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Units and Dimensions

Chapter Notes


Physical Quantities

Base or Fundamental Quantities

Base or Fundamental Units

Derived Units

Systems of Units

The SI system

Fundamental or Base Units

Supplementary Units

Prefixes used for Multiples and Fractions of base unit

Measurement of Length

Normal Distances

Large Distances

Parllax and Parallax Method

Angular Diameter or Apparent Size

Measurement of Very Small Distances

Size of a Molecule

Range of Lengths

Special units of length

For short lengths - Fermi and Angstrom

For large distances

Astronomical Unit

Light Year


Measurement of Mass

Atomic and Subatomic Masses

Large Masses

Range of Masses

Measurement of Time

Atomic standard of time

Range of Time

Accuracy and Precision in Measurement

Error - Uncertainity in Measurement

Types of Errors

Absolute Error

Mean Absolute Error

Relative Error

Percentage Error

Combination of Errors

  1. Error of sum or difference

  2. Error of a product or a quotient

  3. Error in Case of a Quantity Raised to a Power

Significant figures

Scientific Notation and Order of Magnitude

Addition or subtraction with significant figures

Multiplication and division with significant figures

Rounding Off

Dimentional Analysis - Dimensional Formula

Dimensional Constants

Dimensional Variables

Dimensionless Constants

Dimensionless Variables

Importance of Dimentional Analysis

Checking the Dimensional Consistency of Equations

The principle of homogeneity of dimensions

Deducing Relation among the Physical Quantities

Limitations of Dimensional Analysis



Chapter Notes

Measurement of physical quantities

Physics is a quantitative science based on measurement of physical quantities.

Base or Fundamental quantities

There are certain physical quantities, called basic quantities, on which other physical quantities, called derived quantities, are dependent.


Measurement of any physical quantity involves comparison with certain basic arbitrarily chosen and internationally accepted reference standards called UNIT.

Base or Fundamental Units

The units for the fundamental or base quantities are called fundamental or base units. There are also two supplementary units, namely radian (unit of plane angle) and steradian (unit of solid angle).

Derived Units

The units of other physical quantities, namely derived quantities are called derived units and they can be expressed as a combination of the base units.

Systems of Units

A complete set of units, both the base units and derived units, is known as the system of units.

Earlier different systems, the CGS, the FPS (or British) system and the MKS system were in use.

  • In CGS system they were centimetre, gram and second respectively.

  • In FPS system they were foot, pound and second respectively.

  • In MKS system they were metre, kilogram and second respectively

The SI system

To avoid confusion, standard scheme of symbols, units and abbreviations, was developed and recommended by General Conference on Weights and Measures in 1971. This internationally accepted system is called Système Internationale d’ Unites (French for International System of Units), abbreviated as SI.

In SI System, there are seven base units.

Fundamental or Base Units

S. No.

Fundamental Quantity

Fundamental Unit




Length, l



The metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. (1983)


Mass, m



The kilogram is equal to the mass of the international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at international Bureau of Weights and Measures, at Sevres, near Paris, France. (1889)


Time, t



The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. (1967)


Electric current, I



The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2×10–7 newton per metre of length. (1948)


Temperature, T



The kelvin, is the fraction 1/273.16 of the thermo-dynamic temperature of the triple point of water. (1967)


Amount of substance, n



The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon - 12. (1971)


Luminous intensity, Iv



The candela is the luminous intensity, in a given intensity direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979)

Suplplementary Units

S. No.

Supplementary Fundamental Quantities

Supplementary Unit




Plane angle





Solid angle




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π rad = 180o, 1o = π/180o, 1 rad = 180o/π, 1o = 60’, 1 = 60’’

Prefixes used for Multiples and Fractions of base units.




Power of 10



or deca-











































Power of 10










































Measurement of Length

Normal Distances

  • A metre scale is used for lengths from 10–3 m to 102 m.

  • A vernier callipers is used for lengths to an accuracy of 10–4 m.

  • A screw gauge and a spherometer can be used to measure lengths as less as to 10–5 m.

Measurement of Large Distances

Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. One of the methods for such cases is the parallax method.

Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The distance between the two points of observation is called the basis.

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To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time.

The ASB in represented by symbol θ is called the parallax angle or parallactic angle.

D = b/θ [for small θ chord AB = arc AB]

The angular diameter or apparent size is an angular measurement describing how large a sphere or circle appears from a given point of view.

If α is the angular diameter of the sun or star then, the diameter is given by d = α D.

