Physical quantities and units

• Averaging beats random error, never systematic error — and uncertainties only ever add.

• 1.1.1 I can understand — understand that all physical quantities consist of a numerical magnitude and a unit
• 1.1.2 I can — make reasonable estimates of physical quantities included within the syllabus
• 1.2.1 I can recall — recall the following SI base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K)
• 1.2.2 I can — express derived units as products or quotients of the SI base units and use the derived units for quantities listed in this syllabus as appropriate
• 1.2.3 I can use — use SI base units to check the homogeneity of physical equations
• 1.2.4 I can recall — recall and use the following prefixes and their symbols to indicate decimal submultiples or multiples of both base and derived units: pico (p), nano (n), micro (μ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T)
• 1.3.1 I can understand — understand and explain the effects of systematic errors (including zero errors) and random errors in measurements
• 1.3.2 I can understand — understand the distinction between precision and accuracy
• 1.3.3 I can — assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties
• 1.4.1 I can understand — understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus
• 1.4.2 I can — add and subtract coplanar vectors
• 1.4.3 I can — represent a vector as two perpendicular components

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “I took five readings and averaged them, so the zero error is gone.” Correct Averaging reduces random error only. A zero error shifts every one of the five readings by the same amount in the same direction, so their mean carries the same shift. Correct the zero: corrected reading = reading − zero reading.
• “The equation is homogeneous, so it is correct.” Correct Homogeneity cannot see a pure number (½, 2, π), a sign or a missing term with the right units. A homogeneous equation may be correct; only an inhomogeneous one is proved wrong.
• “1 mm2 = 10−3 m2.” Correct The prefix is squared along with the unit: 1 mm2 = (10−3 m)2 = 10−6 m2, and 1 cm3 = 10−6 m3.
• “For a product I add the absolute uncertainties; for a difference they subtract.” Correct Sums and differences: add absolute uncertainties. Products and quotients: add percentage uncertainties. Uncertainties never subtract.
• “Precise means close to the true value.” Correct That is accuracy. Precise means repeated readings agree closely with each other. A micrometer with an uncorrected zero error is precise and inaccurate.
• “A ball hits a wall at 8.0 m s−1 and bounces back at 6.0 m s−1: its velocity changed by 2.0 m s−1.” Correct Velocity is a vector. With the rebound direction positive, Δv = (+6.0) − (−8.0) = +14.0 m s−1.
• “The component of the weight down a slope is W cos θ.” Correct Down the slope it is W sin θ; into the slope it is W cos θ. Check with θ = 0: on flat ground nothing acts along the surface, and sin 0 = 0.
• “The length is 45.” Repair A quantity is a number and a unit: 45 cm. Without the unit the number means nothing.
• “The base unit of mass is the gram.” Repair The kilogram. Convert grams to kilograms before substituting into any equation.
• “The SI base quantities include charge (coulomb) and amount of substance (mole).” Repair Outcome 1.2.1 lists five: mass, length, time, electric current and thermodynamic temperature. The coulomb is derived (A s); the mole joins the list only in Topic 15.
• “In base units, the newton is N.” Repair N = kg m s−2, from F = ma. When base units are asked for, no named unit may remain.
• “The moment is 12 J.” Repair A moment has the same base units as the joule but is not an energy: write 12 N m.
• “It is homogeneous, so it is correct.” Repair Homogeneity cannot check a pure number such as ½, a sign or a missing term. It proves only that an inhomogeneous equation is wrong.
• “1 mm2 = 10−3 m2.” Repair (10−3 m)2 = 10−6 m2; likewise 1 cm3 = 10−6 m3.
• “M and m both mean mega.” Repair m is milli (10−3), M is mega (106), μ is micro (10−6). Case matters.
• “I averaged five readings, so the zero error is gone.” Repair A systematic error shifts every reading the same way and survives averaging. Correct the zero: reading − zero reading.
• “The micrometer reads −0.03 mm when closed, so I subtract 0.03 mm from every reading.” Repair A negative zero reading means every reading is too small. Corrected = reading − (−0.03 mm): add 0.03 mm.
• “Precise means close to the true value.” Repair That is accuracy. Precise means repeated readings agree closely with each other.
• “For L = x2 − x1 the uncertainties subtract.” Repair Absolute uncertainties always add, for a difference as for a sum.
• “For a product I add the absolute uncertainties.” Repair Add percentages for products and quotients; add absolutes for sums and differences.
• “d has a 1% uncertainty, so d3 has 1% too.” Repair Multiply by the power: 3 × 1% = 3%.
• “g = 9.748 ± 0.22 m s−2.” Repair Uncertainty to 1 significant figure, value to the same decimal place: 9.7 ± 0.2 m s−2.
• “Hitting a wall at 8 m s−1 and rebounding at 6 m s−1 is a change of 2 m s−1.” Repair Δv = (+6) − (−8) = 14 m s−1 away from the wall.
• “The weight's component down a slope is W cos θ.” Repair W sin θ; check with θ = 0, where nothing acts along a flat surface.
• “The uncertainty was due to human error.” Repair Name the quantity and the cause: “the time, because of reaction time of about 0.2 s in starting and stopping the stopwatch”.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step from the starting data to the printed result must appear. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• An estimate is also a check. Before you accept any calculated answer, ask whether it is the right order of magnitude. A person of mass 7000 kg, a cyclist at 0.05 m s−1 or a wire of cross-sectional area 1 m2 means a prefix or a power of ten has gone wrong somewhere upstream.
• Convert before you substitute. Put every quantity into base units — mm to m, g to kg, minutes to s, km h−1 to m s−1 — before it goes into an equation. For example 36 km h−1 = 36 × 103 m ÷ 3600 s = 10 m s−1.
• Interleave with the chapters that use this one. When you reach Topic 2, re-answer the change-in-velocity items in section K; in Topic 3, redo the rebound in figure 5 as a change in momentum; in Topic 4, redo studio item 5 (closing the triangle); in Topic 9, re-derive the ohm in base units; and before every practical, re-read section I. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 1 is AS Level content, so it is examined in Papers 1, 2 and 3. AS Level content: examined in Paper 1 (multiple choice), Paper 2 (AS structured) and, as practical context, Paper 3. Assumed knowledge for Papers 4 and 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• A multiple-choice item on this topic turns on one exact step, such as the base units of a derived unit, the factor for a prefix on a powered unit, which uncertainty rule applies, or sin against cos in a component. A structured question asks you to show that an equation is homogeneous, to state base units, to explain the effect of a zero error, or to calculate a resultant with its direction.
• A value and its absolute uncertainty for a derived quantity; a percentage uncertainty; the magnitude and direction of a resultant or a change in velocity; the components of a force. None of the relationships in this topic is printed on the Data and formulas sheet — the uncertainty rules, the component formulas and the cosine rule are all recalled. The only value the sheet supplies here is g = 9.81 m s−2.
• Topic 1 is the toolkit of Papers 3 and 5 rather than one experiment. Its set piece here is the simple pendulum: length varied with a metre rule, the time for 20 oscillations measured with a stopwatch, a graph of T2 against l giving g from its gradient, and reaction time in judging the swing as the largest uncertainty. Every later practical uses the recording, significant-figure and uncertainty rules taught here.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step from the starting data to the printed result must appear. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.