Units and Measurements | ISC Class 11 Physics Notes

This note covers measurement, reference standards, systems of units, SI base and derived units, prefixes, length, mass and time measurements, experimental errors, significant figures, rounding, orders of magnitude, dimensional formulae, dimensional checks and conversion between systems of units.

Why does measurement require a number and a unit?

Definition: Measurement compares a physical quantity, a measurable property, with an accepted reference standard called a unit. A result requires both a numerical value and a unit.

Physics is quantitative: measurements allow physical descriptions to be compared and relationships to be tested. Saying that something is large or small is incomplete unless a standard of comparison is understood. The numerical value alone does not identify the size of a dimensional quantity.

Let Q denote a physical quantity, n its numerical value and u the chosen unit. Their relationship is Q = nu. If the unit changes, the numerical value changes inversely, while the quantity itself remains unchanged.

What makes a useful standard?

A useful standard is well defined, reproducible and stable. It should be accessible for comparison and accepted wherever measurements are exchanged. The choice of standard is conventional, but its consistent use lets different observers communicate results meaningfully.

Base quantities are selected as the independent starting quantities of a system. Their units are base or fundamental units. Derived quantities are related to base quantities through physical definitions or equations; their units are combinations of base units.

A system of units contains both base and derived units. It is unnecessary to establish an unrelated standard for every measurable quantity. Once length and time units are specified, for example, the unit of speed, meaning distance travelled per unit time, follows from their ratio.

A complete measurement also communicates its uncertainty. The unit describes the reference scale; the reported digits describe the precision. Changing the unit does not make an observation more precise or justify extra significant digits.

How are the systems of units and SI base units organised?

The International System of Units, abbreviated SI, provides internationally accepted names and symbols. The centimetre-gram-second, foot-pound-second and metre-kilogram-second systems are abbreviated CGS, FPS and MKS respectively. MKSA extends MKS by including the ampere as a base unit of electric current.

SI has seven base units. Length measures spatial extent, mass measures inertia, the resistance to a change in motion, and time measures duration. Electric current is the rate of flow of electric charge. Thermodynamic temperature specifies temperature on the absolute scale. Amount of substance measures the number of specified elementary entities.

Luminous intensity describes light output in a given direction with the appropriate visual weighting. In the table, square brackets mean “dimensions of”; L, M and T represent length, mass and time. The remaining dimensional symbols follow the corresponding base-unit notation.

What physical standards define the base units?

The second is defined by fixing the frequency of the specified transition between the hyperfine energy levels of an unperturbed caesium-133 atom in its lowest-energy state at 9192631770 hertz. Frequency counts cycles per unit time; hertz, symbol Hz, means one cycle per second and has dimension [T⁻¹].

The metre is defined by fixing the speed of light in vacuum at 299792458 m s⁻¹. The kilogram fixes the Planck constant at 6.62607015 × 10⁻³⁴ J s, where J denotes the joule, the energy unit. The Planck constant relates the energy of a radiation quantum, a discrete energy packet, to its frequency and has dimensions [ML²T⁻¹]. The metre and second are already defined.

The ampere fixes the elementary charge, the magnitude of the charge carried by a proton or electron, at 1.602176634 × 10⁻¹⁹ C. The coulomb, symbol C, is the charge unit: 1 C = 1 A s, with dimension [AT]. The kelvin fixes the Boltzmann constant at 1.380649 × 10⁻²³ J K⁻¹. This constant connects temperature with an energy scale and has dimensions [ML²T⁻²K⁻¹]. These definitions use constants rather than an object's changeable properties.

One mole contains exactly 6.02214076 × 10²³ specified elementary entities. State whether these are atoms, molecules, ions, electrons or another specified group. The candela fixes luminous efficacy, visually weighted light output per unit radiant power, at 683 lm W⁻¹ for monochromatic radiation, meaning radiation of one frequency, at 540 × 10¹² Hz. Here W is watt, the power unit; lm is lumen, the unit of total visually weighted light output, equal to cd sr, with sr denoting steradian, the solid-angle unit.

