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Tools and Inquiry — IB Physics skills

T.3.1 Applying General Mathematics in Physics — IB Physics HL Revision Notes & Topical Questions

10 min read 12 parts Lesson 6 of 22 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
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Lesson summary

T.3.1 Applying General Mathematics in Physics — IB PHYSICS SL notes: Use arithmetic, algebra, proportionality, logarithms, trigonometry, estimation and significant figures fluently in physics.Algebra = applying the laws of arithmetic to expressions and rearranging them to isolate…

  • Other formulae
  • Percentages
  • Reciprocal
  • What the symbols mean
  • Percentage error is not a percentage uncertainty
  • Accepted value can be a label on a component or a reputable data booklet

Part 1 of 10Algebra, areas and volumes

Algebra
Algebra applying the laws of arithmetic to expressions and rearranging them to isolate a variable.
  • Do the same operation on both sides, e.g. from v = u + at, t = (v − u)/a.
Areas of simple shapes.
  • Rectangle A = lw, triangle A = ½bh, circle A = πr2.
Volumes of simple shapes.
  • Cuboid V = lwh, cylinder V = πr2h, sphere V = (4/3)πr3.
Other formulae
  • Other formulae circumference C = 2πr, curved area of a cylinder 2πrh and surface area of a sphere 4πr2.
Most of these formulae are in the data booklet.

Part 2 of 10Decimals, fractions, percentages and ratios

Decimals
  • Decimals most calculations use them; decimal places show the precision of a measurement.
Fractions
  • Fractions used mostly in algebra and uncertainty calculations;
    • use the calculator's conversion key to change a fraction to a decimal.
Percentages
  • Percentages percentage change, percentage difference, percentage error and percentage uncertainty.
Ratios
  • Ratios compare quantities, e.g. efficiency and uncertainties.
Reciprocal
1 divided by the number — e.g. T = 1/f.
What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)
Exponent
Exponent the power a number is raised to — e.g. a = −ω2x.
What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre

Part 3 of 10Percentage formulae

QuantityFormulaUse
Percentage change(final − initial) / initial × 100Relative change between two values
Percentage difference(value 1 − value 2) / average × 100How far two values differ
Percentage error(accepted − experimental) / accepted × 100Result compared with a literature value
Percentage uncertaintyabsolute uncertainty / measured value × 100Significance of an uncertainty
Percentage formulae
Percentage error is not a percentage uncertainty
  • Percentage error is not a percentage uncertainty the smaller the error, the more accurate the result.
Accepted value can be a label on a component or a reputable data booklet
  • Accepted value can be a label on a component or a reputable data booklet e.g. g = 9.81 m s-2.

Part 4 of 10Scientific notation and significant figures

Scientific notation and significant figures
Scientific notation (standard form)
Scientific notation (standard form) a × 10n, with a above 1 and below 10.
  • Large numbers have positive n and small numbers have negative n.
When rounding in standard form, only a is rounded
  • 4.37 × 106 to 2 s.f. is 4.4 × 106.
Significant figure rules.
  • All non-zero digits count, and zeros between non-zero digits count (4107 has 4).
  • Leading zeros do not count (0.00079 has 2).
  • Trailing zeros in a number without a decimal point do not count (57 000 has 2).
  • Trailing zeros after a decimal point count (70.0 has 3, 0.0350 has 3).
Rounding
  • Rounding look at the next digit; if it is 5 or more, increase the previous digit by 1.
Keep at least two extra figures during working
  • Keep at least two extra figures during working round only the final answer.
The answer cannot have more significant figures than the least precise value used.
Constants are limited by their definition
  • Constants are limited by their definition g = 9.8 m s-2 (2 s.f.), c = 3.00 × 108 m s-1 (3 s.f.).

Part 5 of 10Orders of magnitude and estimation

Orders of magnitude and estimation
Order of magnitude
Order of magnitude the nearest power of ten — 3 × 108 is 108, but 6 × 108 rounds up to 109.
One order of magnitude larger means about ten times larger
  • One order of magnitude larger means about ten times larger two orders means 100 times.
Orders of magnitude make very large and very small quantities easier to compare.
  • Football field about 102 m, Earth to Sun about 1011 m — a difference of 9 orders.
  • Ant about 10-3 m, bacterium about 10-6 m, observable universe about 1026 m.
Approximation
Approximation simplifying a calculation to get a value close to the actual one — neglecting small effects or using the small-angle approximation.
Estimation
Estimation a reasoned guess from available data — quick order-of-magnitude checks that a result is reasonable.
  • Typical valuesa person 70 kg, a room 3 m high, a cyclist 5 m s-1.

Part 6 of 10Logarithms, exponentials and trigonometry

Exponential change
Exponential change rate of change proportional to current value — growth (population) or decay (radioactivity), y = ax.
Logarithm
Logarithm inverse of an exponential — y = ax gives log y = x log a.
  • ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(an) = n ln a.
  • ln(a + b) is not ln a + ln b.
Logs straighten curves
  • N = N0e-λt gives ln N = ln N0 − λt, and y = kxn gives ln y = ln k + n ln x.
Trigonometric ratios in a right-angled triangle.
  • sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
Small angles in radians
  • Small angles in radians sin θ ≈ tan θ ≈ θ and cos θ ≈ 1 .
    • (below about 10°)
180°
π rad, and arc length s = rθ.

