Question

Putting the cheapness of fossil fuels into perspective, a gallon of gasoline purchased for \(\$ 4\) might deliver \(6 \mathrm{kWh}\) of mechanical energy after accounting for efficiency of the associated engine. A laborer might be expected to export \(100 \mathrm{~W}\) of mechanical power, on average, and be limited to 8 hours per day. How many hours would it take for the laborer to accomplish the equivalent output of a gallon of gasoline? At a rate of \(\$ 15 / \mathrm{hr}\), how much will this cost you?

Short answer

Answer: It would take the laborer 60 hours to produce the same amount of energy as a gallon of gasoline, and the cost of their work would be $900.

Step-by-step solution

Calculate the laborer's daily mechanical energy output

The laborer can export 100 W of mechanical power and works for 8 hours per day. To find the total mechanical energy output, we need to multiply the power by the time. Daily mechanical energy output = Power × Time = 100 W × (8 hours × 3600 seconds/hour) = 100 W × 28800 seconds = 2,880,000 J = 2.88 MJ

Calculate the number of hours required to produce the same amount of energy as a gallon of gasoline

The gasoline produces 6 kWh of mechanical energy, which we need to convert to Joules. Energy from gasoline = 6 kWh × (3600 seconds/hour) × (1000 J/1 kJ) = 21,600,000 J = 21.6 MJ Now, we divide the energy produced by gasoline by the daily mechanical energy output of the laborer to find the number of days it would take the laborer. Number of days = Energy from gasoline / Daily mechanical energy output = (21.6 MJ) / (2.88 MJ/day) ≈ 7.5 days Since the laborer can only work 8 hours per day, we must multiply this by the number of days. Number of hours = 7.5 days × 8 hours/day = 60 hours

Calculate the cost of the laborer's work

Now that we have the total number of hours required to produce the same amount of energy, we can calculate the cost using the hourly wage ($15/hour). Total cost = Hourly wage × Number of hours = $15/hour × 60 hours = $900 So, it would take the laborer 60 hours to accomplish the equivalent output of a gallon of gasoline, and it would cost $900.

Key concepts

Energy Efficiency

Understanding energy efficiency is fundamental when analyzing how different energy sources are used to perform work. It's a measure of how well we convert one form of energy into another useful form. In the given exercise, we consider the energy efficiency of an engine that runs on gasoline and the human body doing physical labor.

The engine's efficiency is the ratio of the mechanical energy output to the energy stored in the gasoline. As gasoline powers the engine, not all energy translates to useful work due to losses like heat and friction. On the other hand, the human body also has energy losses during physical labor, but the exercise simplifies this to a constant power output.

When calculating how much work a laborer can do compared to a gallon of gasoline, we see that despite the high cost of human labor, the amount of available mechanical energy is significantly less. This accentuates that the gasoline engine, even with its inefficiencies, can deliver more energy output over a shorter period than human labor. Such comparisons highlight the importance of pursuing advancements in energy efficiency, especially in a world trying to balance energy needs with environmental and economic considerations.

Fossil Fuels

The term fossil fuels refers to the natural resources like coal, natural gas, and petroleum that have been formed from the remains of ancient plants and animals over millions of years. These are considered non-renewable resources because their formation takes such a long time on a human timescale.

In the context of our problem, gasoline, a byproduct of petroleum, is the fossil fuel under consideration. It's a popular fuel choice for combustion engines due to its energy density and relative ease of transportation and use. However, the burning of fossil fuels is a major contributor to air pollution and climate change due to carbon dioxide and other greenhouse gases released.

Understanding the actual cost and output of fossil fuels, as compared to human power, serves as a stark reminder of our reliance on these energy sources. It also forces us to reckon with the challenging task of finding sustainable and efficient energy solutions that can ultimately replace them without sacrificing our quality of life.

Work-Energy Principle

The work-energy principle is a cornerstone concept in physics that states that work done on an object is equal to the change in its energy. Essentially, it connects two key concepts in the physical world: work (the product of force and displacement) and energy (the ability to do work).

In the mechanical context, understanding this principle allows us to relate the amount of work done by the laborer or an engine to their respective energy outputs. In the exercise, by calculating the laborer's daily mechanical energy output, we applied the principle to determine that it would take 60 hours for a human to match the energy produced by a single gallon of gasoline in an engine.

Application in Calculating Work

Through the work-energy principle, we see the practical application of physics in everyday tasks and energy consumption. It also reinforces the concept of efficiency, as not all the energy in a gallon of gasoline gets converted to mechanical work, just like not all of a laborer's caloric intake is converted into useful power.

This principle helps demystify complex notions of energy and provides a framework to compare different forms of work, guiding us towards smarter and more efficient energy use.