How do derived units and angle units follow from base units?

Derived units come from the equations defining quantities. Area measures surface extent and has unit m² and dimension [L²]. Volume measures occupied space and has unit m³ and dimension [L³]. Dividing a length by a time gives the speed unit m s⁻¹ and dimension [LT⁻¹].

Acceleration is the rate of change of velocity, where velocity is displacement per unit time and displacement is the directed change of position. Acceleration has unit m s⁻² and dimension [LT⁻²]. Force equals mass times acceleration for a constant-mass body in classical mechanics.

Force, denoted F, has unit newton, symbol N, and dimension [MLT⁻²]. Work transfers energy when a force acts through a displacement. For a constant force along the displacement, work equals their product. Work and energy have unit joule, symbol J, and dimension [ML²T⁻²].

Power is energy transferred per unit time; its unit is watt, symbol W, and dimension [ML²T⁻³]. Pressure is normal force per unit area; its unit is pascal, symbol Pa, and dimension [ML⁻¹T⁻²]. “Normal” means perpendicular to the surface.

Why do dimensionless angles have named units?

A plane angle measures the opening between two rays. For a circular arc of length s and radius r, the angle θ in radians is θ = s/r. A radian, symbol rad, corresponds to an arc length equal to the radius.

Two radii meet at O and bound a curved arc. The radius is labelled r, the small arc ds and the angle dθ. Here d indicates a small element; the diagram gives dθ = ds/r.

See Fig. 1.1(a) in your NCERT textbook

A solid angle measures the opening of a cone of directions in space. If it cuts out area A on a sphere of radius r centred at its vertex, its solid angle Ω in steradians is Ω = A/r². The steradian has symbol sr.

Lines from O bound a cone ending in a curved spherical patch labelled dA. The radius is r and the small solid angle is dΩ. The diagram gives dΩ = dA/r².

See Fig. 1.1(b) in your NCERT textbook

Both ratios cancel their length dimensions, so both angles are dimensionless. Radian and steradian identify the kind of angle being measured. These angle units are traditionally called supplementary units; they are not extra base units.

How should prefixes, special units and unit symbols be used?

A prefix attaches to a unit symbol to represent a decimal multiple or submultiple. The following prefixes cover factors from atto to tera. Case matters: a capital and a lower-case letter can have different meanings.

Which other units are useful?

A fermi equals 10⁻¹⁵ m, the femtometre. An angstrom, symbol Å, equals 10⁻¹⁰ m; this is now an outdated unit. A light year, symbol ly, is the distance light travels in vacuum in one year, so it measures length, not time.

An astronomical unit is a distance scale associated with the mean Earth-Sun separation. A parsec is the distance at which one astronomical unit subtends, or forms at the observer, an angle of one second of arc. A second of arc is an angular subdivision equal to one three-thousand-six-hundredth of a degree.

The unified atomic mass unit, symbol u, is one-twelfth the mass of a carbon-12 atom. Here u is a specific mass-unit symbol, distinct from the generic unit symbol used earlier. Use u, not amu. It is useful for expressing atomic masses.

What are the writing rules?

• Write unit names in lower case in ordinary prose: newton, joule and ampere. Their symbols N, J and A begin with capitals because the names honour scientists.
• Write unit symbols upright. They do not take plural endings or an abbreviation full stop: 25 cm, not 25 cms.
• Attach the prefix directly to the unit symbol: centimetre is cm. Separate multiplied unit symbols with spacing, as in N m.
• Use m s⁻² or m/s² for acceleration. Repeated division signs such as m/s/s are unsuitable.
• Apply a power to the complete prefixed unit: 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³.
• Form multiples and submultiples of mass by attaching prefixes to gram, although the SI base unit is kilogram.

Note: A unit conversion changes the numerical value, not the quantity or its precision. When an area or volume unit changes, square or cube the length conversion factor as appropriate.

How are length, mass and time measured in practice?

Direct measurement compares a quantity with a calibrated instrument. A metre scale measures lengths, a balance measures mass and a clock measures time intervals. Instrument choice depends on the size of the quantity and the precision required.

