Course Content

Units and Measurements  Lesson

Units and Measurements  Quiz 1 and 2
Units and Measurements
Introduction
Physical quantities are those which can be measured i.e., subjected equally to all three elements of scientific study namely; detailed analysis, precise measurement and mathematical treatment (Example: Mass, length, time, volume etc.)
Physical quantities are measured with certain internationally accepted reference standard called unit and the result of a measurement of a physical quantity is stated by a number supplemented by a unit.
Need for measurement
The need for measurement of a physical quantity is shown below sequentially:
 To study a phenomena scientists performed different experiments
 Experiments require measurement of physical quantities such as mass, length, time etc.
 Experiments led to laws and theories
 Experimental verification required precise measurement
 appropriate instruments necessary for precise measurements
Unit for Measurement
The reference standard used for the measurement of a physical quantity is called the unit of that physical quantity. The magnitude of a physical quantity ‘a’ is expressed as:
Magnitude = Numerical value of physical quantity × Size of its unit
i.e., a = nu where,
n = number of times the unit is taken.
u = size of unit of physical quantity
The units are categorized in to fundamental or base units, supplementary base units and derived units. All these type of units put together is known as System of Units. Let us now analyze the system of units followed in earlier times and at present, before getting in to details of category of units.
The system of units that were used in different countries in earlier times, widely, are tabulated below:
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Table 1 – Different System of Units
System  Base units for  

Length  Mass  Time  
CGS  centimeter  gram  second 
FPS  foot  pound  second 
MKS  meter  kilogram  second 
It is easy to imagine the kind of difficulties that the scientists of different countries would have encountered to communicate about the units in the earlier days. To put an end to this perennial problem, scientists recommended in General Conference on Weights and Measures in 1971, to follow International System of Units. The system of units, which is at present internationally accepted for measurement is the International System of Units, in abbreviation known as SI Units. This lesson is prepared based on SI Units. Now, let us further study in detail about SI Units.
International System of Unis (SI Units)
The standard symbols, units and abbreviations developed and recommended in General Conference on Weights and Measures in 1971 for international usage in scientific, technical, industrial and commercial work. SI units are found to be fairly simple and convenient.SI units has the essential characteristics of a good unit as follows:
 Well defined.
 Easily available and reproducible at all places.
 Not perishable.
 Invariable.
 Universally accepted.
 Comparable to the size of the measured physical quantity.
 Easy to form multiples or submultiples of the unit.
Nowadays, SI system has replaced all the other systems of units and is greatly used to exchange scientific data between different parts of the world.
As mentioned earlier, SI units are divided in to three categories as shown below:
 Base.
 Supplementary.
 Derived.
Fundamental or Base Units
In SI units, there are seven base units as given in Table below:
Table 2 – SI base Quantities and Units
Base quantity  SI Units  

Name  Symbol  Definition  
Length  meter  m  The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second. 
Mass  kilogram  kg  The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. 
Time  second  s  The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyper fine levels of the ground state of the cesium 133 atom. 
Electric current  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 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10^{7} newton per meter of length. 
Thermodynamic Temperature  kelvin  K  The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. 
Amount of substance  mole  mol  1. 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; its symbol is “mol.” 2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. 
Luminous intensity  candela  cd  The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 10^{12} hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. 
Supplementary Units
Besides the above seven units, there are two units called supplementary units and are listed below:
Table 3 – Supplementary Units
Quantity  SI Units  

Name  Unit  Symbol  Definition  Formula  
Plane angle  dθ  radian  r  Plane angle is the ratio of length of arc ds to the radius r  dθ =ds/r 
Solid angle  dΩ  steradian  sr  Solid angle is the ratio of intercepted area dA of the spherical surface described about the apex O as center to the square of its radius r  dΩ=dA/r² 
Derived Units
Physical quantities, e.g., volume, velocity, acceleration etc., are derived from these base quantities and can be expressed as a combination of base units and are called derived units. Certain derived units are expressed by means of SI units with special names such as joule, newton, watt etc. Derivative Units are divided in to three different ways as given below:
Now, let us look at the subcategories, sequentially, as given in chart above.
SI derived Units expressed in SI Base Units
Physical quantities, appearing under this category are tabulated below:
Table 4 – SI Derived Units
Physical quantity  SI Unit  

