Chapter 1: Units and Measurement

In the figure : When the arc is as long as the radius, the angle at the centre is exactly 1 radian.

Rule 1: Base and derived units — The 7 base quantities and units are : length (m), mass (kg), time (s), electric current (A), temperature (K), amount of substance (mol), luminous intensity (cd). Every other unit is a derived unit made from these, e.g. newton = kg m s⁻², joule = N m = kg m² s⁻².

• Easy : Speed is measured in m s⁻¹, so it is a derived unit.
• Medium : Convert 72 km/h to m/s : 72 × 1000 ÷ 3600 = 20 m/s.
• Exam : Express 1 dyne in newton (1 dyne = 1 g cm s⁻²). Step 1 : 1 g = 10⁻³ kg and 1 cm = 10⁻² m. Step 2 : 1 dyne = 10⁻³ × 10⁻² kg m s⁻² = 10⁻⁵ N.

Rule 2: Plane angle and solid angle — θ = s ÷ r (s = arc length, r = radius) is measured in radian (rad); Ω = A ÷ r² (A = area cut on a sphere of radius r) is measured in steradian (sr). Both are dimensionless. A full circle = 2π rad; a full sphere = 4π sr.

• Easy : Arc 6 cm on a circle of radius 4 cm ⇒ θ = 6 ÷ 4 = 1.5 rad.
• Medium : 90° = 90 × π ÷ 180 = π ÷ 2 = 1.571 rad.
• Exam : A 2 m² patch on a sphere of radius 5 m subtends Ω = 2 ÷ 25 = 0.08 sr.

Rule 3: Modern definitions and prefixes — Since 2019 SI units are fixed by giving exact values to natural constants (speed of light, Planck constant, elementary charge, Boltzmann constant, Avogadro number and others), so the units never drift. Prefixes : kilo 10³, mega 10⁶, giga 10⁹, milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹².

• Easy : 2.5 GHz = 2.5 × 10⁹ Hz.
• Medium : Light of wavelength 450 nm = 450 × 10⁻⁹ m = 4.5 × 10⁻⁷ m.
• Exam : 3 mm² in m² : 1 mm = 10⁻³ m ⇒ 1 mm² = 10⁻⁶ m² ⇒ 3 mm² = 3 × 10⁻⁶ m².

Spot the difference

Rule 1: Counting significant figures — (a) All non-zero digits are significant. (b) Zeros between non-zero digits are significant. (c) Zeros to the left of the first non-zero digit (as in 0.0045) are not significant. (d) Trailing zeros in a number without a decimal point are not significant. (e) Trailing zeros after a decimal point are significant. Changing the unit never changes the count.

• Easy : 305.2 has 4 significant figures; 0.0045 has 2.
• Medium : 2.50 m = 250 cm = 0.00250 km — each has 3 significant figures.
• Exam : How many significant figures in 0.06700 and 4020? 0.06700 : digits 6, 7, 0, 0 ⇒ 4. 4020 : trailing zero with no decimal point is not counted ⇒ 3.

Rule 2: Scientific notation — Write a number as a × 10ᵇ with 1 ≤ a < 10. Every digit written in a is significant, so there is no confusion about trailing zeros. The power of 10 does not affect the count.

• Easy : 0.000082 = 8.2 × 10⁻⁵ (2 significant figures).
• Medium : 32000 m may mean 3.2 × 10⁴ m (2 figures) or 3.20 × 10⁴ m (3 figures); notation removes the doubt.
• Exam : A rod is 4.700 m long. In mm it is 4700 mm, which looks like 2 figures; the correct way is 4.700 × 10³ mm (4 figures).

Rule 3: Exact numbers — Counted numbers and pure constants in formulae (the 2 in 2πr, the 10 in "10 oscillations", the ½ in ½mv²) are exact and have unlimited significant figures.

• Easy : 20 swings of a pendulum take 36.4 s ⇒ T = 36.4 ÷ 20 = 1.82 s.
• Medium : The 20 is exact, so the answer keeps the 3 figures of 36.4 s.
• Exam : For r = d ÷ 2 with d = 7.40 cm, r = 3.70 cm (3 figures); the 2 never reduces the figures.

Spot the difference

Rule 1: Multiplication and division — Keep as many significant figures as the number with the fewest significant figures.

• Easy : 8.46 g ÷ 3.2 cm³ = 2.64375 ⇒ 2.6 g cm⁻³ (2 figures).
• Medium : Area = 12.35 cm × 4.2 cm = 51.87 ⇒ 52 cm² (2 figures).
• Exam : A cyclist covers 150.0 m in 7.50 s. Fewest figures = 3 ⇒ speed = 150.0 ÷ 7.50 = 20.0 m s⁻¹.

Rule 2: Addition and subtraction — Keep as many decimal places as the number with the fewest decimal places.

• Easy : 12.45 + 3.2 + 0.067 = 15.717 ⇒ 15.7 (1 decimal place).
• Medium : 25.6 − 4.32 = 21.28 ⇒ 21.3.
• Exam : 0.862 − 0.85 = 0.012 ⇒ 0.01. A subtraction of nearly equal numbers loses significant figures.

Rule 3: Rounding off — Look at the first digit to be dropped. If it is more than 5, raise the last kept digit by 1; if it is less than 5, leave it. If it is exactly 5, make the last kept digit even. In long calculations keep one extra digit in the middle steps and round only at the end.