The least count is the smallest increment resolved by an instrument. Vernier callipers combine a main scale with a sliding scale to measure dimensions more finely. A screw gauge uses screw motion and a circular scale for small thicknesses or diameters.

For a screw gauge, pitch is the axial distance advanced in one complete rotation. If N is the number of equal circular-scale divisions and p is the pitch, both p and the least count have length units. Write least count = p/N.

Before measuring, check the zero reading. A zero error occurs when the instrument indicates a non-zero reading for zero input. Subtract the signed zero error from the observed reading. Increasing the number of scale divisions alone does not remove other sources of uncertainty.

How can a small dimension be made measurable?

A thin thread can be wound in closely touching turns around a cylinder. Measure the total width of the turns and divide by their number. This estimates one thread diameter from a larger, more easily measured length. Avoid gaps or overlapping turns.

Draw a cylinder with closely touching turns of thread and a scale along their total width. Label width w, number of turns N and thread diameter d. Show d = w/N; w and d have length units, while N is a count.

Repeated measurements help assess variability. For a recurring motion, measure the total time t for N complete oscillations. An oscillation is one complete repeating cycle; its period P is the time per cycle, so P = t/N, with SI unit s and dimension [T].

Record instrument readings before performing calculations. Retain units at each stage, apply known corrections and report a precision supported by the instrument. An exact count of turns or oscillations does not itself restrict the significant figures of the result.

What causes measurement errors, and how do accuracy and precision differ?

Accuracy describes closeness to the true value. Precision describes the spread of repeated measurements: a smaller spread means greater precision. A tightly grouped set of readings may still be inaccurate if a systematic error shifts all the observations.

An experimental error is an estimated uncertainty in a measurement. The actual error would be the difference between the measured and true values, but the true value is mostly not known. Therefore an uncertainty estimate should not be mistaken for an exactly known correction.

Random errors cannot be avoided; their magnitude may be reduced by repeated measurements and averaging. Systematic errors are those for which corrections can be applied and, in principle, removed. Repetition alone does not identify an incorrect zero setting.

How should repeated readings be summarised?

Let x₁, x₂, up to xₙ be n readings of the same quantity. The arithmetic mean, written x̄, is their sum divided by n: x̄ = (x₁ + x₂ + … + xₙ)/n. The mean has the same unit and dimensions as each reading.

For reading xᵢ, where i identifies a particular observation, its absolute deviation from the mean is Δxᵢ = |xᵢ − x̄|. The vertical bars mean magnitude, without a sign. The mean absolute deviation Δx̄ is the average of these non-negative deviations.

A simple repeated-reading estimate is written x̄ ± Δx̄, with the unit outside the pair. The symbol ± means plus or minus. This estimate describes the observed scatter; any known instrumental corrections and the instrument's resolution must also be considered.

For a directly measured quantity, the least count is generally taken as the maximum error in a simple laboratory estimate. Keep that qualification: it is a working estimate, not a guarantee that every possible experimental error is bounded by the least count.

How are absolute, relative and combined uncertainties calculated?

Absolute uncertainty Δx is an uncertainty expressed in the same unit as a measured value x. Here Δ indicates uncertainty. Relative uncertainty is the ratio of absolute uncertainty to the magnitude of the value: relative uncertainty = Δx/|x|, for non-zero x.

Percentage uncertainty is that ratio multiplied by 100: percentage uncertainty = (Δx/|x|) × 100%. Both relative and percentage uncertainty are dimensionless. Using consistent units before dividing is essential; the unit cancellation is part of the calculation.

Which errors add during calculations?

Let A and B now denote measured quantities, ΔA and ΔB their positive absolute uncertainties, and Z a result calculated from them. For sums or differences of like quantities, the maximum absolute uncertainties add: ΔZ = ΔA + ΔB.

For a product Z = AB or a quotient Z = A/B, the first-order maximum relative uncertainty is ΔZ/|Z| = ΔA/|A| + ΔB/|B|. This approximation assumes small fractional uncertainties and neglects their products. Subtraction or division does not make worst-case uncertainties cancel.

