Efficiency

In §2.9 we assumed every machine was ideal — frictionless, no losses, every joule in equals a joule out. Reality is different. Every real machine wastes some energy.

A typical car engine delivers only about 25% of gasoline’s chemical energy to the crankshaft as work. The rest leaves as waste heat in the exhaust, the radiator, and the engine block. An incandescent light bulb converts about 5% of electrical energy into visible light — the other 95% becomes infrared “heat” radiation. Even your own muscles are only ~20–25% efficient — most of the energy you eat eventually leaves as body heat.

Efficiency tells you what fraction of the energy you put in actually does useful work. Higher efficiency = less waste = lower fuel/electricity costs.

The Efficiency Equation

You can also write this with power: $e = (P_{out}/P_{in}) \times 100\%$, since power is just energy per unit time and the time cancels.

Why Real Machines Are Never 100% Efficient

Every real machine bleeds energy through one (or more) of these channels:

Notice the pattern: almost everything ends up as heat. Sound waves, turbulence, deformation, electrical losses — all of them eventually warm up the surroundings and become “thermal energy” that’s no longer easy to recapture as useful work.

Actual vs. Ideal Mechanical Advantage

Because of friction, the actual force output of a machine is less than the ideal calculation predicts. This gives a second way to express efficiency:

Worked Examples

Example 1. A motor uses 500 J of electrical energy to lift a 30 kg crate 1.5 m. What is the motor’s efficiency? ($g = 10$ m/s².)

• $W_{out} = mgh = (30)(10)(1.5) = 450$ J.
• $e = (450/500) \times 100\% = 90\%$.

The other 50 J became waste heat in the motor windings.

Example 2. A ramp with $IMA = 5$ requires 120 N of force to push a 500 N crate up it. What’s the efficiency?

• $AMA = F_{out}/F_{in} = 500/120 \approx 4.17$.
• $e = (AMA/IMA) \times 100\% = (4.17/5) \times 100\% \approx 83\%$.

The 17% of energy “missing” from the ideal calculation went into friction between the crate and the ramp surface.

Efficiency of Common Systems

The car engine row is brake thermal efficiency: the share of the fuel’s chemical energy that arrives at the crankshaft as useful work. Whole-vehicle figures, sometimes quoted as “tank-to-wheels” efficiency, are lower, because they also subtract drivetrain, idling, and accessory losses. Different denominators, so the two are not interchangeable.

For comparison: a Tesla Model 3’s electric motor + battery system runs around 75–85% efficient, vs. the 20–30% of a gasoline engine. That’s the main reason EVs cost less to fuel per mile.

Cascading Efficiency

When multiple machines are connected in series (the output of one feeds the input of the next), the overall efficiency is the product of the individual efficiencies:

$e_{total} = e_1 \times e_2 \times e_3 \times \ldots$

A 90% efficient motor driving an 80% efficient pump gives an overall system efficiency of $0.9 \times 0.8 = 0.72 = 72\%$. Each stage loses a little, and the losses compound — which is why complex machinery (with many conversion stages) tends to be less efficient than direct ones.