Lesson 10 of 10 · 13 min
Chapter review
Must-know facts
16 facts
- 1Seven SI base units: metre, kilogram, second, ampere, kelvin, mole, candela.
- 2Radian (plane angle, ds/r) and steradian (solid angle, dA/r²) are dimensionless.
- 3CGS: cm, g, s. FPS: foot, pound, second. MKS: m, kg, s.
- 4Significant figures = reliable digits + the first uncertain digit.
- 5Changing units does not change the number of significant figures.
- 6Leading zeros are never significant; trailing zeros count only if there is a decimal point.
- 7Order of magnitude: round a in a × 10ᵇ to 1 if a ≤ 5, to 10 if a > 5.
- 8Product or quotient: fewest significant figures. Sum or difference: fewest decimal places.
- 9Rounding 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
Where marks are lost
Counting the zeros in 0.00230 as significant, and so giving it six significant figures.
Rounding a sum to the fewest significant figures, for example 436.32 + 227.2 + 0.301 = 664.
Treating 4700 mm as four significant figures because it came from 4.700 m.
Letting the exact 2 in 2πr, or the 20 in 'time for 20 oscillations', limit the significant figures of the answer.
Rounding 2.745 to 2.75 by 'always round 5 up'.
Accepting an equation as correct because both sides have the same dimensions.
Trying to find the constant 2π in T = 2π√(l/g), or a relation involving sin or exp, by dimensional analysis.
Adding absolute errors when two measured values are multiplied.
Formulas
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.
Key terms
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.