For Z = AᵖBᑫ, where p and q are exact exponents, the first-order maximum relative uncertainty is |p|ΔA/|A| + |q|ΔB/|B|. A measured quantity raised to a larger power therefore makes a larger contribution for the same fractional uncertainty.

The length and breadth of a rectangular sheet may be written as 16.2 ± 0.1 cm and 10.1 ± 0.1 cm. Their approximate percentage uncertainties are 0.6% and 1%. The product has about 1.6% uncertainty and is reported as 164 ± 3 cm².

These rules estimate a maximum uncertainty. They are not instructions to add signed measurement deviations, which could cancel accidentally. Report the result and absolute uncertainty to compatible decimal places, and keep enough intermediate digits to avoid additional rounding errors.

How are significant figures counted and orders of magnitude estimated?

Significant figures are the reliably known digits together with the first uncertain digit. Normally a reported measurement includes this first uncertain digit. For a pendulum period of 1.62 s, the digits 1 and 6 are reliable and 2 is uncertain.

The number 287.5 cm has four significant figures. Reporting more digits than the measurement supports gives a misleading impression of precision. Decimal places and significant figures describe different features of a number and should not be treated as interchangeable.

Which zeroes count?

• All non-zero digits are significant.
• Zeroes between non-zero digits are significant: 2.308 cm has four significant figures.
• Leading zeroes locate the decimal point and are not significant: 0.02308 m also has four significant figures.
• Trailing zeroes in a decimal measurement are significant: 3.500 and 0.06900 each have four significant figures.
• Trailing zeroes in a whole number without a decimal point are conventionally not significant, but converted measurements can be ambiguous.
• Exact counting numbers and exact conversion factors do not restrict the result's significant figures.

Scientific notation avoids ambiguous trailing zeroes. Write a value as a × 10ᵇ, where a is the coefficient between 1 and 10 and b is an integer exponent. The coefficient retains the significant figures; the power of ten locates the scale.

Thus 4.700 m = 4.700 × 10² cm = 4.700 × 10³ mm. Every form has four significant figures. The conversion changes neither the measurement nor its precision, even if ordinary whole-number notation obscures the significant trailing zeroes.

What does order of magnitude mean?

An order of magnitude states a quantity's approximate scale as a power of ten. In the convention used here, round a to 1 when a ≤ 5 and to 10 when a > 5. The resulting exponent expresses the order in the stated unit.

The Earth's diameter, 1.28 × 10⁷ m, is of order 10⁷ m. The hydrogen atom's diameter, 1.06 × 10⁻¹⁰ m, is of order 10⁻¹⁰ m. Their orders differ by 17. This comparison is a scale estimate, not an exact ratio.

How should arithmetic results be rounded?

Multiplication and division retain as many significant figures as the measured input with the fewest significant figures. Addition and subtraction retain the decimal place of the least precise input, after all values have been expressed in the same unit.

Worked example 1. A sample has mass m = 4.237 g and volume V = 2.51 cm³. Find its mass density ρ, meaning mass per unit volume, with SI unit kg m⁻³ and dimension [ML⁻³]. Formula: ρ = m/V. Substitute: 4.237/2.51 g cm⁻³. Answer: 1.69 g cm⁻³, to three significant figures, limited by the volume.

Worked example 2. Add measured masses 436.32 g, 227.2 g and 0.301 g. Formula: total mass = sum of the masses. Substitute: 436.32 + 227.2 + 0.301 = 663.821. Answer: 663.8 g, because the least precise input reaches only the tenths place.

Subtraction can reduce the number of significant figures sharply. The measured difference 0.307 m − 0.304 m is 0.003 m. Writing 0.00300 m would claim a finer decimal precision than either original length supports.

What happens when the discarded digit is five?

When the discarded digit exceeds 5, increase the preceding digit by one. When it is below 5, leave the preceding digit unchanged. For an exact halfway case ending in 5, use the convention that leaves the retained digit even.