Name  Symbol  
Area  square meter  m^{2} 
Angular velocity  radian per second  rad/s 
Acceleration  meter per second square  m/s^{2} 
Angular acceleration  radian per second square  rad/s^{2} 
Current density  ampere per square meter  A/m^{2} 
Concentration (of amount of substance)  mole per cubic meter  mol/m^{3} 
Density, mass density  kilogram per cubic meter  kg/m^{3} 
Flow rate  cubic meter per second  m^{3}/s 
Kinematic Viscosity  square meter per second  m^{2}/s 
Luminance, intensity of illumination  candela per square meter  cd/m^{2} 
Linear / superficial / volume expansivities  per kelvin  K^{1} 
Magnetic field strength, magnetic intensity, magnetic moment density  ampere per meter  A/m 
Moment of Inertia  kilogram per square meter  kg/m^{2} 
Momentum  kilogram meter per second  kg m/s 
Radius of gyration  meter  m 
Specific volume  cubic meter per kilogram  m^{3}/kg 
Speed, velocity  meter per second  m/s 
Volume  cubic meter  m^{3} 
Wave number  per meter  m^{1} 
Table5
SI Derived Units expressed by means of SI Units with special names
Physical quantity  SI Unit  

Name  Symbol  Expression in terms of other units  Expression in terms of SI units  
Absorbed dose, absorbed dose index  gray  Gy  J/kg  m^{2}/s^{2} 
Activity (of a radio nuclide / radioactive source  becquerel  Bq  –  s^{1} 
Conductance  siemens  S  A/V  m^{2 }kg^{1 }s^{3 }A ^{2} 
Capacitance  farad  F  C/V  A^{2 }s ^{4}kg^{1 }m ^{2} 
Electric resistance  ohm  Ω  V/A  kg m^{2 }s^{3 }A^{2} 
Electric potential, potential difference, electromotive force  volt  V  W/A  kg m^{2 }s^{3 }A^{1} 
Energy, work, quantity of heat  joule  J  N m  kg m^{2 }s^{2} 
Force  newton  N  –  kg m s^{2} 
Frequency  hertz  Hz  –  s^{1} 
Inductance  henry  H  Wb / A  
Illuminance  lux  lx  lm / m^{2}  m^{2 }cd sr ^{1} 
Luminous flux, luminous power  lumen  lm  –  cd / sr 
Magnetic field, magnetic flux density, magnetic induction  tesla  T  Wb / m^{2}  kg s^{–}^{2 }A ^{1} 
Magnetic Flux  weber  Wb  V s or J / A  kg m^{2 }s^{2 }A^{2} 
Pressure, stress  Pascal  Pa  N / m^{2}  kg m^{1 }s^{2} 
Power, radiant flux  watt  W  J / s  kg m^{2 }s^{3} 
Quantity of electricity, electric charge  coulomb  C  –  A s 
Table6
Some SI Derived Units expressed by means of SI Units with special names
Physical quantity  SI Unit  

Name  Symbol  Expression in terms of SI base units  
Dynamic viscosity  poiseulles or pascal second or newton second per square meter  PI or Pa s or N s m^{2}  kg / m s 
Dipole moment  coulomb meter  C m  s A m 
Magnetic moment  joule per tesla  J T^{1}  m^{2 }A 
Powder density  watt per square meter  W / m^{2}  kg s^{3} 
Surface tension  newton per meter  N/ m  kg s^{2} 
Torque, couple, moment of force  newton meter  N m  kg m^{2 }s^{2} 
Measurement of Length
Measurement of length can be done with two methods:
a) Direct method
b) Indirect method
Direct Method
We will first discuss about direct method first. For e.g., some of the instruments used for measurement of length and its level of accuracy are listed in below table:
Table 7
Name  Level of accuracy (m) 

