Answer: (c)

Answer: (b)

Answer: (b) — leading zeros are not significant, but the trailing zero after the decimal point is.

Answer: (b)

Answer: (c)

Answer: (a)

Answer: (b) — relative errors always add, whether multiplying or dividing.

Answer: (b) — closeness to the true value is accuracy.

Answer: (c)

Answer: (b) — the answer must match the fewest decimal places, one, so 16.46 rounds to 16.5.

Answer: kilogram

Answer: [ML²T⁻²]

Answer: dimensions

Answer: never

Answer: gross

Answer: |a_mean − a_i|

Answer: n

Answer: radian

Answer: fewest

Answer: [ML⁻³T⁰]

Answer: False — this is one of its major limitations.

Answer: True

Answer: False — they arise from unpredictable fluctuations and can only be reduced by averaging, not eliminated.

Answer: False — precision and accuracy are independent; a precise instrument can still be wrongly calibrated.

Answer: True — even for subtraction, the absolute errors add.

Answer: True — angle is the ratio of two lengths, arc to radius, so its dimensions cancel.

Answer: False — it cannot handle trigonometric, exponential or logarithmic functions.

Answer: True

Answer: False — without a decimal point the trailing zeros are ambiguous; it could have one, two or three, depending on how it was measured.

Answer: True — weight is simply the force of gravity, [MLT⁻²].

Answer: (a)

Answer: (a)

Answer: Both statements are false — a more precise instrument need not give a more accurate reading, and precision and accuracy are distinct concepts.

Answer: (a)

Answer: (a)

Answer: (d) — the reason is true, but the assertion is false; work and torque share the dimensional formula [ML²T⁻²] yet are entirely different quantities.

Answer: (a)

Answer: (a)

Answer: Both statements are false — even in division, the relative errors add, never subtract, since either quantity’s error can push the result in either direction.

Answer: (a)

Answer: A unit that is independent of other units and cannot be derived from them.

Answer: [M⁰LT⁻¹]

Answer: The dimensions on both sides of a valid physical equation must be identical.

Answer: Four.

Answer: The magnitude of the difference between the mean value and an individual measured value.

Answer: It cannot determine dimensionless constants; equally acceptable is that it fails for trigonometric or exponential relations.

Answer: The mole.

Answer: Accuracy is closeness to the true value; precision is the resolution of the measurement.

Answer: Percentage error = (Δa_mean / a_mean) × 100%

Answer: Carelessness of the observer, such as misreading the instrument.

Answer: Fundamental units are independent of one another and cannot be derived, such as the metre for length. Derived units are obtained by combining fundamental units according to the definition of the quantity, such as the metre per second for speed, which combines the units of length and time.

Answer: Dimensional analysis is used to check whether a given physical equation is dimensionally consistent, to derive a relationship between physical quantities when the form of dependence is known, and to convert a physical quantity from one system of units to another.

Answer: In addition or subtraction, the result should carry as many decimal places as the term with the fewest decimal places, so 12.36 + 4.1 is reported as 16.5. In multiplication or division, the result should carry as many significant figures as the term with the fewest significant figures, so 4.2 × 3.65 with two and three significant figures respectively is reported to two, as 15.

Answer: Systematic errors occur consistently in one direction, arising from instrumental defects, imperfect technique or personal bias, and can be reduced by correcting the technique or instrument. Random errors occur irregularly, sometimes positive and sometimes negative, due to unpredictable fluctuations, and are minimised by taking a large number of readings and averaging them.

Answer: When two measured quantities are multiplied, the relative or fractional errors of each add together to give the relative error of the result, so ΔZ/Z = ΔA/A + ΔB/B. This holds regardless of which quantity is larger, since either measurement’s uncertainty can shift the product in either direction.

Answer: Trigonometric, exponential and logarithmic functions are defined only for dimensionless arguments, and their expansion as a series mixes different powers of that argument. Since a dimensional check requires every term to carry consistent dimensions, and these functions do not correspond to a fixed dimensional power, dimensional analysis simply cannot be applied to them.

Answer: The principle states that only quantities with the same dimensions can be added, subtracted or equated, and both sides of a correct physical equation must have identical dimensions. It is useful because it provides a quick check on any derived formula: if the dimensions on the two sides do not match, the formula must be wrong, even before any numerical calculation is attempted.