Thus 2.746 becomes 2.75 and 1.743 becomes 1.74 to three significant figures. The halfway values 2.745 and 2.735 both become 2.74: the first retains an even 4, while the second raises the odd 3.

Worked example 3. Each side l of a cube measures 7.203 m. Calculate its total surface area A and volume V. Formula: A = 6l²; V = l³. Substitute: A = 6(7.203)² m² and V = (7.203)³ m³. Answer: A = 311.3 m² and V = 373.7 m³, each to four significant figures. The geometric factor 6 is exact.

In a multistep calculation, keep one extra significant digit in intermediate results and round at the end. Premature rounding can change later answers. A calculator's long display does not create precision absent from the measurements.

Worked example 4. A substance has mass m = 5.74 g and volume V = 1.2 cm³. Calculate density ρ. Formula: ρ = m/V. Substitute: 5.74/1.2 g cm⁻³. Answer: 4.8 g cm⁻³. The volume has two significant figures, so the density must be reported to two.

How are dimensional formulae of quantities and constants obtained?

Dimensions describe a physical quantity through the powers of base quantities. They do not include its numerical magnitude. A dimensional formula displays these powers; a dimensional equation equates the dimensions of a quantity to that formula.

For volume V, write [V] = [M⁰L³T⁰], or simply [L³]. A zero power means independence from that base dimension. For speed v, [v] = [LT⁻¹]. Different values and different units of speed retain the same dimensions.

Derivation: Dimensions and base units of force

1. For a constant-mass body in classical mechanics, let F be force, m mass and a acceleration. Use F = ma.
2. Acceleration is change of velocity per unit time. Since velocity has dimensions [LT⁻¹], acceleration has dimensions [LT⁻²].
3. Multiply mass and acceleration dimensions: [F] = [M][LT⁻²]. For SI units, multiply kg by m s⁻² in the same way.

[F] = [MLT⁻²]; 1 N = 1 kg m s⁻². The dimensional exponents are 1 in mass, 1 in length and −2 in time.

How can an equation reveal a constant's dimensions?

The gravitational constant G is the proportionality constant in Newton's gravitational force equation. For point masses m₁ and m₂ separated by distance r, the force magnitude is F = Gm₁m₂/r². The equation gives G = Fr²/(m₁m₂).

Therefore [G] = [MLT⁻²][L²]/[M²] = [M⁻¹L³T⁻²]. Its SI unit is N m² kg⁻², equivalent to m³ kg⁻¹ s⁻². A physical constant can have dimensions; “constant” means fixed in the relevant relationship, not necessarily dimensionless.

Likewise, the Planck constant relates energy to frequency. Dividing energy dimensions [ML²T⁻²] by frequency dimensions [T⁻¹] gives [ML²T⁻¹], with SI unit J s. Distinguish a dimensional constant of this kind from a pure numerical coefficient.

Dimensional analysis uses these expressions to check equations and convert units. Start from a defining equation rather than memorising isolated strings of letters: this makes each exponent explainable and reduces confusion between quantities with similar names.

How does dimensional homogeneity test an equation?

Definition: The principle of homogeneity requires every term added or subtracted in a physical equation to have the same dimensions. Both sides of a valid equation must have matching dimensions.

Consider motion with uniform acceleration, meaning acceleration remains constant. Let x be position, x₀ initial position, v₀ initial velocity, a acceleration and t elapsed time. Positions have unit m, velocity m s⁻¹, acceleration m s⁻² and time s.

The equation is x = x₀ + v₀t + ½at². Check each term separately: [x] and [x₀] are [L]; [v₀t] = [LT⁻¹][T] = [L]; and [at²] = [LT⁻²][T²] = [L]. The numerical factor ½ has no dimensions.

What does passing the check establish?

The equation is dimensionally consistent. That is a necessary test, but it does not prove that the equation is physically correct. Dimensionless coefficients and functions can change without affecting dimensions, so a successful check cannot determine the exact relationship.

