must overcome.

1. a) Express the speed in $\text{m/s}$.
2. b) Since the speed is constant, the driving force equals the total resistance: calculate the power developed by the engine using $P = F \times v$, in $\text{kW}$.

An electric heater has a power rating of $2000\ \text{W}$ and operates for $3\ \text{h}$. The energy consumed is calculated as $E = P \times t$.

1. a) Calculate the energy consumed, in $\text{kWh}$ (use power in $\text{kW}$ and time in hours).
2. 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 $2000\ \text{W}$ hairdryer, 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).

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 summarises the energy balance of the motor:

1. a) Calculate the efficiency of the motor and express it as a percentage.
2. 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 ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

1. a) Calculate its efficiency as a percentage.
2. 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 mains supply and has an efficiency of $80\ \%$. Efficiency can also be expressed using power: $\eta = \frac{P_{util}}{P_{total}}$.

1. a) Calculate the useful power of the motor.
2. b) Calculate the power lost.
3. c) If the motor operates for $4\ \text{h}$, how much useful energy does it deliver? Express it in $\text{kWh}$ and in joules ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

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:

1. a) Reason why the overall efficiency of the chain is the product of the individual efficiencies and calculate it.
2. b) If the water supplies $1\times10^{7}\ \text{J}$ of mechanical energy to the power station, how much electrical energy is obtained?
3. c) Express it in $\text{kWh}$ ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

A jet ski travels at a constant speed of $10\ \text{m/s}$ and its engine develops a useful power of $25\ \text{kW}$.

1. a) What net resistance force does the engine overcome at that speed? First rearrange for force in the expression $P = F \times v$.
2. b) If the efficiency of the engine is $25\ \%$, what total power does it consume?

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 $0.2\ \text{€/kWh}$.

1. a) Calculate the energy consumed by the oven in a month of $30$ days, in $\text{kWh}$.
2. 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 $15\ \text{m}$ higher, in $5\ \text{min}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the useful work done by the pump on the water (remember to convert time to seconds).
2. b) Calculate the useful power, in watts.
3. c) If the efficiency of the pump is $50\ \%$, what electrical power does it absorb?
4. d) If it operates for $2\ \text{h}$ a day and electricity costs $0.15\ \text{€/kWh}$, how much does irrigation cost per day?