Lab skills

Measurement

Build the measurement language for lab work: SI units, metric prefixes, and scientific notation.

Uncertainty

Explore the concept of measurement uncertainty and how it affects the reliability of experimental results.

Measurement

Most physical quantities require units alongside a numerical value. $67\ \text{meters}$ is a length, $67\ \text{seconds}$ is a time, and $67\ \text{kilograms}$ is a mass. The number $67$ (while it does have memetic meaning) does not have any physical meaning without a unit.

A measurement has two parts:

\text{measurement} = \text{number} \times \text{unit}.

The number tells how many. The unit tells how many of what. It is extremely useful, as notated above, to think of this as multiplication of the two. As you perform arithmetic/algebra with measurements, the same operations apply to both the numerical values and corresponding units.

For example, if a car travels $120\ \text{miles}$ in $2\ \text{hours}$ , then its average speed is

v = \frac{120\ \text{miles}}{2\ \text{hours}} = 60\ \text{mi/hr}.

As expected, miles per hour is a standard way of expressing speed. If our work results in units of square miles for a speed, there was an error in our reasoning. Checking units is often an easy way of spotting mistakes.

SI Units

The SI system (Système international d’unités) is the shared measuring language used across physics. There are relatively few base units, and most units are just combinations of these.

Click a row to see how the modern SI defines that unit.

Quantity	SI unit	Symbol

meter (m)

BIPM SI base units

Modern definition: Fixed by setting the speed of light in vacuum to exactly 299,792,458 meters per second; one meter is the path light travels in 1/299,792,458 of a second.
In everyday terms: Most doors are about 2 meters high.

Derived units are combinations of these base units: speed is measured in $\text{m/s}$ and acceleration in $\text{m/s}^2$ . Some combinations are given a special name: force is measured in newtons $\text{N} = \text{kg}\cdot\text{m/s}^2$ , energy in joules $\text{J} = \text{kg}\cdot\text{m}^2/\text{s}^2$ , and momentum (among a small group of people) in langs $\text{L} = \text{kg}\cdot\text{m/s}$ . The special names are convenient, but the base-unit structure often hints at the deeper relationships between quantities and how we define them.

Powers of 10

Metric prefixes are shorthand powers of ten attached to a unit. A millimeter is not a new kind of length; it is $10^{-3}$ meters. A kilometer is $10^3$ meters. Replace the prefix with its power of ten, and the conversion becomes ordinary multiplication.

Click a prefix to see how its power of ten shows up across different SI units.

Prefix	Symbol	Power

milli (m) = 10^-3

Multiply the unit by one thousandth.

250 ms = 250 x 10^-3 s = 0.250 s
A quick human reaction time is often around 250 ms.
20 mA = 20 x 10^-3 A = 0.020 A
A small indicator LED often runs on about 10 to 20 mA.
15 mK = 15 x 10^-3 K = 0.015 K
Climate and lab sensors can resolve temperature changes of only a few millikelvin.

Scientific Notation

Scientific notation writes a number as a coefficient times a power of ten:

a \times 10^n, \qquad 1 \le |a| < 10.

This notation handles both Earth’s radius, about $6.4 \times 10^6\ \text{m}$ , and a visible-light wavelength, about $5.0 \times 10^{-7}\ \text{m}$ , without long strings of zeros.

This also happens to be a convenient way of expressing the precision of a measurement. If a length is reported as

3.10 \times 10^2\ \text{m},

the final digit implies the reliablity of the measurement (i.e. it’s probably between 305 and 315 meters). The numbers $3$ , $1$ , and the final $0$ in the coefficient are said to be significant figures.

Problem Solving

Convert a small length

Problem

A wire is $680\ \mu\text{m}$ thick. Write its thickness in meters and in scientific notation.

Use

Micro means $10^{-6}$ :

680\ \mu\text{m} = 680 \times 10^{-6}\ \text{m}.

Solve

680 \times 10^{-6}\ \text{m} = 0.000680\ \text{m} = 6.80 \times 10^{-4}\ \text{m}.

Check

Micrometers are much smaller than meters, so the meter value should be a small decimal. The trailing zero in $6.80$ preserves the three significant figures from $680$ .

Compare two distances

Problem

Which is larger: $3.2 \times 10^5\ \text{m}$ or $7.5 \times 10^4\ \text{m}$ ?

Solve

Compare powers first. Since $10^5$ is ten times larger than $10^4$ , rewrite the second value with the same power:

7.5 \times 10^4\ \text{m} = 0.75 \times 10^5\ \text{m}.

Now compare coefficients:

3.2 \times 10^5\ \text{m} > 0.75 \times 10^5\ \text{m}.

Check

In ordinary notation these are $320{,}000\ \text{m}$ and $75{,}000\ \text{m}$ , so the first distance is larger.

Report a measured result

Problem

A distance calculation gives $0.004782\ \text{m}$ with an uncertainty of $0.00031\ \text{m}$ . Report the result in scientific notation.

Use

Round the uncertainty to one significant figure, then match the value to that place:

0.00031\ \text{m} \rightarrow 0.0003\ \text{m}.

Solve

The value should stop in the same decimal place:

0.004782\ \text{m} \rightarrow 0.0048\ \text{m}.

Now factor out $10^{-3}\ \text{m}$ from both parts:

0.0048 \pm 0.0003\ \text{m} = (4.8 \pm 0.3) \times 10^{-3}\ \text{m}.

Check

The uncertainty reaches the tenths place in the coefficient, so the measured value also stops at the tenths place: $4.8 \pm 0.3$ .

Units & Notation Checkpoint

Question 1 of 4

What does the unit in $12\ \text{m/s}$ tell you?

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