Measurement of Very Small Distances: Size of a Molecule

To measure a very small distance like that of a size of a molecule (10–8 m to 10–10 m), special methods are adopted, because we cannot use a screw gauge or similar instruments and the microscope has certain limitations.

An optical microscope uses visible light. For visible light the range of wavelengths is from about 4000 Å to 7000 Å (1 angstrom = 1 Å = 10-10 m). Hence an optical microscope cannot resolve particles with sizes smaller than this.

Electronic microscope uses an electron beam. Electron beams can be focussed by properly designed electric and magnetic fields. Electron microscopes with a resolution of 0.6 Å have been built. They can almost resolve atoms and molecules in a material.

With tunnelling microscopy, it is possible to estimate the sizes of molecules.

A simple method for estimating the molecular size of oleic acid - Oleic acid is a soapy liquid with large molecular size of the order of 10–9 m. The idea is to first form mono-molecular layer of oleic acid on water surface.

  • Dissolve 1 cm3 of oleic acid in alcohol to make a solution of 20 cm3. Take 1 cm3 of this solution and dilute it to 20 cm3, using alcohol. So, the concentration of the solution is equal to  120×20  cm3 of oleic acid/cm3 of solution.

  • Lightly sprinkle some lycopodium powder on the surface of water in a large trough and put one drop of this solution in the water. The oleic acid drop spreads into a thin, large and roughly circular film of molecular thickness on water surface. Then, we quickly measure the diameter of the thin film to get its area A.

If we dropped n drops in the water, each drop having volume = V cm3.

So, Volume of n drops of solution = nV cm3

Amount of acid in this solution = nV20×20 cm3

This solution of oleic acid spreads very fast on the surface of water and forms a very thin layer of thickness t.

If this spreads to form a film of area A cm2, then the thickness of the film

t = Volume of the filmArea of the film=nV20×20×A cm

If we assume that the film has mono-molecular thickness, then this becomes the size or diameter of a molecule of oleic acid. The value of this thickness comes out to be of the order of 109 m.

Range of Lengths

Size of the nucleus ≈ order of 1014 m

Size of the extent of the observable universe ≈ order of 1026 m

Special Units of Length

For short lengths -

1 fermi = 1 f = 10–15 m = 1 femtometre

1 angstrom = 1 Å = 10–10 m = 0.1 nm

For large distances -

1 astronomical unit = 1 AU

(average distance of the Sun from the Earth) = 1.496 × 1011 m

1 light year = 1 ly = 9.46 × 1015 m

(distance that light travels with velocity of 3 × 108 m s–1 in 1 year)

1 parsec = 3.08 × 1016 m

(Parsec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second)

Measurement of Mass

Normal masses are measured in Kilogram (kg) – Measured using weighing scales.

Atomic and subatomic masses are measured in Unified Atomic Mass Units (u) – Using mass spectrograph, in which the radius of the trajectory is proportional to the mass of a charged particle moving in uniform electric and magnetic field.

1 u = 112th of the mass of an atom of carbon-12 isotope C612 including the mass of electrons

=1.66 × 10–27 kg

Mass of electron = 9.10938356 × 10-31 kg

Mass of proton = 1.6726219 × 10-27 kg

Mass of neutron = 1.6749 x 10-27 kg

Large masses in the universe like planets, stars, etc., based on Newton’s law of gravitation can be measured by using gravitational method.

Range of Masses

Mass of the electron order of ≈ 10-30 kg

Mass of known universe ≈ 1055 kg

Measurement of Time

Atomic standard of time is based on the periodic vibrations produced in a cesium atom. This is the basis of the cesium clock, called atomic clock, used in the national standards.

In the cesium atomic clock, the second is taken as the time needed for 9,192,631,770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atom.

The cesium atomic clocks are very accurate. They impart the uncertainty in time realisation as ± 1 × 10–13, i.e. 1 part in 1013. They lose or gain no more than 3 μs in one year.

Range of time

From life of most unstable particle (10-24 s) to Age of universe (1017 s)

Accuracy & Precision in Measurement

Accuracy refers to the closeness of a measurement to the true value of the physical quantity.

Precision refers to the resolution or the limit to which the quantity is measured.


The uncertainty in the measurement of a physical quantity is called an error.

The errors in measurement can be classified as (i) Systematic errors and (ii) Random errors

Systematic Errors:

The systematic errors are those errors that tend to be in one direction, either positive or negative.

Sources of systematic Errors

  1. Instrumental errors – Errors that arise due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read 104°C at the boiling point of water at STP whereas it should read 100°C.