• Easy : 6.783 → 6.78 (3 figures).
• Medium : 5.465 → 5.46 (6 is even) but 5.475 → 5.48 (7 is odd, so it is raised).
• Exam : 2.745 → 2.74 and 2.735 → 2.74, because in both cases the kept digit ends up even.

Rule 4: Order of magnitude — Write the number as a × 10ᵇ. If a ≤ 5 the order of magnitude is b; if a > 5 it is b + 1.

• Easy : A car of mass 1.2 × 10³ kg has order of magnitude 3.
• Medium : Height 8.8 × 10³ m : a = 8.8 > 5 ⇒ order 4.
• Exam : An atom of diameter 2 × 10⁻¹⁰ m (order −10) compared with a 1 mm grain of sand, 10⁻³ m (order −3) : sand is larger by −3 − (−10) = 7 orders of magnitude.

Spot the difference

Rule 1: Writing a dimensional formula — Start from the simplest defining formula and replace each quantity by its dimensions. Common ones : velocity [L T⁻¹]; acceleration [L T⁻²]; force [M L T⁻²]; work and energy [M L² T⁻²]; power [M L² T⁻³]; pressure [M L⁻¹ T⁻²]; density [M L⁻³]; momentum [M L T⁻¹]; frequency [T⁻¹].

• Easy : Volume = length × length × length ⇒ [L³].
• Medium : Pressure = force ÷ area = [M L T⁻²] ÷ [L²] = [M L⁻¹ T⁻²].
• Exam : From F = G m₁ m₂ ÷ r², G = F r² ÷ (m₁ m₂) = [M L T⁻²][L²] ÷ [M²] = [M⁻¹ L³ T⁻²].

Rule 2: Dimensionless quantities — Pure numbers, ratios of two like quantities (angle, strain, refractive index, relative density) and the arguments of sin, cos, log and exponential have no dimensions.

• Easy : Angle = arc ÷ radius = [L] ÷ [L] ⇒ dimensionless.
• Medium : In sin(ωt), the argument ωt must be dimensionless, so [ω] = [T⁻¹].
• Exam : Strain = change in length ÷ original length ⇒ [L] ÷ [L] = dimensionless; so stress and Young's modulus have the same dimensions as pressure.

Rule 3: Dimensions do not identify the quantity — Different quantities can share a dimension (work and torque are both [M L² T⁻²]; frequency and angular speed are both [T⁻¹]), so a dimension alone cannot tell which quantity it is.

• Easy : Energy and work have the same dimensions [M L² T⁻²].
• Medium : Torque = force × distance = [M L T⁻²][L] = [M L² T⁻²], same as work, yet they are different quantities.
• Exam : Dimension of power = work ÷ time = [M L² T⁻²] ÷ [T] = [M L² T⁻³].

Spot the difference

Rule 1: Checking an equation — Write the dimensions of every term. If they differ, the equation is wrong. If they match, it may be right, but this is not a proof.

• Easy : s = ut + ½at² : [L] = [L T⁻¹][T] + [L T⁻²][T²] = [L] + [L] — consistent.
• Medium : v = u + at² : [L T⁻¹] against [L T⁻²][T²] = [L]. Not equal ⇒ the equation is wrong.
• Exam : P = ½ρv² + ρgh as pressure. ½ρv² : [M L⁻³][L² T⁻²] = [M L⁻¹ T⁻²]; ρgh : [M L⁻³][L T⁻²][L] = [M L⁻¹ T⁻²]; both equal pressure — consistent.

Rule 2: Deriving a relation — If a quantity depends on up to three others as a product, write it as k aˣ bʸ cᶻ (k = pure number), equate dimensions on both sides and solve for the powers.

• Easy : Speed of a wave on a string depends on tension F [M L T⁻²] and mass per unit length μ [M L⁻¹] : v = k Fˣ μʸ. Mass : x + y = 0; length : x − y = 1; time : −2x = −1 ⇒ x = ½, y = −½ ⇒ v = k √(F ÷ μ).
• Medium : Centripetal force F depends on m [M], v [L T⁻¹] and r [L] : F = k mᵃ vᵇ rᶜ. Mass : a = 1; time : −b = −2 ⇒ b = 2; length : b + c = 1 ⇒ c = −1 ⇒ F = k m v² ÷ r.
• Exam : Time period T of a mass m on a spring of stiffness κ [M T⁻²] : T = k mˣ κʸ. Mass : x + y = 0; time : −2y = 1 ⇒ y = −½, x = ½ ⇒ T = k √(m ÷ κ).

Rule 3: Converting units — Use n₁u₁ = n₂u₂ together with the dimensional formula.

• Easy : 5 N in dyne : 5 kg m s⁻² = 5 × 10³ g × 10² cm s⁻² = 5 × 10⁵ dyne.
• Medium : 1 J in erg (1 erg = 1 g cm² s⁻²) : 1 kg m² s⁻² = 10³ g × 10⁴ cm² s⁻² = 10⁷ erg.
• Exam : A pressure of 1 N m⁻² in dyne cm⁻² : [M L⁻¹ T⁻²] ⇒ 1 kg m⁻¹ s⁻² = 10³ g ÷ 10² cm ÷ s² = 10 dyne cm⁻².

Limits of the method : It cannot give pure numbers such as ½ or 2π (so k stays unknown); it cannot tell apart quantities with the same dimensions; it cannot deal with formulae that are sums of terms (such as s = ut + ½at²) or with trigonometric, logarithmic and exponential functions, and it fails when a quantity depends on more than three others as a simple product.

Spot the difference