SI units and why they matter
Physics rests on seven base units, and at this level three of them do nearly all the work: the metre for length, the kilogram for mass and the second for time. Everything else is built from these. A newton is a kilogram metre per second squared; a joule is a newton metre.
Working in base units is not bureaucratic fussiness. Nearly every equation you will use assumes them, so a length left in centimetres or a time left in minutes produces an answer that is wrong by a factor of a hundred or sixty — large enough to be nonsense, but not always large enough to look obviously wrong.
Prefixes let us write very large and very small quantities without strings of zeros. Learn the common ones by their powers of ten rather than by their names, because the power is what you actually substitute.
| Prefix | Symbol | Multiplier |
|---|---|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
The area trap
A centimetre is 10⁻² m, so a square centimetre is 10⁻⁴ m² and a cubic centimetre is 10⁻⁶ m³. The prefix gets raised to the same power as the unit. Forgetting this is the single most common conversion error in the whole course.
Choosing the right instrument
Every measuring instrument has a precision — the smallest division it can read. Choosing an instrument means matching that precision to the size of what you are measuring.
A metre rule reads to the nearest millimetre. Measuring a 2 m corridor with it is fine, since 1 mm out of 2000 is negligible. Measuring the thickness of a coin with it is useless, because the uncertainty is a large fraction of the answer.
For small lengths, vernier callipers read to 0.1 mm and a micrometer screw gauge to 0.01 mm. For time, a stopwatch reads to 0.01 s, but human reaction time of roughly 0.2 s swamps that — which is why timing a single pendulum swing is poor practice and timing twenty and dividing is good practice.
| Instrument | Reads to | Suitable for |
|---|---|---|
| Metre rule | 1 mm | lengths from centimetres to metres |
| Vernier callipers | 0.1 mm | diameter of a rod, internal width of a tube |
| Micrometer | 0.01 mm | thickness of a wire or sheet |
| Stopwatch | 0.01 s | intervals of seconds or more |
| Measuring cylinder | 1 cm³ | volume of a liquid or irregular solid |
The two sliders separate the two ideas. Systematic error shifts the whole cluster off the true value — accurate no longer, however tight the grouping. Random error spreads the cluster out, so the mean can still be right while no single reading is.
Accuracy, precision and error
Accuracy — How close a measurement is to the true value.
Accuracy and precision are different things, and an instrument can have one without the other. Accuracy is closeness to the true value. Precision is how tightly repeated readings agree with each other.
A balance that reads every mass 5 g too high is precise but not accurate: repeat the measurement and you get the same answer every time, and it is wrong every time. A worn stopwatch operated by a distracted student may be accurate on average but not precise, because the readings scatter either side of the truth.
The two failures come from two kinds of error. A systematic error shifts every reading in the same direction — a zero error on an instrument is the classic case, and it can be corrected once found. A random error scatters readings either side, and it is reduced by taking repeats and averaging.
Key points
- Accuracy is closeness to the truth; precision is closeness to each other.
- Systematic errors shift everything one way and can be corrected.
- Random errors scatter both ways and are reduced by averaging repeats.
- Check for a zero error before you start, and subtract it if there is one.
- Repeating cannot fix a systematic error, however many times you do it.
Measuring density
Density — The mass per unit volume of a substance, ρ = m/V. Measured in kg m⁻³ or g cm⁻³.
Density is a property of the material, not of the object. A steel nail and a steel girder have the same density; they differ only in how much of the material there is.
For a regular solid the method is direct: measure the mass on a balance, calculate the volume from the dimensions, and divide. For a liquid, find the mass of an empty measuring cylinder, add the liquid, weigh again, and subtract.
For an irregular solid the volume comes from displacement. Lower the object into a measuring cylinder of water and record the rise in level, or use a displacement can and collect the overflow. The volume of water moved is exactly the volume of the object.
Whether an object floats is decided by density, not by weight. A steel ship floats because its overall density — steel plus the air inside the hull — is less than that of water, even though a solid lump of the same steel would sink.
- ρ
- densitykg m⁻³
- m
- masskg
- V
- volumem³
A stone of mass 78 g is lowered into a measuring cylinder containing 50 cm³ of water. The level rises to 80 cm³. Calculate the density of the stone in g cm⁻³ and in kg m⁻³.
- Volume of stone
= 80 − 50 = 30 cm³.The rise in level is the volume displaced. ρ = m/V = 78 / 30.ρ = 2.6 g cm⁻³.Denser than water, so it sinks — consistent with the method working.- To convert, multiply by 1000:
1 g cm⁻³ = 1000 kg m⁻³.A gram is 10⁻³ kg and a cm³ is 10⁻⁶ m³, giving a factor of 10³. ρ = 2600 kg m⁻³.
2.6 g cm⁻³ = 2600 kg m⁻³