Measuring very large and very small lengths, masses and times

A ruler will not do

A ruler is fine for a book. But how would you measure the height of a mountain, the distance to the Moon, or the size of an atom? You cannot lay a ruler on any of them.

So physicists measure most lengths indirectly: they measure something easy, like an angle or a volume, and calculate the length.

From the very small to the very large

Lengths in nature cover more than $40$ powers of ten. A proton is about $10^{-15}\ \text{m}$ across, an atom about $10^{-10}\ \text{m}$, you about $1\ \text{m}$, the Earth about $10^{7}\ \text{m}$, and the part of the universe we can see about $10^{26}\ \text{m}$. Masses and times have the same huge spread.

Large distances: parallax

Hold up a finger and look at it with your left eye, then with your right. It jumps against the background. That jump is parallax.

Two observers stand a known distance $b$ apart: the baseline. Each looks at the same far object. The two lines of sight meet at the object at a small angle $\theta$, the parallax angle. For a small angle, the arc of the baseline is almost straight, so

Try it with a tree.

You should find that a far tree gives a tiny angle, and a longer baseline gives a bigger angle that is easier to measure. That is why astronomers use the widest baseline they have: the Earth's orbit, $3 \times 10^{11}\ \text{m}$ across, measured six months apart.

Watch out. $\theta$ must be in radians. Change degrees with $\times \dfrac{\pi}{180}$, and seconds of arc with $\times \dfrac{1}{3600} \times \dfrac{\pi}{180}$.

A worked example

Two friends stand $50\ \text{m}$ apart on a straight road. The lines of sight to a water tower meet at $1.2^\circ$. How far is the tower?

1. Angle in radians. $1.2^\circ = 1.2 \times \dfrac{\pi}{180}\ \text{rad} = 0.020\,94\ \text{rad}$.
2. Distance. $D = \dfrac{50\ \text{m}}{0.020\,94\ \text{rad}} \approx 2390\ \text{m}$, about $2.4\ \text{km}$.

Think first. $1.2^\circ$ is small but not tiny, so the tower should be about $50$ times $\dfrac{180}{\pi \times 1.2} \approx 48$ baselines away. Yes, about $2.4\ \text{km}$.

Very small sizes: one molecule thick

A drop of oil spreads on water into a very thin film. Oleic acid spreads until the film is one molecule thick. So:

The drop must be tiny, so the oil is first mixed with alcohol, say $1$ part in $500$. The alcohol dissolves in the water and leaves only the oil. The answer comes out near $10^{-9}\ \text{m}$, a nanometre.

Masses

A kitchen scale weighs grams; a lorry scale weighs tonnes. For the very large and the very small we again calculate:

• The mass of the Earth comes from gravity: how fast things fall and how the Moon moves.
• The mass of an atom is measured in atomic mass units: $1\ \text{u}$ is $\dfrac{1}{12}$ of the mass of a carbon-12 atom, $1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$. A water molecule is about $18\ \text{u}$.

Times

A pendulum or a quartz watch is good for everyday times. The most accurate clocks are atomic clocks, which count the vibrations of caesium atoms. They lose less than a second in millions of years. Very short times, like the life of an unstable particle, are found from how far the particle travels before it decays.

Practise with new numbers

Try it in Python

The program works out the distance of the water tower from the worked example.

Remember these

• Most lengths are measured indirectly: measure something easy, then calculate.
• Parallax: $D = \dfrac{b}{\theta}$, with $\theta$ in radians. A longer baseline gives a bigger, easier angle.
• An oil film one molecule thick: size $=$ volume of oil $\div$ area, about $10^{-9}\ \text{m}$.
• $1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$; atomic clocks keep the second.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A distance gives a parallax of $0.01\ \text{rad}$ with a $3\ \text{m}$ baseline. How far is it?

2. Why must the oil be mixed with alcohol first?

3. A water molecule is about $18\ \text{u}$. What is its mass in kilograms?

4. Why do astronomers measure a star's position six months apart?