Foundations — Efficiency

Before you can use the parent note Efficiency, you need every symbol it throws at you to feel obvious. This page builds each one from absolutely nothing — a picture, a plain-words meaning, and why the topic needs it — in an order where each idea rests on the one before.

0. Energy — the thing that flows

Picture a bucket of water. The amount of water = the amount of energy. You can pour it, split it, move it around — but (this is the key rule below) you can never create extra water out of nothing.

Look at the bucket figure: the level of water is $E$. Two buckets side by side let us compare — that comparison is the whole game.

1. The subscript notation — tagging different piles of energy

Before we split energy into pieces, we need a way to name each piece. Physicists attach a small lowered tag.

We now have the tools ($E$, and the subscript trick) to write the most important rule of the whole topic.

2. Conservation of energy — the "no water vanishes" rule

Picture the leaky bucket from the parent note. You pour water in the top. It comes out of two holes: the useful spout at the bottom, and leaks in the sides. The water in the sides didn't disappear — it's on the floor. Add up spout-water + floor-water and you get exactly what you poured in. Using our subscript tags:

$\underset{\text{poured in}}{\underbrace{E_{\text{in}}}}=\underset{\text{spout}}{\underbrace{E_{\text{useful}}}}+\underset{\text{leaks}}{\underbrace{E_{\text{wasted}}}}$

See Conservation of Energy for the full law.

In this figure the total poured-in arrow (magenta) splits into a useful arrow (orange) and a wasted arrow (violet). They always add back to the input — that is conservation drawn as arrows.

3. Useful vs wasted energy — the split that matters

See Heat and Internal Energy (waste usually becomes heat) and Friction (a top cause of the leak).

4. The ratio and the fraction bar — comparing two piles

Picture two buckets. The useful bucket holds 300 units, the input bucket holds 500. The fraction $\frac{300}{500}$ measures how full the useful bucket is compared to what you paid for.

$\frac{E_{\text{useful}}}{E_{\text{in}}}=\frac{300}{500}=0.6$

5. $\eta$ — the Greek letter "eta"

6. Dimensionless numbers and percentages — the 0-to-1 scale

Picture a ruler marked 0 at one end and 1 at the other. Every real machine's $\eta$ is a mark somewhere between 0 and 1 — never past the 1. Below we'll see why the 1 is a wall.

The number line figure: 0 = totally useless (all leaks), 1 = perfect (impossible), and real machines live in the shaded middle. The red wall at 1 is conservation of energy saying "no further".

7. Power $P$ and time $t$ — the "per second" version

Picture the bucket again but now watch the flow rate — litres per second through the spout — instead of the total litres.

8. $mgh$ — where "useful output" often comes from

Many efficiency problems lift something, and the useful energy is the height gained. But why is that energy exactly $mgh$? Let us build it, not just quote it.

Now assemble the picture: to raise the box steadily you push up with force equal to its weight $mg$, through the height $h$ you lift it. So the energy spent is force $\times$ distance:

$E=\underset{\text{force you apply}}{\underbrace{(mg)}}\times \underset{\text{distance lifted}}{\underbrace{h}}=mgh$

In the figure the orange arrow is the upward push (size $mg$), the box travels the height $h$, and the shaded bucket beside it fills to $mgh$ — the useful "height energy" you bought.

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, reread that section.

Connections

• Efficiency — the parent topic these foundations feed.
• Conservation of Energy — the law that makes $\eta \le 1$.
• Work and Power — source of the power form and of force $\times$ distance.
• Gravitational Potential Energy — where useful $mgh$ comes from.
• Heat and Internal Energy — where wasted energy usually ends up.
• Friction — a chief cause of the leak.
• Heat Engines & Carnot Efficiency — the deeper thermodynamic limit.