Problem 2 - University Physics with Modern Physics

Question

An aircraft engine takes in 9000 \(\mathrm{J}\) of heat and discards 6400 \(\mathrm{J}\) each cycle. (a) What is the mechanical work output of the engine during one cycle? (b) What is the thermal efficiency of the engine?

Step-by-Step Solution

Answer

(a) Work output is 2600 J. (b) Thermal efficiency is approximately 28.89%.

Step 1: Understanding the Given Information

We are given that the heat absorbed by the engine, \( Q_1 \), is 9000 Joules. The heat discarded by the engine, \( Q_2 \), is 6400 Joules. We need to find both (a) the mechanical work output and (b) the thermal efficiency.

Step 1: Mechanical Work Output Calculation

Mechanical work output, \( W \), of the engine can be calculated using the formula: \( W = Q_1 - Q_2 \). Substitute the given values: \( W = 9000 \, \text{J} - 6400 \, \text{J} = 2600 \, \text{J} \). Hence, the work output is 2600 Joules.

Step 2: Thermal Efficiency Calculation

The thermal efficiency, \( \eta \), of an engine is calculated using the formula: \( \eta = \frac{W}{Q_1} \times 100 \% \). We already found \( W = 2600 \, \text{J} \) and \( Q_1 = 9000 \, \text{J} \). Substitute these values into the formula: \( \eta = \frac{2600}{9000} \times 100 \% = 28.89\% \). Hence, the thermal efficiency is approximately 28.89%.

Key Concepts

Mechanical Work Output

Mechanical work output is a fundamental concept in thermodynamics, especially in the study of engines. In simple terms, it refers to the amount of energy converted from heat into mechanical energy or work. This transformation is central to the functioning of all engines and can be quantified using a straightforward formula.

For an engine, the mechanical work output (\( W \)) is calculated by the difference between the heat absorbed (\( Q_1 \)) and the heat discarded (\( Q_2 \)). In our example, this means:

• The absorbed heat is 9000 Joules.
• The discarded heat is 6400 Joules.
• Thus, the mechanical work output is \( W = Q_1 - Q_2 = 9000 \, \text{J} - 6400 \, \text{J} = 2600 \, \text{J} \).

By understanding the output, we get insight into how efficiently a machine converts energy, helping engineers improve energy systems.

Thermal Efficiency

Thermal efficiency measures how well an engine converts heat into work. It's a vital concept for assessing engine performance. Simply put, thermal efficiency tells us what percentage of heat energy is converted into useful work.

To calculate thermal efficiency (\( \eta \)), use the formula:

• \( \eta = \frac{W}{Q_1} \times 100 \% \)
• Where \( W \) is the work output, and \( Q_1 \) is the heat absorbed.
• For our engine: \( \eta = \frac{2600}{9000} \times 100 \% = 28.89\% \)

This calculation tells us that only about 28.89% of the heat energy is used to do work. The rest is lost, usually as waste heat. By understanding thermal efficiency, we can work towards designing engines that lose less energy.

Heat Engines

Heat engines are devices that convert heat energy into mechanical work. They play a crucial role in various technologies, from powering vehicles to generating electricity.

A heat engine operates by exploiting the flow of heat from a high-temperature source to a low-temperature sink, and in this process, it converts some of this heat into work. This process involves:

• Absorbing heat from a high-temperature source (e.g., combustion in engines).
• Performing work (e.g., moving pistons).
• Releasing unused heat to a cold reservoir or sink.

The efficiency and work output of a heat engine are significant indicators of its effectiveness. By understanding these elements, engineers can innovate to create more sustainable and powerful engines, thereby advancing energy technology.