Efficiency

WHY do we even need "efficiency"?

WHAT is efficiency? (Definition)

HOW do we derive it from first principles?

Formula Building

$\eta$ from energy conservation Start with conservation of energy for any machine: $E_{\text{in}} = E_{\text{useful}} + E_{\text{wasted}}$ Why this step? Energy cannot vanish; everything that enters must come out either as useful work or as waste (heat, sound, light you didn't want).

Divide both sides by $E_{\text{in}}$ : $1 = \frac{E_{\text{useful}}}{E_{\text{in}}} + \frac{E_{\text{wasted}}}{E_{\text{in}}}$ Why this step? Normalising by the input turns absolute joules into fractions — easier to compare machines of different sizes.

Define the first term as efficiency: $\boxed{\eta = \frac{E_{\text{useful}}}{E_{\text{in}}} = 1 - \frac{E_{\text{wasted}}}{E_{\text{in}}}}$ Why this matters: This shows $\eta = 1$ (100%) ONLY if $E_{\text{wasted}}=0$ , which never happens in reality. So $\boxed{\eta < 1}$ always for real machines.

Power version

Since power $P = \dfrac{E}{t}$ and both input and output share the same time $t$ : $\eta = \frac{E_{\text{useful}}/t}{E_{\text{in}}/t} = \frac{P_{\text{useful out}}}{P_{\text{in}}}$ Why this step? The $t$ cancels, so you can use energies OR powers — whichever the question gives you.

Worked Examples

Common Mistakes (Steel-man + Fix)

Flashcards

What is efficiency in words?

The fraction of total input energy (or power) that comes out as useful energy (or power).

Write the efficiency formula.

\eta = E_{\text{useful out}}/E_{\text{total in}} = P_{\text{useful out}}/P_{\text{in}}

Why can efficiency never exceed 100%?

Because energy is conserved — you cannot get more useful energy out than you put in.

What happens to the "wasted" energy?

It is still conserved, but converted to unwanted forms (heat, sound, vibration), not destroyed.

For machines in series, how do efficiencies combine?

They multiply:

\eta_{\text{total}} = \eta_1 \times \eta_2 \times \dots

A motor uses 500 J and gives 300 J useful. Efficiency?

300/500 = 60\%

Express efficiency in terms of wasted energy.

\eta = 1 - E_{\text{wasted}}/E_{\text{in}}

Is efficiency dimensioned or dimensionless?

Dimensionless (a ratio); often shown as a %.

Why does a lever NOT have efficiency > 100%?

It multiplies force not energy; you move a greater distance, so energy out ≤ energy in.

Recall Feynman: explain to a 12-year-old

Imagine you have £1 of pocket money (the energy you put in). You want sweets, but the shop charges you a "service fee" you can't avoid — maybe 30p. You only get 70p of sweets. Efficiency is "how much of your money turned into sweets" — here 70%. No shop can ever give you more sweets than your money is worth, so efficiency can't beat 100%. The lost 30p didn't vanish — the shopkeeper has it (it became heat and noise in the machine!).

Connections

Conservation of Energy — efficiency is just energy conservation rewritten as a ratio.
Work and Power — the power form $\eta = P_{out}/P_{in}$ comes straight from $P=E/t$ .
Gravitational Potential Energy — useful output is often $mgh$ .
Heat and Internal Energy — wasted energy almost always becomes heat.
Heat Engines & Carnot Efficiency — the thermodynamic limit on $\eta$ for engines.
Friction — a major real-world cause of wasted energy.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho yaar, efficiency ka matlab simple hai: jitni energy tum machine mein daalte ho (input), uska kitna hissa kaam ka (useful output) ban kar nikla. Baaki sab heat, sound aur friction mein waste ho jaata hai. Formula hai $\eta = \frac{\text{useful output}}{\text{total input}}$ . Yeh ek pure ratio hai, isliye usually percentage mein likhte hain. Yaad rakho — efficiency kabhi bhi 100% se zyada nahi ho sakti, kyunki energy conservation kehta hai ki jitna andar gaya utna hi (ya usse kam) bahar aa sakta hai.

Ab "wasted energy gayab ho gayi" yeh galat soch hai. Energy kabhi destroy nahi hoti — bas form badal jaati hai. Jaise motor garam ho jaata hai, ya machine awaaz karti hai — woh sab "waste" energy hi hai jo tumhe chahiye nahi thi. Isliye good machine thandi rehti hai, ghatiya machine garam hoti hai.

Ek important trick: agar do machines series mein lagi hain (ek ka output doosre ka input), toh unki efficiencies multiply hoti hain. Maan lo A ki 80% aur B ki 50% hai, toh overall $0.8 \times 0.5 = 0.4 = 40\%$ . Hamesha sabse chhoti stage se bhi kam aata hai — isliye chaining mein efficiency girti chali jaati hai. Bas itna dhyaan rakhna ki "useful" output sirf wahi hai jo tumne chaha tha, total output nahi. Exam mein yahi galti sabse zyada hoti hai!

WHY do we even need "efficiency"?

WHAT is efficiency? (Definition)

HOW do we derive it from first principles?

Power version

Worked Examples

Common Mistakes (Steel-man + Fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Work, Energy & Power

Connections