For example, kinetic energy is energy associated with motion. Its dimensions are [ML²T⁻²]. Both ½mv² and (3/16)mv² have these dimensions, although dimensional analysis cannot choose the correct coefficient. In classical mechanics the kinetic-energy expression is ½mv².

The expression ma has force dimensions [MLT⁻²], so it cannot by itself represent energy. Nor can ma be added to ½mv² in an energy formula: the two terms have different dimensions. Check sums term by term before comparing sides.

Arguments of trigonometric, logarithmic and exponential functions must be dimensionless. These are functions such as sine, logarithm and the exponential. A ratio of like quantities can supply a dimensionless argument even when its numerator and denominator each carry units.

The method does not distinguish different physical quantities sharing dimensions, and it cannot supply a missing dimensionless coefficient. An equation failing the consistency test is wrong; an equation passing it is not thereby proved right.

How can dimensions convert units and check a calculation?

The same physical quantity has the same magnitude whichever unit system is used. If its dimensional formula is [MᵃLᵇTᶜ], the exponents a, b and c tell us how mass, length and time conversion factors enter.

Derivation: Conversion between two unit systems

Let n₁ and n₂ be numerical values in two systems. Let M₁, L₁, T₁ be the first system's actual mass, length and time units, and M₂, L₂, T₂ those of the second. Here these symbols denote units, not dimensional labels.

1. Express the quantity in the first system as n₁M₁ᵃL₁ᵇT₁ᶜ.
2. Express the same quantity in the second system as n₂M₂ᵃL₂ᵇT₂ᶜ and equate the two expressions.
3. Divide by the second system's compound unit. Each ratio of like base units becomes a numerical conversion factor raised to the appropriate power.

n₂ = n₁(M₁/M₂)ᵃ(L₁/L₂)ᵇ(T₁/T₂)ᶜ. A negative dimensional exponent requires the corresponding inverse conversion factor.

Worked example 5. Convert 1 J into the CGS energy unit erg, defined as 1 g cm² s⁻². Given 1 J = 1 kg m² s⁻², 1 kg = 10³ g and 1 m = 10² cm. Formula: energy conversion factor = mass factor × length factor squared × time factor to power −2. Substitute: 10³ × (10²)² × 1. Answer: 1 J = 10⁷ erg. The second is unchanged.

How do units and significant figures work together?

Worked example 6. Find one light year using light speed c = 3.00 × 10⁸ m s⁻¹, a year of 365.25 days and 86400 s per day. Let t be the time in seconds and d the distance. Formula: t = number of days × seconds per day; d = ct. Substitute: t = 365.25 × 86400 s, or approximately 3.1557 × 10⁷ s; d = (3.00 × 10⁸)(3.1557 × 10⁷) m. Answer: d = 9.47 × 10¹⁵ m, to three significant figures.

Worked example 7. A microscope magnifies a hair 100 times. Across 20 observations, its mean apparent width is 3.5 mm. Find the estimated thickness d. Magnification M is the ratio of apparent width w to actual thickness. Formula: d = w/M. Substitute: d = 3.5/100 mm. Answer: 0.035 mm. The 20 observations have already been averaged, so do not divide by 20 again.

The time unit cancels in the distance calculation. That cancellation checks the kind of result, while significant-figure rules determine its reported precision. Neither check replaces the other: a correctly rounded number with the wrong dimensions is still invalid.

A reliable sequence is to identify the required quantity, define every symbol, select the relationship, convert incompatible units, substitute the given data and inspect the result. Finish by checking dimensions and rounding only to a precision supported by those data.

Glossary

• Unit — An accepted reference standard against which the magnitude of a physical quantity is compared.
• Base unit — A unit assigned to one of the independent base quantities chosen for a system.
• Derived unit — A unit expressed as a combination of base units through a physical relationship.
• Least count — The smallest increment that a measuring instrument can resolve in its readings.
• Accuracy — Closeness of a measured value to the true value of the quantity.
• Precision — The degree of agreement indicated by the spread of repeated measurements.
• Systematic error — An error associated with an identifiable instrumental, procedural or observational cause that can in principle be corrected.
• Random error — Untraceable residual variation whose magnitude may be reduced by repeated measurement and averaging.
• Relative uncertainty — Absolute uncertainty divided by the magnitude of the measured value, giving a dimensionless ratio.
• Significant figures — All reliably known digits of a measurement together with its first uncertain digit.
• Order of magnitude — An approximate scale expressed through a power of ten in a stated unit.
• Dimensional formula — An expression showing the powers of base quantities that represent a physical quantity.
• Dimensional homogeneity — The requirement that all terms combined by addition or subtraction have the same dimensions.

