CBSE NOTES CLASS 11 PHYSICS CHAPTER 2
UNITS OF MEASUREMENT
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.
Unit
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 
Symbol 
Definition 
1. 
Length, l 
Metre 
m 
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) 
2. 
Mass, m 
kilogram 
kg 
The kilogram is equal to the mass of the international prototype of the kilogram (a platinumiridium alloy cylinder) kept at international Bureau of Weights and Measures, at Sevres, near Paris, France. (1889) 
3. 
Time, t 
Second 
s 
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 cesium133 atom. (1967) 
4 
Electric current, I 
Ampere 
A 
The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular crosssection, 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) 
5 
Temperature, T 
Kelvin 
K 
The kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967) 
6 
Amount of substance, n 
Mole 
mol 
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) 
7 
Luminous intensity, I_{v} 
Candela 
cd 
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 
Symbol 
Remarks 
1 
Plane angle 
Radian 
rad 
$$\mathrm{}=\frac{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}}$$ 
2 
Solid angle 
Steradian 
Sr 
$$\mathrm{}=\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}}{{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}}^{2}}$$ 
π rad = 180^{o}, 1^{o} = π/180^{o}, 1 rad = 180^{o}/π, 1^{o} = 60’, 1^{’} = 60^{’’}
Prefixes used for Multiples and Fractions of base units.
Multiples  
Prefix 
Abbreviation 
Power of 10 
Equivalent 
deka or deca 
da 
10^{1} 
ten 
hecto 
h 
10^{2} 
hundred 
kilo 
k 
10^{3} 
thousand 
mega 
M 
10^{6} 
million 
giga 
G 
10^{9} 
billion 
tera 
T 
10^{12} 
trillion 
peta 
P 
10^{15} 
quadrillion 
exa 
E 
10^{18} 
quintillion 
zetta 
Z 
10^{21} 
sextillion 
yotta 
Y 
10^{24} 
septillion 
Fractions  
Prefix 
Abbreviation 
Power of 10 
Equivalent 
deci 
d 
10^{1} 
tenth 
centi 
c 
10^{2} 
hundredth 
milli 
m 
10^{3} 
thousandth 
micro 
μ 
10^{6} 
millionth 
nano 
n 
10^{9} 
billionth 
pico 
p 
10^{12} 
trillionth 
femto 
f 
10^{15} 
quadrillionth 
atto 
a 
10^{18} 
quintillionth 
zepto 
z 
10^{21} 
sextillionth 
yocto 
y 
10^{24} 
septillionth 
Measurement of Length
Normal Distances
 A metre scale is used for lengths from 10^{–3} m to 10^{2} 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 semiangle of inclination between those two lines. The distance between the two points of observation is called the basis.
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 monomolecular layer of oleic acid on water surface.
 Dissolve 1 cm^{3} of oleic acid in alcohol to make a solution of 20 cm^{3}. Take 1 cm^{3} of this solution and dilute it to 20 cm^{3}, using alcohol. So, the concentration of the solution is equal to$\mathrm{}\mathrm{}\frac{1}{20\times 20}\mathrm{}$ $\mathrm{c}{\mathrm{m}}^{3}$ of oleic acid/cm^{3 }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 cm^{3}.
So, Volume of n drops of solution = nV cm^{3}
$$\mathrm{Amount\; of\; acid\; in\; this\; solution\; =}\frac{\mathrm{n}\mathrm{V}}{20\times 20}\mathrm{}\mathrm{c}{\mathrm{m}}^{3}$$
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 cm^{2}, then the thickness of the film
$$\mathrm{t}\mathrm{}=\mathrm{}\frac{\mathrm{Volume\; of\; the\; film}}{\mathrm{Area\; of\; the\; film}}=\frac{\mathrm{n}\mathrm{V}}{20\times 20\times \mathrm{A}}\mathrm{}\mathrm{c}\mathrm{m}$$
If we assume that the film has monomolecular 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 10^{–9} m.
Range of Lengths
Size of the nucleus ≈ order of 10^{–14} m
Size of the extent of the observable universe ≈ order of 10^{26} 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 × 10^{11} m
1 light year = 1 ly = 9.46 × 10^{15} m
(distance that light travels with velocity of 3 × 10^{8} m s^{–1} in 1 year)
1 parsec = 3.08 × 10^{16} 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 = $\frac{1}{12}$^{th} of the mass of an atom of carbon12 isotope $}_{6}{}^{12}\mathrm{C$ 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 1027 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 ≈ 10^{55} 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 cesium133 atom.
The cesium atomic clocks are very accurate. They impart the uncertainty in time realisation as ± 1 × 10^{–13}, i.e. 1 part in 10^{13}. 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 (10^{17} 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.
ErrorUncertainity
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
 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.
 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.
 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 setups, 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 a_{1}, a_{2}, a_{3}...., a_{n}. 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,
$${\mathrm{a}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}=\mathrm{}\frac{{\mathrm{a}}_{1}+{\mathrm{a}}_{2}+{\mathrm{a}}_{3}+\dots +{\mathrm{a}}_{\mathrm{n}}\mathrm{}}{\mathrm{n}}\mathrm{}=\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}}\mathrm{}$$
Δa_{1} = a_{mean} – a_{1},
Δa_{2} = a_{mean} – a_{2},
.... .... .... .... .....