Part 7 of 10Rates of change, mean and range

Rates of change, mean and range
Δ means a change in a quantity
  • ΔT = Tfinal − Tinitial.
Average rate of change
Δy / Δx — the gradient of a graph of y against x.
Mean
Mean sum of values divided by the number of values.
Range
Range highest value − lowest value — a measure of spread, for numerical data only.
  • It can be written as a range of values (8.4–9.2) or one value (0.8).
Anomalous highest or lowest values distort both the mean and the range.

Part 8 of 10Equations, derivations and proportions

1The data booklet gives constants and equations
  • The data booklet gives constants and equations you must select, rearrange and derive them.
2
Selecting an equation.
  1. List the known quantities with symbols.
  2. Find the data-booklet equation connecting them.
  3. Rearrange for the unknown quantity.
  4. Substitute the values and calculate.
3Deriving a relationship
  • Deriving a relationship identify the underlying principle, list relevant equations, then combine them.
    • (forces, energy, momentum)
  • In a "show that" question the final form is given, so manipulate the algebra until you reach it.
4
Key phrases give information.
  • "Stationary" or "starts from rest" means u = 0.
  • "Constant velocity" means Fnet = 0 and a = 0.
  • "Rebounds in the opposite direction" changes the sign of velocity, Δv = v + u.
5Effect of changing variables
  • Effect of changing variables write the equation as a ratio so that constants cancel.
  • Charles' lawV1/T1 = V2/T2.
6Continuous variables take any value (time, mass)
  • Continuous variables take any value (time, mass) discrete variables are counted whole numbers.
    • (number of particles)

Part 9 of 10Proportionality and negligible effects

Directly proportional
  • Directly proportional y = kx, a straight line through the origin; if x doubles, y doubles.
  • A straight line not through the origin is proportional in change but not directly proportional.
Inversely proportional
  • Inversely proportional y = k/x; if x doubles, y halves, with a curve that does not cross either axis.
Positive correlation
  • Positive correlation gradient slopes upward; negative correlation — gradient slopes downward.
A negligible effect is so small that it has no significant effect on the outcome.
  • Air resistance on a small ball over a short distance at low speed.
  • Friction on a smooth surface.
  • Thermal losses much smaller than the energy transfers in specific heat capacity work.
  • Internal resistance much smaller than the load resistance.
  • The mass of the string in a simple pendulum compared with the bob.

Part 10 of 10More from the specification

1Scalars vs Vectors in Physics
  • Scalar quantities possess only magnitude,
    • (e.g. speed, mass, time, energy)
  • whereas vector quantities possess both magnitude and a defined spatial direction.
    • (e.g. displacement, velocity, force, acceleration)
2Resultant Vector Addition
  • Combining multiple vectors requires vector addition; perpendicular vector components combine via Pythagoras (R = √(Ax² + Ay²)) and trigonometry (tan θ = Ay / Ax).
    • (triangle or parallelogram method)
3SI Base and Derived Units
  • Fundamental physical dimensions are quantified using SI base units,
    • (kg, m, s, A, K, mol, cd)
  • from which all other mechanical and electrical units are derived.
    • (such as newtons N = kg m s⁻² and joules J = N m)
4Graphical Error Analysis and Worst Acceptable Gradient
  • When evaluating experimental graphs with error bars, plotting lines of maximum and minimum acceptable gradient (worst gradient) allows calculation of the uncertainty in the experimental slope.
5Vector
  • Quantity with magnitude and direction.
6Scalar
  • Quantity with magnitude only.
7Resultant vector
  • Single vector equivalent to a combination of vectors.
8Derived unit
  • A unit formed by multiplying or dividing SI base units.
9Scalar quantity
  • A quantity with magnitude only, such as potential
10Vector quantity
  • A quantity with both magnitude and direction.
11Scale drawing
  • Accurate drawing with a chosen scale.
12Parallelogram method
  • Vector addition using a parallelogram of vectors.

Key terms

TermMeaning
Standard formA number written as a × 10^n with 1 ≤ a < 10.
Order of magnitudePower of ten closest to a quantity.
Percentage errorDifference from the accepted value as a percentage of it.
Natural logarithmln x, the inverse function of e^x.
RadianAngle unit equal to arc length divided by radius.
Significant figuresDigits carrying meaningful information in a number.
Directly proportionaly = kx; a straight line through the origin.

Avoid these mistakes

1
Forgetting to convert units (cm2 to m2 before substituting.
2
Using the calculator in the wrong mode — radians instead of degrees for trigonometry.
3
Writing ln(a + b) = ln a + ln b — the log rule applies to products only.
4
Rounding at every step instead of keeping extra digits until the end.
5
Calling any rising straight line directly proportional — it must pass through the origin.

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