Question

What is the mathematical relationship between efficiency, input work, and useful output work?

Answer 1

Efficiency is defined as the ratio of useful output work to input work, often expressed as a percentage to indicate performance. Efficiency cannot exceed 100% as real-world applications involve energy losses such as friction. The efficiency formula is Efficiency (Eff) = Wo / Wi.

Step-by-step explanation:

The mathematical relationship between efficiency, input work (Wi), and useful output work (Wo) is such that efficiency is the ratio of useful output work to input work. The correct formula is designated as Efficiency (Eff) = Wo / Wi. It is important to note that efficiency can never be greater than 100%, because in real-world scenarios, systems often lose energy to friction, sound, heat, and other forms of non-useful work.

By this definition, the statements claiming that efficiency is the sum (option a), the product (option c), or the difference (option d) between useful energy and input energy are incorrect. The correct answer is option b, which states that efficiency is the proportion of useful work to input work. This value is often multiplied by 100 to express it as a percentage.

Thus, when we apply this formula, we understand that to achieve high efficiency, the goal is to minimize energy losses during the work process. However, it's practically impossible to create a perfectly efficient machine due to inherent energy losses like friction.

Answer 2

The efficiency of a machine is defined as the ratio between the useful output work and the input work:

`\eta = (W_(out))/(W_(in))`
where

`\eta` is the efficiency

`W_(out)` is the useful output work

`W_(in)` is the input work
We see from the formula that as the useful output work approaches the input work, the efficiency gets closer to 1 (this is the situation where no energy is wasted by the machine, so it is very efficient). On the contrary, when the useful output work is very small, the efficiency of the machine is very low.