Answer: Percentage error is the relative error expressed as a percentage, given by (Δa_mean/a_mean) × 100%. A small percentage error indicates that the individual readings were close together and close to the mean, suggesting the measurement was carried out carefully and consistently, though it does not by itself guarantee that the mean value is close to the true value.

Answer: Work and torque both have the dimensional formula [ML²T⁻²], yet work is a scalar quantity while torque is a vector quantity. Similarly, the impulse of a force and momentum both have the dimensional formula [MLT⁻¹], though they arise in different physical contexts.

Answer: The small variation between readings is most likely due to random errors, from slight differences in positioning the scale or reading the marking each time. The correct approach is to calculate the mean of all five readings as the best estimate of the true length, and to report the mean absolute error alongside it to indicate the uncertainty in the measurement.

Answer: The seven SI base units are the metre for length, kilogram for mass, second for time, ampere for electric current, kelvin for temperature, mole for amount of substance, and candela for luminous intensity, along with the supplementary units radian for plane angle and steradian for solid angle. Fundamental units are independent of one another and form the building blocks of the system, whereas derived units are formed by combining these base units according to the definition of a quantity: speed, defined as distance over time, has the derived unit metre per second, and force, defined through mass times acceleration, has the derived unit kilogram metre per second squared, later named the newton.

Answer: Dimensional analysis is used to check the consistency of an equation, since both sides must have identical dimensions; to derive a relationship between quantities, such as finding that the time period of a pendulum must depend on the square root of its length divided by g; and to convert units from one system to another by comparing the dimensional formula in each system. Its limitations are equally important: it cannot fix dimensionless numerical constants, such as the factor 2π in the pendulum formula, which must come from experiment or derivation rather than dimensions alone; it cannot handle equations with trigonometric, exponential or logarithmic terms, since these require a dimensionless argument; it cannot distinguish between a scalar and a vector quantity of the same dimensions, such as work and torque; and it breaks down if a quantity depends on more than three independent unknowns, since only three equations can be written from the powers of M, L and T.

Answer: Significant figures are all the digits in a measured value that are known with certainty, plus one estimated digit, and they communicate how precisely the quantity was measured. All non-zero digits are significant; zeros between two non-zero digits are significant; leading zeros before the first non-zero digit are never significant; and trailing zeros are significant only if the number contains a decimal point, otherwise they are ambiguous unless scientific notation is used. In addition or subtraction the result is rounded to the least number of decimal places among the terms, while in multiplication or division it is rounded to the least number of significant figures among the terms, ensuring the final answer never claims more precision than the least precise measurement that went into it.

Answer: Systematic errors occur consistently in one direction due to identifiable causes such as a poorly zeroed instrument, faulty calibration or a consistent bias in the observer’s technique; they are reduced by correcting or replacing the instrument and by refining the experimental method. Random errors arise from unpredictable fluctuations in conditions during the experiment and appear irregularly in either direction; since they cannot be traced to a single cause, they are minimised statistically by taking a large number of readings and computing the mean, which tends to average the fluctuations out. Gross errors result from outright carelessness, such as misreading a scale or recording a wrong digit; the only remedy is greater attention and, where possible, repeating and cross-checking readings, since these are mistakes rather than genuine limitations of measurement.

Answer: For a quantity built from products, quotients and powers of measured quantities, the general rule is that the relative error in Z equals the sum of the relative errors of A, B and C, each multiplied by the magnitude of its power: ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C). This follows because taking the logarithm of Z converts the powers into multiplying factors and the products and quotients into sums and differences, so that differentiating gives exactly this combination, and because an error in either the numerator or the denominator can shift the final result in the same direction, the terms are always added rather than subtracted regardless of whether the quantity appears above or below the line. The physical reasoning is that uncertainty never cancels by coincidence; every measured input contributes its own share of uncertainty to the final answer, and a quantity raised to a higher power contributes proportionally more, since a small relative error in a squared or cubed term is magnified by that power.

Answer:
[v] = [u] = [M⁰LT⁻¹]
[at] = [M⁰LT⁻²][T] = [M⁰LT⁻¹]
Both sides give [M⁰LT⁻¹], so the equation is dimensionally consistent.

