Question

Concept Check Engine 1 does twice the work of engine \(2 .\) Is it correct to conclude that engine 1 produces twice as much power as engine 2? Explain.

Short answer

Only if both engines take the same time to do their work.

Step-by-step solution

Understand the Relationship between Work and Power

Work is defined as the amount of energy transferred by a force over a distance, while power is the rate at which this work is done. This means that power also depends on the time over which the work is done. The formula for power is given by \( P = \frac{W}{t} \), where \( P \) is power, \( W \) is work, and \( t \) is time.

Given Information

We know that engine 1 does twice the work of engine 2. Let's denote the work done by engine 1 as \( W_1 = 2W_2 \), where \( W_2 \) is the work done by engine 2.

Establish power relationship

The power produced by each engine can be represented as \( P_1 = \frac{W_1}{t_1} \) for engine 1 and \( P_2 = \frac{W_2}{t_2} \) for engine 2, where \( t_1 \) and \( t_2 \) are the times taken by each engine to do the work. Since \( W_1 = 2W_2 \), this implies \( P_1 = \frac{2W_2}{t_1} \).

Compare the power outputs

To conclude that engine 1 produces twice as much power as engine 2, \( \frac{2W_2}{t_1} \) must equal \( 2 \times \frac{W_2}{t_2} \). This simplifies to \( \frac{2}{t_1} = \frac{2}{t_2} \), meaning \( t_1 = t_2 \). Therefore, it is only correct to conclude engine 1 produces twice as much power if both engines take the same amount of time to perform their respective works.

Key concepts

Understanding Work Efficiency

Work efficiency is a measure of how effectively an engine converts the energy it uses into work output. The concept of work efficiency is crucial in determining how much of the input energy is converted into useful work. It is calculated as the ratio of useful work done to the total energy input, often expressed as a percentage.
Efficiency can be affected by factors such as friction, heat losses, and other non-productive uses of energy. A highly efficient engine will use most of its energy to perform work, while a less efficient engine wastes more energy.
Understanding work efficiency can help in evaluating the performance of different engines. It allows engineers to improve mechanical systems by maximizing output while minimizing waste.

• Efficiency = (Useful Work Output / Total Energy Input) x 100%
• Factors affecting efficiency: friction, heat loss, energy waste

Exploring Power Output

Power output refers to the amount of work an engine can perform in a specific timeframe. It is a crucial metric as it tells us how quickly an engine can perform its task. The power output is determined using the formula: \( P = \frac{W}{t} \), where \( P \) is power, \( W \) is work, and \( t \) is the time in which the work is done.
In the context of our original exercise, engine 1 does twice the work of engine 2. But whether it produces twice the power depends on the time each engine takes to perform this work.
For accurate comparison and understanding, the times \( t_1 \) and \( t_2 \) during which engine 1 and engine 2 perform their work must be the same. If \( t_1 \) is equal to \( t_2 \), then indeed, engine 1 produces twice the power. Thus, the power output is not just about gathering work done, but also the rate at which it is done.

• Power = Work/Time
• Comparison requires equal time intervals

Understanding Energy Transfer

Energy transfer is the process through which energy is moved from one place or system to another. In engines, this is the transfer of chemical energy from fuel into mechanical energy, which is then used to perform work.
There are various forms of energy that can be transferred, including mechanical, thermal, and electrical energies. Efficient energy transfer is vital for good engine performance, as it ensures maximum energy is used for productive work.
In the context of engines such as those in our exercise, how efficiently energy is transferred affects the overall efficiency and power of the engines. Energy transfer processes must minimize losses to maximize work efficiency and power output. This can be achieved by optimizing mechanisms such as fuel consumption and by reducing waste heat and friction.

• Types of energy transfer: mechanical, thermal, electrical
• Efficient transfer improves work and power efficiency