Common errors and misconceptions

• Misconception: A light year measures time. Correct: It measures the distance travelled by light in vacuum in one year.
• Misconception: Repeating a measurement removes its zero error. Correct: Repetition may reduce random error, while a known zero error needs a correction.
• Misconception: Precise readings must be accurate. Correct: Closely grouped readings can still be shifted from the true value by systematic error.
• Misconception: Every zero in a measurement is insignificant. Correct: Internal zeroes and significant trailing decimal zeroes convey precision.
• Misconception: All arithmetic uses the fewest-significant-figures rule. Correct: Addition and subtraction use decimal-place precision; multiplication and division use significant figures.
• Misconception: A dimensionally consistent equation must be correct. Correct: Dimensional analysis cannot determine dimensionless numerical coefficients or establish the complete physical relationship.
• Misconception: A volume conversion uses the length factor once. Correct: The complete length conversion factor must be cubed, so 1 cm³ = 10⁻⁶ m³.

Exam-style questions with model answers

Q1. Distinguish accuracy from precision in measurement. [2 marks]

1. Accuracy describes how close a measured value is to the true value of the quantity.
2. Precision describes the spread of repeated measurements; a smaller spread means greater precision, even if systematic error makes the readings inaccurate.

Q2. A sample has mass 4.237 g and volume 2.51 cm³. Calculate its density, defined as mass divided by volume, with appropriate significant figures. [3 marks]

1. Write ρ = m/V, where ρ is density, m is the mass and V is the volume. Substitution gives ρ = 4.237 g ÷ 2.51 cm³.
2. The mass has four significant figures but the volume has three. A division result must retain the smaller number, so the answer requires three significant figures.
3. Rounding the quotient accordingly gives ρ = 1.69 g cm⁻³. The unit is mass per volume and must accompany the numerical answer.

Q3. Add the measured masses 436.32 g, 227.2 g and 0.301 g. Explain the reporting rule and state the final mass. [3 marks]

1. All three masses already use grams, so they can be added directly. Their arithmetic sum is 436.32 + 227.2 + 0.301 = 663.821 g.
2. For addition, the limiting feature is decimal-place precision. The value 227.2 g is reported only to one decimal place, making the tenths place the limit for the result.
3. Round the sum to that place to obtain 663.8 g. Applying a multiplication-style significant-figure rule would misrepresent the precision of this addition.

Q4. A cube has measured side l = 7.203 m. Its total surface area is A = 6l² and its volume is V = l³. Calculate both quantities and explain their significant figures. [4 marks]

1. The measured side has four significant figures. Both powers of this length therefore support four significant figures in their final results.
2. Substituting into the area formula gives A = 6(7.203)² m², which rounds to 311.3 m².
3. Substituting into the volume formula gives V = (7.203)³ m³, which rounds to 373.7 m³.
4. The factor 6 is an exact geometric number, so it does not limit precision. The units must be squared for area and cubed for volume.

Q5. Test x = x₀ + v₀t + ½at² for dimensional homogeneity. Here x and x₀ are final and initial positions, v₀ is initial velocity, a is constant acceleration and t is elapsed time. Use [x] = [x₀] = [L], [v₀] = [LT⁻¹], [a] = [LT⁻²] and [t] = [T]. State the limitation of your conclusion. [5 marks]

1. The left-hand term x has dimension [L], where L represents length. Every term added on the right must therefore also have the dimension of length.
2. The initial-position term x₀ has dimension [L]. It is dimensionally compatible with x, as both quantities describe position.
3. The product v₀t has dimension [LT⁻¹][T] = [L]. Its time powers cancel, leaving a length.
4. The product ½at² has dimension [LT⁻²][T²] = [L]. The coefficient ½ is dimensionless and does not affect the test.
5. All terms have matching dimensions, so the equation is dimensionally homogeneous. This does not prove the exact physical equation or establish the numerical coefficient; dimensional consistency is necessary but not sufficient.

