Physical Quantities

Physics is an experimental science: every law in this book began as a number read off an instrument. That number is never exact. This unit is about how much of a measured number you are entitled to believe — the language of precision, error and significant figures — and about the hidden structure that every physical quantity carries with it, its dimensions. Dimensions let you check an unfamiliar equation in ten seconds, convert a constant between unit systems, and sometimes guess a formula you were never taught.

1.1 Precision and significant figures

Measurement, units and standards

A physical quantity is any property of matter or of a phenomenon that can be measured. Measuring it means comparing it with a standard, so every measurement is a numerical value multiplied by a unit:

For a given quantity the product nu is fixed, so a smaller unit always gives a larger number. This one relation is the whole of unit conversion:

Quantities that are independent of one another are called fundamental (base) quantities; everything else is derived from them. The SI system takes seven base quantities.

Base quantity	SI unit	Symbol	Dimension
Length	metre	m	[L]
Mass	kilogram	kg	[M]
Time	second	s	[T]
Electric current	ampere	A	[I]
Thermodynamic temperature	kelvin	K	[K]
Amount of substance	mole	mol	[N]
Luminous intensity	candela	cd	[J]

Plane angle (radian) and solid angle (steradian) are supplementary units; both are ratios of two lengths or two areas and are therefore dimensionless.

Accuracy, precision and least count

Precision is capped by the least count of the instrument — the smallest change it can display.

Instrument	Least count
Metre rule	1 mm = 0.1 cm
Vernier callipers (10 divisions)	0.1 mm = 0.01 cm
Screw gauge (pitch 1 mm, 100 divisions)	0.01 mm
Spherometer	0.01 mm
Mechanical stop-watch	0.1 s
Laboratory thermometer	1 °C

For a vernier scale, the least count is the difference between one main-scale division (MSD) and one vernier-scale division (VSD):

For a screw gauge, L.C. = pitch ÷ number of circular-scale divisions.

Errors in measurement

	Systematic error	Random error
Cause	faulty instrument, zero error, wrong technique, personal bias	unpredictable fluctuations: judgement, draughts, supply ripple
Sign	always in the same direction	equally likely either way
Effect	spoils accuracy	spoils precision
Remedy	calibrate, correct for zero error, change method	repeat and take the mean

If a quantity is measured n times giving a_1, a_2, \dots, a_n, the best estimate is the arithmetic mean \bar{a}. Then

The result is quoted as a = \bar{a} \pm \Delta a_{mean}.

Significant figures

Rule	Example	Count
All non-zero digits count	2345	4
Zeros between non-zero digits count	2.005	4
Leading zeros never count	0.0025	2
Trailing zeros after a decimal point count	2.500	4
Trailing zeros in a whole number are ambiguous — use standard form	4700 → 4.70\times10^{3}	3
Changing the unit never changes the count	2.50 cm = 25.0 mm = 0.0250 m	3
Exact (counted) numbers and pure numbers such as \pi	12 eggs, 2\pi	infinite

Rounding off. If the digit to be dropped is greater than 5, raise the preceding digit by one; if it is less than 5, leave the preceding digit unchanged; if it is exactly 5 with nothing after it, round so that the preceding digit becomes even (4.25 \to 4.2, 4.35 \to 4.4).

Arithmetic. Two different rules, and mixing them up is the commonest error:

• Addition and subtraction — the answer keeps as many decimal places as the term with the fewest decimal places.
• Multiplication and division — the answer keeps as many significant figures as the factor with the fewest significant figures.

Combining uncertainties

Let \Delta A and \Delta B be the absolute errors in A and B. For the maximum possible error in a result Z:

Operation	Rule
Z = A + B or Z = A - B	\Delta Z = \Delta A + \Delta B (absolute errors add)
Z = AB or Z = A/B	\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}
Z = A^{p}B^{q}/C^{r}	\dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B} + r\dfrac{\Delta C}{C}

