Theory Exercises

Measurement in science requires correct units, conversion methods, and consistent notation.

1. SI Base Magnitudes (common in 2 ESO)

MagnitudeSI unitSymbol
Lengthmeterm
Masskilogramkg
Timeseconds
TemperaturekelvinK
Electric currentampereA

2. Metric Prefixes

PrefixSymbolFactor
gigaG\(10^9\)
megaM\(10^6\)
kilok\(10^3\)
hectoh\(10^2\)
decada\(10^1\)
unit-\(10^0\)
decid\(10^{-1}\)
centic\(10^{-2}\)
millim\(10^{-3}\)
micromu\(10^{-6}\)
nanon\(10^{-9}\)

3. Scientific Notation

Any number can be written as:

\[a \times 10^b\]

with \(1 \le a < 10\) and integer \(b\).

Examples:

  • \(4200000 = 4.2 \times 10^6\)
  • \(0.00056 = 5.6 \times 10^{-4}\)

4. Unit Conversion by Conversion Factors

Multiply by fractions equal to 1 so units cancel.

Example:

\[180\,\mathrm{km/h} \times \frac{1000\,\mathrm{m}}{1\,\mathrm{km}} \times \frac{1\,\mathrm{h}}{3600\,\mathrm{s}} = 50\,\mathrm{m/s}\]

Useful equalities:

  • \(1\,\mathrm{L} = 1\,\mathrm{dm^3}\)
  • \(1\,\mathrm{mL} = 1\,\mathrm{cm^3}\)
  • \(1\,\mathrm{m^3} = 1000\,\mathrm{L}\)

5. Areas and Volumes in Conversions

When converting area and volume, the conversion factor is squared or cubed:

\[1\,\mathrm{m^2} = (100\,\mathrm{cm})^2 = 10^4\,\mathrm{cm^2}\]
\[1\,\mathrm{m^3} = (100\,\mathrm{cm})^3 = 10^6\,\mathrm{cm^3}\]

6. Geometry Formulas for Volume

ObjectVolume
Cube\(V=a^3\)
Rectangular prism\(V=lwh\)
Cylinder\(V=\pi r^2 h\)
Sphere\(V=\frac{4}{3}\pi r^3\)
Cone\(V=\frac{1}{3}\pi r^2 h\)

7. Density and Temperature

Density:

\[\rho = \frac{m}{V}\]

Temperature conversion:

\[T(\mathrm{K}) = T(^\circ\mathrm{C}) + 273.15\]
\[T(^\circ\mathrm{C}) = T(\mathrm{K}) - 273.15\]

8. Good Measurement Practice

  • Always include units in final answers.
  • Keep significant figures consistent with data precision.
  • Convert to SI before applying formulas when possible.
  • Check if the final value is physically reasonable.