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IMAT Physics · Formula sheet

Measurement and Units formulas

3 formulas, 2 reference tables and 8 common traps for IMAT Physics Measurement and Units, grouped by subtopic.

Full notes with worked examples

SI Units, Prefixes and Significant Figures

Learn this subtopic in the notes

Prefixes, powers of ten and scientific notation

Changing a prefix

1 nm=10−9 m=10−3 μm=103 pm1\ \text{nm} = 10^{-9}\ \text{m} = 10^{-3}\ \mu\text{m} = 10^{3}\ \text{pm}
  • nm\text{nm}nanometre, 10−910^{-9} m
  • μm\mu\text{m}micrometre, 10−610^{-6} m
  • pm\text{pm}picometre, 10−1210^{-12} m

SI base units and the common derived units

QuantityUnitSymbolIn SI base units
Lengthmetrembase unit
Masskilogramkgbase unit
Timesecondsbase unit
Electric currentampereAbase unit
TemperaturekelvinKbase unit
Amount of substancemolemolbase unit
ForcenewtonNkg m s−2\text{kg m s}^{-2}
Energy, workjouleJ = N mkg m2 s−2\text{kg m}^2\text{ s}^{-2}
PowerwattW = J/skg m2 s−3\text{kg m}^2\text{ s}^{-3}
PressurepascalPa = N/m²kg m−1 s−2\text{kg m}^{-1}\text{ s}^{-2}
ChargecoulombC = A sA s\text{A s}
Potential differencevoltV = J/Ckg m2 s−3 A−1\text{kg m}^2\text{ s}^{-3}\text{ A}^{-1}
ResistanceohmΩ = V/Akg m2 s−3 A−2\text{kg m}^2\text{ s}^{-3}\text{ A}^{-2}
FrequencyhertzHzs−1\text{s}^{-1}
The last column is what you reach when you break a derived unit all the way down.

Common traps

The base unit of mass is the kilogram

The kilogram is the only base unit with a prefix in its name. A density in g/cm³ or an energy worked out with a mass in grams is not in SI, and an answer computed that way is out by a factor of 1000.

A prefix on a squared or cubed unit is raised to that power

1 cm31\ \text{cm}^3 is 10−6 m310^{-6}\ \text{m}^3, not 10−2 m310^{-2}\ \text{m}^3, and 1 mm21\ \text{mm}^2 is 10−6 m210^{-6}\ \text{m}^2, not 10−3 m210^{-3}\ \text{m}^2. Convert the length first, then square or cube the whole factor.

Milli and micro are a thousand apart

milli (m) is 10−310^{-3} and micro (μ\mu) is 10−610^{-6}. A list of sizes in mm, μ\mum, nm and pm is easiest to order after rewriting every value with the same prefix.

Leading zeros are not significant

In 0.0072 the zeros only show where the decimal point is; the value has 2 significant figures. Writing it as 7.2×10−37.2 \times 10^{-3} makes this obvious.

Dimensional Analysis, Scalars and Vectors

Learn this subtopic in the notes

Dimensional analysis: checking units in an equation

Working a unit down to base units

J=N m=(kg m s−2)(m)=kg m2 s−2\text{J} = \text{N m} = (\text{kg m s}^{-2})(\text{m}) = \text{kg m}^2\,\text{s}^{-2}
  • J\text{J}joule, the unit of energy and work
  • N\text{N}newton, the unit of force

Adding vectors and splitting them into components

Components and the resultant

Fx=Fcos⁡θ,Fy=Fsin⁡θ,R=Fx2+Fy2F_x = F\cos\theta, \quad F_y = F\sin\theta, \quad R = \sqrt{F_x^2 + F_y^2}
  • θ\thetaangle between the vector and the horizontal
  • Fx,FyF_x, F_yhorizontal and vertical components
  • RRsize of the resultant

Scalars and vectors: which quantities have a direction

QuantityScalar or vectorNote
Distancescalarlength of the path travelled
Displacementvectorstraight-line change of position
Speedscalardistance per unit time
Velocityvectordisplacement per unit time
Accelerationvectorchange of velocity per unit time
Massscalaramount of matter
Weight, any forcevectorweight points towards the centre of the Earth
Momentumvectorsame direction as the velocity
Energy, work, powerscalarno direction, even though forces cause them
Pressurescalarthe force it makes on a surface is a vector
Temperaturescalara minus sign is not a direction
Electric chargescalarpositive or negative, never a direction
Electric fieldvectordirection of the force on a positive charge
Electric currentscalarcurrents at a junction add as numbers

Common traps

Consistent units do not prove an equation is right

s=ut+at2s = ut + at^2 has consistent units but is wrong: the factor 12\tfrac{1}{2} is missing. A units check can rule an equation out, but it can never confirm a numerical factor.

A resistance and its reciprocal cannot be added

1/R1/R is measured in Ω−1\Omega^{-1}, so an expression like R1+1/R2R_1 + 1/R_2 mixes units and cannot be right. The same goes for any sum that mixes, say, an energy and a power.

Mass is a scalar, weight is a vector

Weight is the gravitational force on a mass, so it has a direction (down). Mass has none. An option listing weight as a scalar confuses the two.

Vector sizes add only when the vectors are parallel

3 N and 4 N give 7 N only if they point the same way. At right angles they give 5 N, and in opposite directions 1 N. The resultant is always between the difference and the sum of the sizes.

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