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Heat engines are fundamental devices in thermodynamics that convert thermal energy into mechanical work, which is essentially the energy associated with motion. At its core, a heat engine interacts with two reservoirs of thermal energy at different temperatures: a hot and a cold reservoir.

The engine absorbs heat energy from the hot reservoir and uses part of this energy to do work, while the remainder is released as waste heat into the cold reservoir. A classic example of a heat engine is the steam engine, where steam is used to drive pistons and perform work.

Heat engines operate on a cycle, returning to their initial state at the end of each cycle. During this process, it is crucial to know that not all absorbed heat is converted into work due to inherent inefficiencies and the laws of thermodynamics, which dictate that some energy must always be lost as waste heat.

Efficiency is a measure of how well a heat engine converts the heat absorbed from its hot reservoir into useful work. It is represented as a dimensionless number or a percentage.

The efficiency (\( \text{η} \)) of a heat engine can be calculated using the formula:
\[ \text{η} = \frac{\text{Work Done}}{\text{Heat Absorbed}} \times 100\text{%} \tag{1}\]
In the context of our exercise, an efficiency of 0.30, or 30%, signifies that for every 500 J of heat absorbed from the hot reservoir, only 30% is converted into work.

Also, we can derive the work done by the engine per cycle using the information about the heat absorbed per cycle and the engine’s efficiency:
\[ \text{Work Done per cycle} = \text{Efficiency} \times \text{Heat Absorbed per cycle} \tag{2}\]
This fundamental efficiency calculation allows us to understand the performance of heat engines, guiding the design and optimization of various mechanical systems.

The work-energy principle is a critical concept in physics that states the transfer of energy into or out of a system can be accounted for in terms of work done on or by the system.

In the realm of thermodynamics and specifically for heat engines, this principle is observed as the engine performs work on the environment, such as lifting a weight or turning a shaft. The energy that enables this work to be done comes from the heat absorbed by the engine.

According to the work-energy principle, the total energy remains constant in a closed system. Hence, the heat energy not transformed into work during the engine’s cycle must be accounted for—as is our heat discharged in the exercise. This balance is expressed as:
\[ \text{Heat Absorbed} = \text{Work done} + \text{Heat discharged} \tag{3}\]
Therefore, the work done by the engine and the waste heat discharged are directly linked to the energy balance within the heat engine cycle. If we know two of these values, we can always calculate the third.