Exercises: 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.

Two assembly line robots perform exactly the same work when welding a part, but robot A takes $20\ \text{s}$ and robot B takes $40\ \text{s}$.

1. a) Which of the two performs more work? Justify your answer.
2. b) Which is more powerful? How many times more?

The driving force of a car engine moving at a constant speed of $20\ \text{m/s}$ is equal to $1500\ \text{N}$. When the force and velocity have the same direction, power can be calculated as $P = F \times v$. Calculate the power developed by the engine, in watts and 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.

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.

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?

A cyclist pedals with the same net force on the bicycle in two situations: at $18\ \text{km/h}$ on a flat section and at $36\ \text{km/h}$ on a gentle descent.

1. a) Using the expression $P = F \times v$, reason in which of the two situations more power is developed and how many times more.
2. b) Does that mean they perform more work in the same amount of time? Explain.

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).

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 chairlift lifts $30$ skiers, of average mass $75\ \text{kg}$ with their equipment, to an elevation $100\ \text{m}$ higher, in $5\ \text{min}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done by the chairlift on the skiers (remember to convert time to seconds).
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 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 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.

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.

In a fitness test, a soldier weighing $75\ \text{kg}$ with equipment climbs $6\ \text{m}$ up a rope in $15\ \text{s}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work performed against weight.
2. b) Calculate the average power developed.
3. c) If her female comrade, of mass $60\ \text{kg}$ with equipment, takes the same amount of time, does she develop more, less, or equal power? Justify your answer by calculating it.

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 ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

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?

The motor of a crane draws an electrical power of $10\ \text{kW}$ from the power grid and has an efficiency of $60\ \%$. It lifts a load of $500\ \text{kg}$ at constant speed for $30\ \text{s}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the useful power of the motor.
2. b) Calculate the useful work done in the $30\ \text{s}$.
3. c) How much height has the load gained in that time?

A passenger car of $1200\ \text{kg}$ drives up a mountain pass with an $8\ \%$ grade (for every $100\ \text{m}$ traveled horizontally, it rises by $8\ \text{m}$; you may take $\sen \alpha \approx 0.08$) at a constant speed of $20\ \text{m/s}$. Frictional and aerodynamic resistance forces add up to $600\ \text{N}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the component of weight parallel to the road.
2. b) Calculate the total force that the engine must develop to maintain speed.
3. c) Calculate the useful power of the engine, in $\text{kW}$.
4. d) If the efficiency of the engine is $30\ \%$, what power does it consume? Comment on why consumption on an uphill stretch is much higher than on flat ground.

The escalator in a subway station lifts passengers through a vertical height of $6\ \text{m}$. At peak times it transports a flow of $900$ people per hour, with an average mass of $70\ \text{kg}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done by the escalator on the passengers in one hour.
2. b) Calculate the required useful power.
3. c) If the motor-chain assembly has an efficiency of $70\ \%$, what power rating should the installed motor have?

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?

James Watt defined horsepower ($\text{CV}$) as the power required to lift $75\ \text{kg}$ at a rate of $1\ \text{m}$ every second, at constant speed (take $g = 9.8\ \text{m/s}^2$).

1. a) Calculate the work done against weight in $1\ \text{s}$ under these conditions and the corresponding power.
2. b) Compare the result with today's equivalence, $1\ \text{CV} = 736\ \text{W}$, and comment on the reason for the small difference.
3. c) A well-trained cyclist can maintain around $300\ \text{W}$ for one hour: approximately how many $\text{CV}$ is this equivalent to? Comment on the result.

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 high-speed train develops a power of $8\ \text{MW}$ when traveling at $288\ \text{km/h}$.

1. a) Express the speed in $\text{m/s}$.
2. b) Calculate the total resistance force that the train overcomes at that speed, rearranging from $P = F \times v$.
3. c) If the train traveled at half speed while maintaining the same power, what maximum force could it overcome? Reason your answer and comment on why freight locomotives, which haul huge loads, travel slowly.

Two weightlifters compete in the snatch event. Weightlifter A lifts $100\ \text{kg}$ to a height of $2\ \text{m}$ in $1\ \text{s}$; weightlifter B lifts $150\ \text{kg}$ to the same height in $1.5\ \text{s}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done by each against gravity.
2. b) Calculate the power developed by each. Who is more powerful? Comment on the result.
3. c) How much time should weightlifter B have taken to complete his lift to develop a power of $2500\ \text{W}$?

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.