1.2 Dimensions and uses of dimensional analysis

For example, force = mass \times acceleration, so

Quantity	Defining relation	Dimensional formula	SI unit
Velocity	displacement/time	[LT^{-1}]	m s⁻¹
Acceleration	velocity/time	[LT^{-2}]	m s⁻²
Force	mass × acceleration	[MLT^{-2}]	N
Momentum, impulse	mass × velocity	[MLT^{-1}]	kg m s⁻¹
Work, energy, torque	force × distance	[ML^{2}T^{-2}]	J
Power	work/time	[ML^{2}T^{-3}]	W
Pressure, stress, modulus	force/area	[ML^{-1}T^{-2}]	Pa
Density	mass/volume	[ML^{-3}]	kg m⁻³
Frequency	1/period	[T^{-1}]	Hz
Gravitational constant G	Fr^{2}/m_1m_2	[M^{-1}L^{3}T^{-2}]	N m² kg⁻²
Surface tension	force/length	[MT^{-2}]	N m⁻¹
Coefficient of viscosity	F/(A\,dv/dx)	[ML^{-1}T^{-1}]	Pa s
Strain, angle, refractive index	ratio	[M^{0}L^{0}T^{0}]	none

Use 1 — checking an equation

Test v^{2} = u^{2} + 2as. Left side [LT^{-1}]^{2} = [L^{2}T^{-2}]. First term on the right [L^{2}T^{-2}]; second term [LT^{-2}][L] = [L^{2}T^{-2}]. All three agree, so the equation is dimensionally correct. Note that the pure number 2 is invisible to this test — dimensional analysis can prove an equation wrong but never fully right.

Use 2 — converting between systems of units

Use 3 — deriving a relation

If a quantity is known to depend on at most three others, assume a product of powers and match dimensions on both sides.

Limitations of dimensional analysis

1. It cannot supply dimensionless constants. The 2\pi in T = 2\pi\sqrt{l/g}, and the \tfrac{1}{2} in \tfrac{1}{2}mv^{2}, must come from theory or experiment.
2. It fails if the quantity depends on more than three base quantities, because M, L and T give only three equations.
3. It cannot handle equations that are a sum of terms, such as s = ut + \tfrac{1}{2}at^{2} — it can only check them, not build them.
4. It cannot deal with trigonometric, exponential or logarithmic functions, though it does tell you their arguments must be dimensionless.
5. It cannot distinguish quantities with the same dimensions (work and torque, both [ML^{2}T^{-2}]), and cannot tell a scalar from a vector.

Chapter summary

• A measurement is Q = nu, and n_1u_1 = n_2u_2; SI has seven base quantities.
• Accuracy = closeness to the true value; precision = closeness of repeated readings, limited by the least count.
• Vernier L.C. = 1\ \text{MSD} - 1\ \text{VSD}; screw-gauge L.C. = pitch ÷ number of circular divisions.
• Systematic errors spoil accuracy and are removed by calibration; random errors spoil precision and are reduced by averaging. Quote results as \bar{a} \pm \Delta a_{mean}.
• Significant figures = certain digits + first uncertain digit. Addition keeps the least decimal places; multiplication keeps the least significant figures.
• Errors combine as \Delta Z = \Delta A + \Delta B for sums, and \Delta Z/Z = p\,\Delta A/A + q\,\Delta B/B for products and powers.
• The principle of homogeneity says every term of a physical equation has the same dimensions. Dimensional analysis checks equations, converts units via n_2 = n_1 (M_1/M_2)^{a}(L_1/L_2)^{b}(T_1/T_2)^{c}, and derives simple relations.
• It cannot give dimensionless constants, handle more than three variables, build equations with several terms, or separate work from torque.

Practice questions

Group A — Multiple choice (1 mark each)

1. The number of significant figures in 0.00250\ \text{m} is [1]a2b3c5d6
2. Which pair has the same dimensional formula? [1]awork and powerbforce and momentumcwork and torquedpressure and force
3. The dimensional formula of the universal gravitational constant is [1]a[M^{-1}L^{3}T^{-2}]b[ML^{3}T^{-2}]c[M^{-1}L^{2}T^{-2}]d[ML^{2}T^{-1}]
4. The radius of a sphere is measured with an error of 2 %. The error in its volume is [1]a2 %b4 %c6 %d8 %
5. A vernier callipers has 10 vernier divisions matching 9 main-scale divisions of 1 mm each. Its least count is [1]a1 mmb0.5 mmc0.1 mmd0.01 mm
6. Which of the following is dimensionless? [1]astressbstraincsurface tensiondforce constant

Answers to Group A

1. (b) — leading zeros are place-holders; the 2, 5 and the trailing 0 count.