Meter scale  10^{3} to 10^{3} 
Vernier calipers  10^{4} 
Spherometer  10^{5} 
Indirect Method
Measurement of Large Distances
To measure lengths beyond these ranges, we make use of some special indirect methods. Large distances such as distance of a planet or a star from the earth cannot be measured directly with meter scale. The method used here is parallax method.
a. What is parallax method?
Imagine that a pencil is held in front of you against some specific point on the background (a wall) and look at the pencil through your left eye A (closing the right eye) and then look at the pencil through your right eye B (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax. The distance between the two points of observation is called basis. In this case, the basis is the distance between the eyes.
To measure the distance D of a faraway planet S by parallax method, we observe from two different positions (observatories) A and B on Earth, separated by a distance AB = B at the same time as shown in Fig below:
The angle between the two directions along which the planet is viewed at these two points shall then be measured. The∠ASB in Fig 1 above represented by symbol θ is called the parallax angle.
As the planet is very far away, b/D« 1 is very small. Then we approximately take AB as an arc of length b of a circle with center at S and the distance D as the radius AS = BS so that AB = b = Dθ where θ is in radians. Therefore,
D = b/θ……………………………………………… (2.1)
Having determined D, we can employ a similar method to determine the size or angular diameter of the planet. If d is the diameter of the planet, and α the angular size of the planet (the angle subtended by d at the earth, we have
α = d/D …………………………………………………….. (2.2)
The angle α can be can be measured from the same location on the earth. It is angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since D is known, the diameter of the planet can be determined using Eq. (2.2).
Let us now look at some examples:
Example 1 Calculate the angle of (a) 1^{0 }(degree) (b) 1’ (minute of arc) and (c) 1’’ (second of arc) in radians. Use 360^{0} = 2π rad, 1^{0} = 60’ and 1’ = 60’’.
Ans.:
a) We have,
360^{0} = 2π rad
1^{0} = (π/180 rad = 1.745 x 10^{2} rad
b) 1^{0} = 60’ = 1.745×10^{2} rad
c) 1’ = 60’’ = 2.908×10^{4} rad ; 2.91×10^{4} rad
1” = 4.847×10^{6} rad; 4.85×10^{6} rad
Example 2 A man wishes to estimate the distance of a nearby tower from him. He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100m, and looks at O and C again. Since O is very distant, the direction BO is practically the same as AO; but he finds the line of sight of C shifted from the original line of sight by an angle = 40^{0} ( is known as ‘parallax) estimate the distance of the tower C from his original position A.
Ans.: We have, parallax angle θ = 40^{0}
From Fig. 2, AB = AC tanθ
AC = AB/tanθ = 100 m/tan40^{0}
= 100 m/0.8391 = 119 m
Example3 The moon is observed from two diametrically opposite points A and B on earth. The angle θ subtended at the moon by the two directions of observation is 1^{0}54’. Given the diameter of earth to be about 1.276 x 10 ^{7} m, compute the distance of the moon from the earth.
Ans.: We have, parallax angle θ = 1^{0} 54’ = 114’
= (114 x 60)’ x 4.85×10^{6} rad,
= 3.32 x 10^{2} rad
Therefore, 1” = 4.85 x 10^{6 }rad
Also b = AB = 1.276 x 10^{7}
Hence from equation (2.1), we have the earth moon distance,
D = b/θ
=(1.276 xE7/3.32 x E2)
= 3.84 x 10^{8 }m
Example 2.4 The Sun’s angular diameter is measured to be 1920″. The distance D of the Sun from the Earth is 1.496 × 10^{11} m. What is the diameter of the Sun ?
Answer Sun’s angular diameter a
= 1920″
= 1920× 4.85× 10^{6} rad
= 9.31×10^{3} rad
Sun’s diameter
d =α D
= (9.31×10^{3} )×(1.496× 10^{11}) m
=1.39 ×10^{9}m
3.2 Estimation of Very Small Distances: Size of a Molecule
To measure a very small size like that of a molecule (10^{8} m to 10^{10}) m, we have to adopt special methods. We cannot use screw gauge or similar instruments, even a microscope has certain limitations. An optical microscope uses visible light to look at the system under investigation. As light has wave like feature, the resolution to which an optical microscope can be used is the wavelength of light. For visible light, the range of wave length is from about 4000A to 7000A (1 angstrom = 1A = 10_{} ^{10 }m). Hence, an optical microscope cannot resolve particles with sizes smaller than this. Instead of visible light, we can use an electron beam. Electron beams can be focused by properly designed electric and magnetic field. The resolution of such an electron microscope is limited finally by the fact that electrons can also behave as waves. The wave length of an electron can be as small as a fraction of an angstrom. Such electron microscopes with a resolution of 0.6 A have been built. They can almost resolve atoms and molecules in a material. In recent times, tunneling microscopy has been developed in which again the limit of resolution is better than angstrom. It is possible to estimate the sizes of molecules.
A simple method for estimating the molecular size of oleic acid is given below. 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. We dissolve 1 cm^{3} of oleic acid in alcohol to make a solution of 20 cm^{3}. Then we 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 { 1/(20×20)} cm^{3 of} oleic acid/cm^{3 }of solution. Next we lightly sprinkle some lycopodium powder on the surface of water in a large trough and we put one drop of this solution in the water. The oleic acid drop spreads in to 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. Suppose we have dropped n drops in the water, initially, we determine the approximate volume of each drop (V cm ^{3}).
Volume of n drops of solution = nV cm^{3}
Amount of oleic acid in this solution = nV{1/(20×20)} cm^{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
t = Volume of film/Area of film or,
t ={ nV/(20×20)} cm ……………………………………………….(2.3)
If we assume that the film has monomolecular thickness, then this becomes the size or diameter of a molecule of oleic acid. This value of this thickness comes out to be the order of 10^{9}m.
Example 5 The size of a nucleus (in the range of 10 ^{15 }to 10^{14}m) is scaled up to the tip of a sharp pin, what roughly is the size of an atom? Assume tip of the pin in the range 10 ^{5 }m to 10^{4 }m.
Answer: The size of a nucleus is in the range of 10^{15 }m and 10^{14} m. The tip of a sharp pin is taken to be in the range of 10 ^{5 }m and 10^{4 }m. Thus we are scaling up by a factor of 10^{10}. An atom roughly of size 10^{10 }m will be scaled up to a size of 1 m. Thus a nucleus in an atom is as small in size as the tip of a sharp pin placed at the center of a sphere of radius about a meter long,
3.3 Range of lengths
The sizes of the objects we come across in the universe vary over a wide range. These may vary from the size of the order of 10 ^{14}m of the tiny nucleus of an atom to the size of the order of 10^{26 }m of the extent of the observable universe. Table8 below gives the range and order of lengths and sizes of some of these objects.
Table 8 – Range and order of lengths
Size of object or distance  Length (m) 