  2. Imperfection in experimental technique or procedure - To determine the temperature of a human body a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment, may systematically affect the measurement.

  3. Personal errors - Arise due to an individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions, etc.

    Example: Holding the head a bit too far to one of the sides while reading the position of a needle on the scale.

Random Errors

The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc..

Least Count Error

The smallest value that can be measured by the measuring instrument is called its least count.

Least count error is the error associated with the resolution of the instrument.

For example, a Vernier Callipers has the least count as 0.01 cm; a Spherometer may have a least count of 0.001 cm. Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors. If we use a metre scale for measurement of length, it may have graduations at 1 mm division scale spacing or interval.

Using instruments of higher precision, improving experimental techniques, etc., we can reduce the least count error. Repeating the observations several times and taking the arithmetic mean of all the observations, the mean value would be very close to the true value of the measured quantity.

Absolute Error

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement, |Δa|.

Suppose the values obtained in several measurements are a1, a2, a3...., an. The arithmetic mean of these values is taken as the best possible value (or true value) of the quantity under the given conditions of measurement,

amean= a1+a2+a3++an n =1ni=1nai 

Δa1 = amean – a1,

Δa2 = amean – a2,

.... .... .... .... .....

.... .... .... .... .....

Δan = amean – an

The Δai may be positive or negative, but absolute errors |Δai| will always be positive.

Mean Absolute Error

The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Δamean.

Δamean =Δa1+Δa2 +Δa3++ Δann

If we do a single measurement, the value we get may be in the range

amean ± Δamean i.e. a = amean ± Δamean

That is,

amean – Δamean ≤ a ≤ amean + Δamean

Relative error - It is the ratio of the mean absolute error to the true value.

Relative ErrorΔameanamean

Percentage Error - When the relative error is expressed in percent, it is called the percentage error (δa).

Percentage Error = Δameanamean×100

Combination of Errors

Error of sum or difference

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Let us take two two physical quantities A and B having measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors.

Z = A + B.

We have by addition,

Z ± ΔZ = (A ± ΔA) + (B ± ΔB).

The maximum possible error in Z will be

ΔZ = ΔA + ΔB

For the difference Z = A – B, we have

Z ± ΔZ = (A ± ΔA) – (B ± ΔB)

= (A – B) ± ΔA ± ΔB

Or, ± ΔZ = ± ΔA ± ΔB

The maximum value of the error ΔZ is again ΔA + ΔB.

  1. Error of a product or a quotient

When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then,

Z ± ΔZ = (A ± ΔA) (B ± ΔB)

= AB ± B A ± AΔB ± ΔAΔB.

Dividing LHS by Z and RHS by AB we have,

1 ± ΔZZ= 1 ± ΔAA± ΔBB±  ΔAAΔBB

Since ΔA and ΔB are small, we can ignore their product. Hence the maximum relative error will be


⇒ δZ = δA + δB

Error in Case of a Quantity Raised to a Power

The relative error in a physical quantity raised to the power k is k times the relative error in the individual quantity.

In general, if Z = Ap BqCr, then, 

ΔZZ= p ΔAA+ q ΔBB+ r ΔCC

⇒ δZ = pδA + qδB + rδC

Significant figures

The reported result of measurement is a number that includes all digits in the number that are known reliably plus the last digit that is uncertain. The significant figures are those digits in a measured quantity which are known reliably plus one additional digit that is uncertain.

  • All the non-zero digits are significant.

  • All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.

  • If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. In 0.002308, the underlined zeroes are not significant.

  • The terminal or trailing zero(s) in a number without a decimal point are not significant. Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.

  • The trailing zero(s) in a number with a decimal point are significant. The numbers 3.500 or 0.06900 have four significant figures each.

  • For a number greater than 1, without any decimal, the trailing zero(s) are not significant.

  • For a number with a decimal, the trailing zero(s) are significant.

  • A choice of change of different units does not change the number of significant digits or figures in a measurement. For example, the length 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 mm or 23080 μm.

  • The digit 0 on the left of a decimal for a number less than 1 (like 0.1250) is never significant. However, the zeroes at the end of such number are significant in a measurement.

  • The multiplying or dividing factors which are neither rounded numbers nor numbers representing measured values are exact and have infinite number of significant digits. For example in r = d2 or s = 2πr, the factor 2 is an exact number and it can be written as 2.0, 2.00 or 2.0000 as required. Similarly, in t = Tn, n is an exact number.