2. (c) — both are force × distance, [ML^{2}T^{-2}].

3. (a) — G = Fr^{2}/m_1m_2 = [MLT^{-2}][L^{2}]/[M^{2}].

4. (c) — V \propto r^{3}, so the percentage error is tripled.

5. (c) — L.C. = 1 - 0.9 = 0.1 mm, i.e. 1 mm ÷ 10.

6. (b) — strain is a ratio of two lengths.

Group B — Short answer (5 marks each)

1. Distinguish between accuracy and precision, giving one example of each. State what limits the precision of an instrument. [5]
2. State the rules for counting significant figures. Hence give the number of significant figures in (i) 0.00602, (ii) 6.023\times10^{23}, (iii) 4.700\ \text{m}. [5]
3. The length, breadth and thickness of a metal block are 5.12\ \text{cm}, 2.34\ \text{cm} and 1.05\ \text{cm}. Calculate its volume to the correct number of significant figures. [5]
4. Check the correctness of the equation T = 2\pi\sqrt{l/g} by the method of dimensions, and explain why the test cannot confirm the factor 2\pi. [5]
5. A physical quantity is given by P = a^{3}b^{2}/(\sqrt{c}\,d). If the percentage errors in a, b, c and d are 1 %, 2 %, 3 % and 4 % respectively, find the maximum percentage error in P. [5]

Answers to Group B

1. Outline: accuracy = closeness to the true value (a zero-error instrument is inaccurate); precision = closeness of repeated readings, set by the least count.

2. (i) 3 — leading zeros do not count. (ii) 4 — only the mantissa 6.023 counts. (iii) 4 — the trailing zeros follow a decimal point.

3. V = 5.12 \times 2.34 \times 1.05 = 12.57984\ \text{cm}^{3}. The fewest significant figures among the factors is 3 (in 1.05), so V = 12.6\ \text{cm}^{3}.

4. LHS = [T]. RHS = \sqrt{[L]/[LT^{-2}]} = \sqrt{[T^{2}]} = [T]. The two sides agree, so the equation is dimensionally correct. 2\pi is a pure number with dimensions [M^{0}L^{0}T^{0}], so replacing it by any other number leaves the check unchanged — dimensional analysis cannot detect it.

5. \dfrac{\Delta P}{P}\times100 = 3(1) + 2(2) + \tfrac{1}{2}(3) + 1(4) = 3 + 4 + 1.5 + 4 = 12.5\ \%.

Group C — Long answer (8 marks each)

1. (a) Define absolute error, mean absolute error, relative error and percentage error. [3](b) In an experiment the mass of a body is found to be 20.00 \pm 0.05\ \text{g} and its volume 5.00 \pm 0.02\ \text{cm}^{3}. Calculate its density, the percentage error and the absolute error, and quote the result properly. [5]
2. (a) State the principle of homogeneity of dimensions and describe three uses of dimensional analysis. [4](b) The value of the gravitational constant is G = 6.67\times10^{-11}\ \text{N m}^{2}\ \text{kg}^{-2} in SI units. Convert it into the CGS system. [4]

Answers to Group C

1. (a) See the definitions in §1.1.

1. (b) \rho = m/V = 20.00/5.00 = 4.00\ \text{g cm}^{-3}.

\frac{\Delta\rho}{\rho} = \frac{0.05}{20.00} + \frac{0.02}{5.00} = 0.0025 + 0.0040 = 0.0065

so the percentage error is 0.65\ \% and \Delta\rho = 0.0065 \times 4.00 = 0.026 \approx 0.03\ \text{g cm}^{-3}. The result is \rho = 4.00 \pm 0.03\ \text{g cm}^{-3}.

2. (a) Outline: state homogeneity, then the three uses — checking equations, converting units, deriving relations.

2. (b) [G] = [M^{-1}L^{3}T^{-2}], so a = -1, b = 3, c = -2 with M_1/M_2 = 10^{3}, L_1/L_2 = 10^{2}, T_1/T_2 = 1:

n_2 = 6.67\times10^{-11}\,(10^{3})^{-1}(10^{2})^{3} = 6.67\times10^{-11}\times10^{-3}\times10^{6}

giving G = 6.67\times10^{-8}\ \text{dyne cm}^{2}\ \text{g}^{-2}.