Physical quantities
Introduced| English | Meaning |
|---|---|
| physical quantity/ˈfɪzɪkl ˈkwɒntɪti/ | physical quantity |
| magnitude/ˈmæɡnɪtjuːd/ | The size or numerical value of a quantity. |
| estimate/ˈestɪmət/ | To find an approximate value using available information or simplifying assumptions. |
| free fall/friː fɔːl/ | Motion under gravity alone, with other forces neglected. |
| order of magnitude/ˈɔːdə ɒv ˈmæɡnɪtjuːd/ | A scale of size expressed in powers of ten. |
A 125-million-dollar mistake
- In 1999, NASA lost the Mars Climate Orbiter at Mars.
- One team used pounds, the other used newtons — and nobody converted.
- The spacecraft flew too low and burned up.
- It was a unit mistake, not a maths mistake.
Physical quantity lab
Classify measurement examples by the kind of quantity they represent.
What is a physical quantity 物理量?
- In physics we measure things.
- Every measurement is a physical quantity.
It always has two parts
- A quantity = a magnitude 大小 (the number) + a unit.
- The magnitude says how much; the unit says of what.
Every physical quantity has two parts: a number (its ____) and a unit.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The two parts are the magnitude (the number) and the unit.
Match each term to the definition the examiner marks.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Both halves of the first one score. A number with no unit is not a physical quantity, which is exactly how the Mars orbiter was lost.
A number alone means nothing
- "The speed is 90" — 90 what?
- $90\ \dfrac{\text{km}}{\text{h}}$, $90\ \dfrac{\text{m}}{\text{s}}$ and 90 mph are all different.
- Always write the unit too.
Saying "the speed is 90" — with no unit — gives a complete physical quantity.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Without a unit the number is ambiguous. "$90\ \dfrac{\text{km}}{\text{h}}$" is complete; "90" is not.
An answer of "5" to a question asking for a length is complete as long as the working is shown.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
A physical quantity is a magnitude AND a unit. 5 metres, 5 centimetres and 5 kilometres are three different lengths, and the mark scheme asks for the unit.
"$90\ \dfrac{\text{km}}{\text{h}}$" is a complete quantity. "90" on its own is not.
Which statement gives a complete physical quantity?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
A physical quantity needs both a magnitude (the number, 70) and a unit (kg). A number alone, or a unit alone, is not enough.
Estimate 估算 everyday sizes
- You should know roughly how big common things are.
- Learn these standard values by heart.
Select all the estimates that are about right.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
All are right except the speed of sound: in air it is about $340\ \dfrac{\text{m}}{\text{s}}$. $3 \times 10^{8}\ \dfrac{\text{m}}{\text{s}}$ is the speed of light.
Put the method for an order-of-magnitude estimate in order.
Difficulty 3/5 · 200 coins + a small bonus on first completion today
An order-of-magnitude answer is only claiming the right power of ten, so carrying three significant figures through it is wasted work.
How to estimate
- Compare with something you know (a person $\approx 70\ \text{kg}$).
- Round to one figure — a car is about $1000\ \text{kg}$, not $1043$.
- Getting the power of ten right matters most.
Estimate the acceleration of free fall $g$ near the Earth's surface.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Near the Earth's surface $g \approx 9.81\ \dfrac{\text{m}}{\text{s}^2}$ — roughly $10\ \dfrac{\text{m}}{\text{s}^2}$.
A human hair is about 0.1 mm across. What is the order of magnitude of that width in metres, as a power of ten? (Give the exponent, e.g. -3)
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$0.1\ \text{mm} = 1 \times 10^{-4}\ \text{m}$. Convert the prefix before taking the power of ten; treating $0.1\ \text{mm}$ as $0.1$ is the standard slip.
Worked example: the air in your classroom
- Estimate the mass of air in a classroom. Air has a density of about $1.2\ \dfrac{\text{kg}}{\text{m}^{3}}$. Rough sizes: $10\ \text{m}$ long, $8\ \text{m}$ wide, $3\ \text{m}$ high, so the volume is $10 \times 8 \times 3 = 240\ \text{m}^{3}$.
- Mass $= \text{density} \times \text{volume} = 1.2 \times 240 \approx 300\ \text{kg}$: about four adults. The order of magnitude is $10^{2}\ \text{kg}$, and that is all an estimate claims.
A heart beats about 70 times a minute, for a life of about 75 years. Estimate the number of heartbeats in a lifetime and give its order of magnitude as a power of ten. (Give the exponent, e.g. 6)
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$70 \times 60 \times 24 \times 365 \times 75 \approx 3 \times 10^{9}$ beats, so the order of magnitude is $10^{9}$. Step by step: about $10^{5}$ beats a day, $4 \times 10^{7}$ a year, $3 \times 10^{9}$ in 75 years.
Order of magnitude 数量级
- A good estimate has the right power of ten.
- $70\ \text{kg}$ ✓ · $7\ \text{kg}$ ✗
Put these masses in order, smallest first.
Difficulty 3/5 · 200 coins + a small bonus on first completion today
Ordering by size checks your sense of order of magnitude: $0.1\ \text{kg}$, then $1\ \text{kg}$, then $70\ \text{kg}$.
The scale of the universe
- The same idea runs from an atom to the Sun.
- Each step along the line is ten times bigger.
You've got it
- a physical quantity = a magnitude + a unit
- learn the standard estimates (human, apple, sound, light, $g$)
- a good estimate has the right order of magnitude
Next — the SI units these are all built from.