Answer:
[T] = [M⁰L⁰T¹], [l] = [M⁰LT⁰], [g] = [M⁰LT⁻²]
T = k l^xg^y → [T] = [L]^x[LT⁻²]^y = [L^x+yT^−2y]
Comparing powers of L: x + y = 0. Comparing powers of T: −2y = 1, so y = −1/2
Then x = 1/2
T ∝ √(l/g), matching the known formula T = 2π√(l/g).

Answer:
G = Fr²/(m₁m₂)
[G] = [MLT⁻²][L²] / [M][M] = [ML³T⁻²] / [M²]
[G] = [M⁻¹L³T⁻²]

Answer:
(a) Four significant figures: 24.65 (the dropped digit 8 rounds the preceding 4 up to 5)
(b) Three significant figures: 24.6

Answer:
Raw sum = 4.2 + 0.362 + 12.1 = 16.662
The term with the fewest decimal places (4.2 and 12.1) has only one decimal place.
Answer: 16.7 m

Answer:
Raw product = 3.24 × 2.0 = 6.48
The term with fewer significant figures (2.0) has 2 significant figures.
Answer: 6.5

Answer:
Mean = (25.2+25.4+25.1+25.3+25.0)/5 = 126.0/5 = 25.2 cm
Absolute errors: 0.0, 0.2, 0.1, 0.1, 0.2
Mean absolute error = (0.0+0.2+0.1+0.1+0.2)/5 = 0.6/5 = 0.12 cm
Result: (25.2 ± 0.12) cm

Answer:
Percentage error = (Δa_mean/a_mean) × 100 = (0.12/25.2) × 100
≈ 0.48%

Answer:
Area A = l × b = 5.0 × 4.0 = 20 cm²
ΔA/A = Δl/l + Δb/b = 0.1/5.0 + 0.2/4.0 = 0.02 + 0.05 = 0.07
ΔA = 0.07 × 20 = 1.4 cm²
Result: A = (20 ± 1.4) cm²

Answer:
ρ = m × r⁻³ (constants like 4/3 and π contribute no error)
Δρ/ρ = Δm/m + 3(Δr/r) = 2% + 3(1%) = 2% + 3%
Maximum percentage error in density = 5%

Answer: Mean = (40.2+40.4+40.1+40.5+40.3)/5 = 201.5/5 = 40.3 s.

Answer: Absolute errors: 0.1, 0.1, 0.2, 0.2, 0.0. Mean absolute error = 0.6/5 = 0.12 s.

Answer: The slight variation among the student’s own five readings is due to random errors, from small differences in reaction time each time she starts and stops the watch. Taking multiple readings and averaging reduces the effect of this random error, which is exactly why her mean is more trustworthy than any single reading.

Answer: Less precise. Precision depends on the resolution of the instrument, and the wristwatch can only resolve to the nearest whole second while the stopwatch resolves to a hundredth of a second, so the stopwatch’s readings carry far more significant information about the true time.

Answer: A single reading carries the full effect of whatever random error happened to occur at that instant, with no way to know whether it lies above or below the true value. Repeating the measurement and averaging causes the random fluctuations, which are equally likely to be positive or negative, to largely cancel out, giving a mean that lies closer to the true value than any one reading is likely to.

Answer: [F] = [MLT⁻²]

Answer: [mv/t²] = [M][LT⁻¹]/[T²] = [MLT⁻³]

Answer: No. The left side has dimensions [MLT⁻²] while the right side has [MLT⁻³], and these do not match even if k is taken as a pure number. By the principle of homogeneity, the equation as written cannot be correct.

Answer: [k] must supply the missing [T] to turn [MLT⁻³] into [MLT⁻²], so [k] = [M⁰L⁰T¹], meaning k would have to carry the dimension of time.

Answer: Dimensional analysis can only confirm that the equation is dimensionally consistent, that is both sides reduce to [ML²T⁻²]. It cannot confirm the numerical constant of proportionality, here exactly 1, nor can it show that the underlying physical relationship between energy, mass and the speed of light is actually true; that confirmation came from experiment and from the theory of relativity, not from dimensions alone.

