Question

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

Answer 1

The efficiency of a machine is defined as the ratio between the useful output work and the input work:
$\eta = \frac{W_{out}}{W_{in}}$
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.