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Physics (AS & A Level) 9702 · Chapter 1

Physical quantities and units

Magnitudes and units, reasonable estimates, SI base units and prefixes, checking equations for homogeneity, systematic and random errors, precision and accuracy, combining uncertainties, and adding and resolving vectors.

7 min read

1.1 Physical quantities

Every physical quantity is a numerical magnitude and a unit. "The length is 3" means nothing; "the length is 3 m" does.

Reasonable estimates

You are expected to know roughly how big things are, so that you can spot an answer that cannot be right.

QuantityRough value
Mass of an adult70 kg
Mass of a car1000 kg
Walking speed1.5 m/s
Speed of sound in air330 m/s
Speed of light3×1083 \times 10^{8} m/s
Atmospheric pressure1×1051 \times 10^{5} Pa
Diameter of an atom10−1010^{-10} m
Wavelength of visible light5×10−75 \times 10^{-7} m
Power of a domestic kettle2 kW
Mains voltage230 V (in many countries)

1.2 SI units

Base quantities

QuantityUnitSymbol
Masskilogramkg
Lengthmetrem
Timeseconds
CurrentampereA
TemperaturekelvinK

(Amount of substance, the mole, is added in chapter 15.)

Derived units

Every other unit is a product or quotient of base units. Build them from a defining equation.

UnitFromIn base units
newton, NF=maF = makg m s⁻²
joule, JW=FsW = Fskg m² s⁻²
watt, WP=W/tP = W/tkg m² s⁻³
pascal, Pap=F/Ap = F/Akg m⁻¹ s⁻²
coulomb, CQ=ItQ = ItA s
volt, VV=W/QV = W/Qkg m² s⁻³ A⁻¹
ohm, ΩR=V/IR = V/Ikg m² s⁻³ A⁻²

Homogeneity

An equation is homogeneous if every term has the same base units. A correct equation must be homogeneous, but a homogeneous equation need not be correct: numbers such as the 12\tfrac{1}{2} in Ek=12mv2E_k = \tfrac{1}{2}mv^2 have no units, so this test cannot detect them.

Prefixes

PrefixSymbolFactor
teraT101210^{12}
gigaG10910^{9}
megaM10610^{6}
kilok10310^{3}
decid10−110^{-1}
centic10−210^{-2}
millim10−310^{-3}
microμ10−610^{-6}
nanon10−910^{-9}
picop10−1210^{-12}

Be careful with powers: 1 cm2=10−4 m21\ \text{cm}^2 = 10^{-4}\ \text{m}^2 and 1 mm3=10−9 m31\ \text{mm}^3 = 10^{-9}\ \text{m}^3.

1.3 Errors and uncertainties

Systematic and random errors

  • A systematic error shifts every reading the same way by the same amount or proportion. A zero error on a micrometer is the classic example. Repeating readings does not reduce it. Remove it by correcting the instrument or the readings.
  • A random error scatters readings either side of the true value, for example reaction time with a stopwatch. Repeating and averaging does reduce its effect.

Target diagrams showing accurate and precise, precise but not accurate, and accurate but not precise readings

Target diagrams showing accurate and precise, precise but not accurate, and accurate but not precise readings

Precision and accuracy

  • Accuracy is how close a result is to the true value.
  • Precision is how close repeated readings are to each other (small scatter).

A set of readings can be precise but inaccurate: tightly grouped, all with the same systematic error.

Combining uncertainties

CalculationWhat to add
y=a+by = a + b or y=a−by = a - babsolute uncertainties
y=aby = ab or y=a/by = a/bpercentage uncertainties
y=any = a^nnn times the percentage uncertainty in aa

Uncertainties always add, even when the quantities are subtracted.

1.4 Scalars and vectors

A scalar has magnitude only: distance, speed, mass, time, energy, temperature, charge. A vector has magnitude and direction: displacement, velocity, acceleration, force, momentum, field strength.

Adding vectors

Draw the vectors head to tail. The resultant goes from the start of the first to the end of the last. To subtract a vector, reverse it and add it. For two vectors at right angles, use Pythagoras and trigonometry; otherwise a scale drawing works.

Two vectors added head to tail to give a resultant, and a force resolved into horizontal and vertical components

Two vectors added head to tail to give a resultant, and a force resolved into horizontal and vertical components

Components

A vector FF at angle θ\theta to the horizontal has two perpendicular components:

Fx=Fcos⁡θFy=Fsin⁡θF_x = F\cos\theta \qquad F_y = F\sin\theta

The components act together in exactly the same way as the single vector. Problems become easy when you resolve everything along two perpendicular directions and treat each direction separately.

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Every past-paper question on these topics, with the mark scheme.