Answer: No. Dimensions describe only how a quantity is built from mass, length and time, not its full physical character. Work and torque both carry the dimensional formula [ML²T⁻²], yet work is a scalar representing energy transferred, while torque is a vector representing a turning effect. Dimensional agreement is a necessary condition for two quantities to be added or equated, but it is not sufficient to establish that they mean the same thing physically.

Answer: Dimensional analysis can identify the correct combination of powers of the relevant variables, but it is blind to any purely numerical, dimensionless factor multiplying that combination, such as 2π or 1/2, since these have no dimensions to constrain them. It also assumes in advance which variables are relevant; if the true relationship depends on a fourth quantity the analyst did not think to include, the method will confidently produce a formula that is simply wrong. The method is a powerful check and a useful shortcut, but it can never substitute for an actual derivation or experimental verification.

Answer: Each original reading has only two significant figures, so no arithmetic operation performed on them can manufacture extra genuine precision; reporting six decimal places falsely claims a level of certainty the measuring instrument never provided. The correct mean, 13.0/3 = 4.333…, should be rounded to match the precision of the original data, giving 4.3 cm, consistent with the two significant figures in each input.

Answer: Error propagation concerns how uncertain the final result is, not how large it is. Whether a quantity is multiplied or divided, an uncertainty in either input can push the final answer away from its true value in either direction, and there is no reason for the uncertainty from the numerator to partially cancel the uncertainty from the denominator, since the two are independent and unrelated measurements. The size of the result changes with multiplication or division, but the fractional uncertainty always accumulates, which is why the rule for relative error is identical in both cases.

Answer: [v] = [LT⁻¹], [P] = [ML⁻¹T⁻²], [ρ] = [ML⁻³]. Writing [LT⁻¹] = [ML⁻¹T⁻²]^x[ML⁻³]^y and comparing powers of M gives x + y = 0; comparing powers of T gives −2x = −1, so x = 1/2 and y = −1/2, giving v ∝ √(P/ρ). What dimensional analysis cannot supply is the numerical constant k, nor can it tell us whether the real physical law requires an additional factor, such as the ratio of specific heats γ that actually appears in the true formula v = √(γP/ρ); since γ is dimensionless, dimensional analysis has no way of detecting its presence at all.

Answer: Random error fluctuates unpredictably in sign from one reading to the next, so averaging many readings allows the positive and negative deviations to cancel, bringing the mean closer to the true value. Systematic error, by contrast, shifts every single reading in the same direction by roughly the same amount, for instance because a scale is miscalibrated by a fixed offset; averaging a hundred readings that are all wrong in the same way simply reproduces that same wrongness in the average. No amount of repetition can reveal or correct a systematic error, since repetition only cancels effects that vary from trial to trial.

Answer: Error propagation must account for the worst-case combination of uncertainties, since we do not know in advance whether each individual error will push its quantity above or below its true value. If A and B are overestimated and C is underestimated at the same time, all three mistakes conspire to push the final answer in the same direction, so the absolute errors must be added regardless of whether the corresponding quantity was added or subtracted in the formula. Subtracting a quantity in the formula does not subtract its uncertainty.

Answer: Writing 4200 leaves the number of significant figures ambiguous, since the trailing zeros could be placeholders or could be genuinely measured digits, and there is no way for a reader to tell which. Writing 4.20 × 10³ removes all ambiguity: every digit shown in the coefficient, including both the two and the zero, is explicitly stated to be significant, so the reader knows immediately that the measurement is trusted to three significant figures.

Answer: Distance has dimension [L], mass has dimension [M], and time has dimension [T]; these are three entirely independent base quantities with no common footing. The principle of homogeneity requires that only quantities of the same dimension can be added, and adding a length to a mass is as meaningless as trying to add three kilograms to five seconds and calling the result a number of anything in particular — the operation simply has no physical interpretation.

Answer: The first instrument is more precise, since it resolves to three decimal places against the second instrument’s two, giving finer detail in every reading. However, the first instrument is less accurate here, since its reading of 9.812 differs from the true value of 9.780 by 0.032, whereas the second instrument’s 9.75 differs by only 0.030, putting it slightly closer to the true value despite its coarser resolution. This illustrates precisely why precision and accuracy must be judged separately: a finer instrument is not automatically a more correct one.