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Physical quantities and units

A-Level Physics 9702 · Topic 1 revision notes · AS

The language of the subject: SI units, homogeneity, estimates, errors and uncertainties, and adding and resolving vectors.

16 definitions to learn word for word, 12 equations in their own column, 6 diagrams and graphs. Read it, cover it, test yourself.

  1. 1.1 Physical quantities
  2. 1.2 SI units
  3. 1.3 Errors and uncertainties
  4. 1.4 Scalars and vectors
Definitions (mark-scheme wording)EquationsExam traps

1.1Physical quantities

Definitions

Physical quantity
a quantity that can be measured, stated as a numerical magnitude and a unit.
Estimate
an approximate value, found by reasoning, correct to the nearest order of magnitude.
Order of magnitude
the power of ten nearest to the value of a quantity.

Key points

  • Every quantity = number × unit; a bare number answers no physics question.
  • Useful estimates: adult mass ~70 kg, walking speed ~1.5 m s-1, door height ~2 m.
  • More: atom ~10-10 m, nucleus ~10-15 m, room air mass ~50 kg, domestic lamp ~10-60 W.
  • Give estimates to 1 significant figure; the power of ten is what matters.
  • Check every final answer for a sensible size before moving on.
length / m 10⁻¹⁵ 10⁻¹⁰ 10⁻⁵ 10⁰ 10⁵ nucleus atom cell person Earth 1 sig. fig. + the right power of ten
Orders of magnitude of length: each step on the scale is a factor of 105.

Watch out!

  • An answer wrong by several powers of ten (a runner at 106 m s-1) signals a unit or prefix slip.
  • Quoting an estimate to 4 significant figures implies precision you do not have.

1.2SI units

Definitions

SI base quantities
mass (kg), length (m), time (s), electric current (A), thermodynamic temperature (K) and amount of substance (mol).
Derived unit
a unit formed by multiplying and/or dividing base units.
Homogeneous equation
an equation in which every term has the same base units.

Key points

  • Syllabus base units to recall: kg, m, s, A, K (and mol).
  • Express derived units in base units via a defining equation: N = kg m s-2 from F = ma.
  • J = N m = kg m2 s-2; W = J s-1 = kg m2 s-3; Pa = N m-2 = kg m-1 s-2.
  • C = A s; V = W A-1 = kg m2 s-3 A-1.
  • Homogeneity check: reduce each term to base units; they must all match.
  • A homogeneous equation can still be wrong: pure numbers (½, 2π) carry no units.
  • Prefixes: p 10-12, n 10-9, μ 10-6, m 10-3, c 10-2, d 10-1, k 103, M 106, G 109, T 1012.
kg, m, s F=ma N = kg m s⁻² × m (W=Fs) J = kg m² s⁻² ÷ s (P = W/t) W = kg m² s⁻³
Building a derived unit: follow the defining equations back to base units.

Watch out!

  • Temperature base unit is the kelvin, not °C; charge is not a base quantity (current is).
  • Writing the mass base unit as g instead of kg.
  • Prefix powers: 1 cm2 = 10-4 m2 and 1 cm3 = 10-6 m3, not 10-2.

1.3Errors and uncertainties

Definitions

Systematic error
an error that shifts every reading by the same amount or in the same direction (e.g. a zero error); not reduced by repeating.
Random error
an error that causes readings to scatter unpredictably about the mean; reduced by repeating and averaging.
Precision
how close repeated measurements are to each other (small spread).
Accuracy
how close a measurement (or the mean) is to the true value.
Absolute uncertainty
the range within which the true value is expected to lie, ±Δx, in the unit of the quantity.
Fractional / percentage uncertainty
Δx / x (or × 100%).

Key points

  • Systematic errors: zero error, wrongly calibrated scale, parallax always from one side, reaction time.
  • Systematic error shows as a straight-line graph that misses the origin when it should pass through it.
  • Random errors: fluctuating readings, reaction time varying, judging a swing or a meniscus.
  • Uncertainty of a mean: half the range of the repeats.
  • Adding or subtracting: add absolute uncertainties.
  • Multiplying or dividing: add fractional (percentage) uncertainties.
  • Power n: multiply the fractional uncertainty by |n|. Constants (π, 2) add no uncertainty.
  • Give the final uncertainty to 1 (or 2) sig. figs; round the value to the same decimal place.
precise, not accurate accurate, not precise
Precise but not accurate (systematic shift) versus accurate but not precise (random scatter).
readingtrue value zero error ideal (through 0)
Zero error: a systematic error shifts every point equally, so the line has the right gradient but misses the origin.

Watch out!

  • Repeating reduces random error only; a systematic error must be found and corrected.
  • Subtracting two close values: the absolute uncertainties add, so the percentage uncertainty can become huge.
  • For r in V = (4/3)πr3: % uncertainty in V is 3 × that in r.

1.4Scalars and vectors

Definitions

Scalar
a quantity that has magnitude only.
Vector
a quantity that has both magnitude and direction.
Resultant
the single vector that has the same effect as two or more vectors combined.
Resolving
splitting a vector into two perpendicular components.

Key points

  • Scalars: distance, speed, mass, time, energy, power, temperature, charge.
  • Vectors: displacement, velocity, acceleration, force, weight, momentum, field strength.
  • Add vectors tip-to-tail; the resultant runs from the first tail to the last tip.
  • Scale drawing: draw to scale, measure the resultant length and angle.
  • Perpendicular vectors: Pythagoras for magnitude, tan for direction.
  • Subtracting a vector = adding it reversed (e.g. change in velocity Δv = v - u).
  • Components along perpendicular axes can be treated independently.
F θ F cos θ F sin θ
Resolving: cos θ goes with the axis the angle touches, sin θ with the other.
A B R = A + B α tail of A to tip of B
Adding vectors tip-to-tail: the resultant R closes the triangle.

Watch out!

  • If θ is measured from the vertical, the vertical component is F cos θ.
  • Adding magnitudes of non-parallel vectors: 3 N + 4 N at right angles is 5 N, not 7 N.
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