.... .... .... .... .....
Δa_{n} = a_{mean} – a_{n}
The Δa_{i} may be positive or negative, but absolute errors Δa_{i} 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 Δa_{mean}.
$${\mathrm{\Delta}\mathrm{a}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}\mathrm{}=\frac{\left\mathrm{\Delta}{\mathrm{a}}_{1}\right+\left\mathrm{\Delta}{\mathrm{a}}_{2\mathrm{}}\right+\left\mathrm{\Delta}{\mathrm{a}}_{3}\right+\dots +\mathrm{}\left\mathrm{\Delta}{\mathrm{a}}_{\mathrm{n}}\right}{\mathrm{n}}$$
If we do a single measurement, the value we get may be in the range
a_{mean} ± Δa_{mean }i.e. a = a_{mean} ± Δa_{mean}
That is,
a_{mean} – Δa_{mean} ≤ a ≤ a_{mean} + Δa_{mean}
Relative error  It is the ratio of the mean absolute error to the true value.
$$\mathbf{Relative\; Error}\frac{{\mathrm{\Delta}\mathrm{a}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}{{\mathrm{a}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}$$
Percentage Error  When the relative error is expressed in percent, it is called the percentage error (δa).
$$\mathbf{Percentage\; Error\; =}\frac{{\mathrm{\Delta}\mathrm{a}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}{{\mathrm{a}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}\times 100$$
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.
 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,
$$1\mathrm{}\pm \mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{Z}}{\mathrm{Z}}\right)=\mathrm{}1\mathrm{}\pm \mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{A}}{\mathrm{A}}\right)\pm \mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{B}}{\mathrm{B}}\right)\pm \mathrm{}\mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{A}}{\mathrm{A}}\right)\left(\frac{\mathrm{\Delta}\mathrm{B}}{\mathrm{B}}\right)$$
Since ΔA and ΔB are small, we can ignore their product. Hence the maximum relative error will be
$$\frac{\mathrm{\Delta}\mathrm{Z}}{\mathrm{Z}}=\mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{A}}{\mathrm{A}}\right)+\mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{B}}{\mathrm{B}}\right)$$
⇒ 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.
$$\mathrm{I}\mathrm{n}\mathrm{}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l},\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{Z}\mathrm{}=\mathrm{}\frac{{\mathrm{A}}^{\mathrm{p}}\mathrm{}{\mathrm{B}}^{\mathrm{q}}}{{\mathrm{C}}^{\mathrm{r}}},\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n},\mathrm{}$$
$$\frac{\mathrm{\Delta}\mathrm{Z}}{\mathrm{Z}}=\mathrm{}\mathrm{p}\mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{A}}{\mathrm{A}}\right)+\mathrm{}\mathrm{q}\mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{B}}{\mathrm{B}}\right)+\mathrm{}\mathrm{r}\mathrm{}\left(\frac{\mathrm{\Delta}\mathrm{C}}{\mathrm{C}}\right)$$
⇒ 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 nonzero digits are significant.
 All the zeros between two nonzero 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 nonzero 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 = $\frac{\mathrm{d}}{2}$ 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 = $\frac{\mathrm{T}}{\mathrm{n}}$, 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 × 10^{2} cm
= 4.700×10^{3} mm
= 4.700×10^{–3} km
The power of 10 is irrelevant to the determination of significant figures.
In the number a × 10^{b}, 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 10^{b} 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 10^{3} 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 2.51cm^{3},
Then, density = $\frac{4.327}{2.51}\mathbf{}\mathbf{}$should be reported with three significant digits, that is 1.69 g/cm^{3}.
 Similarly, if the speed of light is given as 3.00 × 10^{8} ms^{1} (three significant figures) and one year (1 y = 365.25 d) has 3.1557 × 10^{7}s (five significant figures), the light year is 9.47 × 10^{15 }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,
 If the preceding digit is even, the insignificant digit is simply dropped and,
 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
 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.
 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] = [M^{0} L^{3} T^{0}]
[v] = [M^{0} L T^{–1}]
[F] = [M L T^{–2}]
[ρ] = [M L^{–3} T^{0}]
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.
 Unitfactor 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 l^{x} g^{y} m^{z},
where k is dimensionless constant and x, y and z are the exponents.
By considering dimensions on both sides, we have
[L^{o}M^{o}T^{1}] = [L^{1} ]^{x}[L^{1} T^{–2} ]^{y} [M^{1} ]^{z }= L^{x+y }T^{–2y }M^{z}
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
$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n},\mathrm{}\mathrm{T}\mathrm{}=\mathrm{}\mathrm{k}\mathrm{}{\mathrm{l}}^{\mathbf{\xbd}}\mathrm{}{\mathrm{g}}^{\u2013\mathbf{\xbd}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathrm{T}\mathrm{}=\mathrm{k}\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}\mathbf{}$$

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.