Lesson 10 of 10 · 13 min

Chapter review

Must-know facts

16 facts

01Seven SI base units: metre, kilogram, second, ampere, kelvin, mole, candela.
02Radian (plane angle, ds/r) and steradian (solid angle, dA/r²) are dimensionless.
03CGS: cm, g, s. FPS: foot, pound, second. MKS: m, kg, s.
04Significant figures = reliable digits + the first uncertain digit.
05Changing units does not change the number of significant figures.
06Leading zeros are never significant; trailing zeros count only if there is a decimal point.
07Order of magnitude: round a in a × 10ᵇ to 1 if a ≤ 5, to 10 if a > 5.
08Product or quotient: fewest significant figures. Sum or difference: fewest decimal places.
09Rounding a final 5: even preceding digit stays, odd preceding digit goes up (2.745 → 2.74, 2.735 → 2.74).
10Carry one extra digit through intermediate steps; round at the end.
11In a product or quotient, percentage errors add.
12Exact numbers (2 in 2πr, a count) have unlimited significant figures.
13Force [M L T⁻²], velocity [L T⁻¹], acceleration [L T⁻²], density [M L⁻³], energy and work [M L² T⁻²].
14Only like dimensions can be added; arguments of sin, log and exp are dimensionless.
15Dimensionally wrong means wrong; dimensionally right does not mean right.
16Pendulum by dimensions: T ∝ √(l/g), independent of mass; k = 2π comes from theory.

Common traps

8 to know

Plane angle

dθ = ds/r

Unit radian (rad); dimensionless.

Solid angle

dΩ = dA/r²

Unit steradian (sr); dimensionless.

Scientific notation

value = a × 10ᵇ, 1 ≤ a < 10

All digits of a are significant; the order of magnitude is 10ᵇ (or 10ᵇ⁺¹ if a > 5).

Percentage error

percentage error = (Δa/a) × 100%

Δa is the uncertainty in the measured value a.

Error in a product or quotient

if Z = A × B or A/B, then ΔZ/Z = ΔA/A + ΔB/B

Relative errors add.

Dimensions of force

[F] = [M L T⁻²]

From F = ma.

Dimensions of density

[ρ] = [M L⁻³ T⁰]

Mass per unit volume.

Period of a simple pendulum

T = 2π√(l/g)

Dimensions fix T ∝ √(l/g); the factor 2π comes from theory.

13 terms

Unit: An agreed reference standard with which a physical quantity is compared.
Base quantity: One of the seven quantities chosen as independent starting points, such as length, mass and time.
Derived unit: A unit built from base units, such as m s⁻¹ or kg m s⁻².
SI: The internationally accepted system of units, with seven base units defined by fixed constants of nature.
Radian: Unit of plane angle: arc length divided by radius.
Steradian: Unit of solid angle: intercepted area on a sphere divided by the square of its radius.
Significant figures: The reliable digits of a measurement together with the first uncertain one.
Least count: The smallest value an instrument can read directly; it limits the precision.
Order of magnitude: The power of ten nearest a quantity, found by rounding a in a × 10ᵇ to 1 or 10.
Relative error: The uncertainty in a value divided by the value, often given as a percentage.
Dimensions: The powers to which base quantities are raised to express a physical quantity.
Dimensional formula: An expression such as [M L T⁻²] showing the base quantities in a quantity and their powers.
Principle of homogeneity: Every term of a physically meaningful equation has the same dimensions.