Problem — Ranking orders of magnitude

Problem

Ranking orders of magnitude. Order from smallest to largest: (A) the mass of a human being; (B) the mass of an apple; (C) the mass of a mosquito; (D) the mass of a blue whale. For each, state the order of magnitude in $\text{kg}$.

Solution

Step 1 — Estimating each order of magnitude. We reason about each object separately, keeping only the power of ten.

• (C) Mosquito. A mosquito weighs a few milligrams: $m_C \approx 2{,}5\;\text{mg} = 2{,}5\cdot 10^{-6}\;\text{kg}$, order $10^{-6}\;\text{kg}$.
• (B) Apple. An apple fits in one hand and weighs about a hundred grams: $m_B \approx 0{,}1\;\text{kg}$, order $10^{-1}\;\text{kg}$.
• (A) Human being. An adult weighs a few tens of kg: $m_A \approx 70\;\text{kg}$, order $10^{2}\;\text{kg}$.
• (D) Blue whale. The largest living animal: $m_D \approx 1{,}5\cdot 10^{5}\;\text{kg}$ (about $150$ tonnes), order $10^{5}\;\text{kg}$.

Step 2 — Comparison. Arranging the exponents in ascending order, $-6 < -1 < 2 < 5$, we get

$\ev{\text{C} < \text{B} < \text{A} < \text{D}}$

Between the mosquito and the blue whale there are about $11$ orders of magnitude: the whale is more massive than a mosquito by a factor of $\sim 10^{11}$, i.e. a hundred billion times.

Linked notes

Topics: Grandezze fisiche e misura Concepts: Notazione scientifica e ordini di grandezza Skills: Stime di Fermi