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.

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: $1\ \text{CV} = 736\ \text{W}$.

1. a) Express the car's power in watts.
2. b) Express it in kilowatts, the unit used by industry today.

A passenger car maintains a constant speed of $90\ \text{km/h}$ on a highway. 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 $2000\ \text{W}$ 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).

Unit challenge. Are you more efficient than an electric motor? Design a biomechanical and economic audit of your own motive power at home.

1. Data collection and research: Measure the height of a flight of stairs ($h \pm 0.01\text{ m}$) and the mass of a load ($m \pm 0.1\text{ kg}$) and your own body mass ($M \pm 0.1\text{ kg}$). Look up today's average $\text{kWh}$ price on the U.S. Energy Information Administration (EIA) website.

2. Experimentation: Carry the load upstairs at four different paces, timing each ascent with its uncertainty ($t \pm 0.2\text{ s}$, performing three trials to find the average).

3. Data processing: Create a data table showing the useful mechanical work ($W = m \times g \times h$), the power output ($P = W/t$), and their uncertainties. Plot $P$ against $1/t$.

4. Consumption and efficiency: Assuming a human muscular efficiency of $\eta = 22\,\%$, calculate the actual metabolic energy consumed (in $\text{kJ}$ and $\text{kWh}$), taking into account that you also lift your own body ($E_{\text{metab}} = (m+M) \cdot g \cdot h/\eta$), and compare it with the financial cost if a small domestic motor with $\eta = 85\,\%$ performed the same task (lifting only the load). Write a brief report with your conclusions.

Evaluation criteria:

• Rigor in estimating uncertainties and quality of the table and graph.

• Accuracy in applying the concepts of work, power, and efficiency.

• Critical analysis of efficiency and proper use of EIA data.

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:

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 power line 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) Explain 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?