Q6. Convert 1 J to ergs. Use 1 J = 1 kg m² s⁻², 1 erg = 1 g cm² s⁻², 1 kg = 10³ g and 1 m = 10² cm; the second is the same in both systems. Explain how the dimensional exponents control the conversion. [5 marks]

1. A joule has the compound unit kg m² s⁻². Its mass, length and time exponents are respectively 1, 2 and −2.
2. Replace the mass unit using 1 kg = 10³ g. Because mass occurs to the first power, this contributes a factor of 10³.
3. Replace the squared length unit using 1 m² = (10² cm)² = 10⁴ cm². The length conversion factor must be squared.
4. The second is unchanged, so its conversion factor is one, including when raised to the power −2. The combined factor is therefore 10³ × 10⁴ = 10⁷.
5. Hence 1 J = 10⁷ g cm² s⁻² = 10⁷ erg. The numerical value changes to match the smaller energy unit, while the physical energy remains unchanged.

Q7. A rectangular sheet has length 16.2 ± 0.1 cm and breadth 10.1 ± 0.1 cm. For small uncertainties, add percentage uncertainties when multiplying; use their rounded values 0.6% and 1%. Find the approximate area uncertainty and report the area. [4 marks]

1. Let A denote area, l length and b breadth. The central value follows from A = lb = 16.2 × 10.1 = 163.62 cm².
2. The maximum percentage uncertainty in this product is approximately 0.6% + 1% = 1.6%, using the supplied rounded uncertainties.
3. The corresponding absolute uncertainty is approximately 1.6% of 163.62 cm², or 2.6 cm², which rounds to 3 cm².
4. Report the central value to the same decimal place as the uncertainty: A = 164 ± 3 cm². This is an approximate maximum-uncertainty estimate.

Key takeaways

• A measurement combines a numerical value with an accepted unit; changing that unit does not change the physical quantity.
• SI has seven base units, while derived units follow from relationships among physical quantities.
• Radian and steradian are named angle units even though plane angle and solid angle are dimensionless.
• Accuracy concerns closeness to the true value; precision concerns the spread of repeated measurements.
• For small uncertainties, products and quotients add maximum relative uncertainties; sums and differences add maximum absolute uncertainties.
• Significant figures include reliable digits and the first uncertain digit, so unsupported calculator digits should not be reported.
• Addition uses decimal-place precision; multiplication uses significant figures; keep an extra intermediate digit before final rounding.
• Dimensional homogeneity can reject an inconsistent equation, but cannot establish the correct dimensionless numerical coefficient.

Test yourself

Why is a numerical value without a unit incomplete?

The unit supplies the reference standard; without it, the number does not specify the magnitude of a dimensional quantity.

What are the SI units of amount of substance and luminous intensity?

Amount of substance uses the mole, symbol mol; luminous intensity uses the candela, symbol cd.

Why is a plane angle dimensionless?

Its radian measure is arc length divided by radius, so the length dimensions cancel.

How many significant figures are present in 0.06900?

There are four significant figures: the leading zeroes do not count, but both trailing decimal zeroes do.

Round 2.745 to three significant figures using the even-digit convention.

The result is 2.74, because the discarded halfway digit is 5 and the retained preceding digit is even.

Does changing 4.700 m into centimetres change its precision?

No. Writing 4.700 × 10² cm preserves all four significant figures and the same measurement precision.

Why can repeated measurements remain inaccurate?

A systematic error may shift the readings together, even when their spread is small and their precision is high.

What can dimensional analysis say about a dimensionless coefficient?

It cannot determine the coefficient's value, because changing that number leaves the dimensions unchanged.