Scientific Notation and Order of Magnitude

To remove ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (give examples).

4.700 m = 4.700 × 102 cm

= 4.700×103 mm

= 4.700×10–3 km

The power of 10 is irrelevant to the determination of significant figures.

In the number a × 10b, round off the number a to 1 (for a ≤ 5) and to 10 (for 5 < a ≤ 10). Then the number can be expressed approximately as 10b in which the exponent (or power) b of 10 is called order of magnitude of the physical quantity. Order of magnitude of 6.2 x 103 is 4.

Addition or subtraction with significant figures

In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

For example if A = 334.5 kg; B = 23.43kg then,

A + B = 334.5 kg + 23.43 kg = 357.93 kg

The result with significant figures is 357.9 kg

Multiplication and division with significant figures

In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.

  • For example if the mass of an object is 4.237g and the volume is 2.51cm3,

    Then, density = 4.3272.51 should be reported with three significant digits, that is 1.69 g/cm3.

  • Similarly, if the speed of light is given as 3.00 × 108 ms-1 (three significant figures) and one year (1 y = 365.25 d) has 3.1557 × 107s (five significant figures), the light year is 9.47 × 1015 m (three significant figures).

Rounding Off

While rounding off measurements the following rules are applied

Rule I: If the digit to be dropped is less than 5, then the preceding digit should be left unchanged. For example, 9.32 is rounded off to 9.3

Rule II: If the digit to be dropped is greater than 5, then the preceding digit should be raised by 1 For example 8.27 is rounded off to 8.3

Rule III: If the digit to be dropped is 5 then,

  1. If the preceding digit is even, the insignificant digit is simply dropped and,

  2. If the preceding digit is odd, the preceding digit is raised by 1.
  • For example the number 2.745 rounded off to three significant figures becomes 2.74. On the other hand, the number 2.735 rounded off to three significant figures becomes 2.74.

Dimensions, Dimensional Formula and Dimensional Equation

  1. Dimensions of a derived unit are the powers to which the fundamental units of mass, length and time etc. must be raised to represent that unit.

  2. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity.

Q: What are the dimensional formulae of volume, density, acceleration, work, energy etc.?

[V] = [M0 L3 T0]

[v] = [M0 L T–1]

[F] = [M L T–2]

[ρ] = [M L–3 T0]

Categories Physical Quantities

Dimensional Constants: These are the quantities which possess dimensions and have a fixed value. For example - Gravitational Constant

Dimensional Variables: These are the quantities which possess dimensions and do not have a fixed value. Examples - velocity, acceleration.

Dimensionless Constants: these are the quantities which do not possess dimensions and have a fixed value. Examples: π.

Dimensionless Variables: These are the quantities which are dimensionless and do not have a fixed value. Examples: Strain, Specific Gravity etc.

Dimensional Analysis

Dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.

  • Unit-factor method, is used for conversion from one system of units to other, using the rules of algebra

Importance of Dimentional Analysis

Checking the Dimensional Consistency of Equations

The principle of homogeneity of dimensions: The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions.

  • A given physical relation is dimensionally correct if the dimensions of the various terms on either side of the relation are the same.

  • If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong.

Deducing Relation among the Physical Quantities

The method of dimensions can be used to deduce relation among the physical quantities. For this we should know the dependence of the physical quantity on other quantities (upto three physical quantities or linearly independent variables) and consider it as a product type of the dependence.

Powers of the units are treated algebraically.

Example - The dependence of time period T on the quantities l, g and m as a product may be written as,

T = k lx gy mz,

where k is dimensionless constant and x, y and z are the exponents.

By considering dimensions on both sides, we have

[LoMoT1] = [L1 ]x[L1 T–2 ]y [M1 ]z = Lx+y T–2y Mz

On equating the dimensions on both sides, we have

x + y = 0; –2y = 1; and z = 0

So that x = ½, y = -½, z = 0

Then, T = k l½ g½ T =klg

  • The value of constant k cannot be obtained by the method of dimensions.

Limitations of Dimensional Analysis

  • It supplies no information about dimensionless constants. They have to be determined either by experiment or by mathematical investigation.

  • This method applicable only in the case of power functions. It fails in case of exponential and trigonometric relations.

  • It fails to derive a relation which contains two or more than two quantities of like nature.

  • It can only check whether a physical relation is dimensionally correct or not. It cannot tell whether the relation is absolutely correct or not.

  • It cannot identify all the factors on which the given physical quantity depends upon.