Size of proton  10^{15} 
Size of atomic nucleus  10^{14} 
Size of hydrogen atom  10^{10} 
Length of typical virus  10^{8} 
Wavelength of light  10^{7} 
Size of red blood corpuscle  10^{5} 
Thickness of paper  10^{4} 
Height of Mount Everest above sea level  10^{4} 
Radius of Earth  10^{7} 
Distance of moon from Earth  10^{8} 
Distance of Sun from Earth  10^{11} 
Distance of Pluto from the Sun  10^{13} 
Distance of galaxy  10^{21} 
Distance to the boundary of observable universe  10^{26} 
We also use certain special length units for short and large lengths. These are,
1 fermi = 1f = 10^{15 }m
1 angstrom = 1 A = 10^{10 }m
1 astronomical unit = 1AU (average distance of the Sun from the Earth)
= 1.496 x 10^{11 }m
1 light year = 1 ly = 9.46 x 10^{5 }m (distance that light travels with velocity of 3 x 10^{8 }m s^{1 }in 1 year)
1 parsec = 3.08 x 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
 Mass is a fundamental property of matter. It does not depend on temperature, pressure or location of the object in space. The SI unit of mass is kilogram (kg).
 Mass of an object can be measured directly by comparison with multiples of the kilogram using a beam balance.
 For atomic levels, unified atomic mass unit is used.
1 unified atomic mass unit = 1u
= (1/12) of the mass of an atom of carbon 12 isotope ( C) including the mass of electrons
= 1.66 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.
 For measurement of small masses of atomic/subatomic particles etc., we make use of mass spectrograph in which radius of trajectory is proportional to the mass of a charged particle moving in uniform electric and magnetic field. The table below gives range and order of masses.
 The prototypes of the International standard kilogram supplied by the International Bureau of Weights and Measures (BIPM) are available in many other laboratories of different countries. In India, this is available in National Physical Laboratories (NPL) New Delhi.
Table9
Object  Length (m) 

Electron  10^{30} 
Proton  10^{27} 
Uranium atom  10^{25} 
Red blood cell  10^{13} 
Dust particle  10^{9} 
Rain drop  10^{6} 
Mosquito  10^{5} 
Grape  10^{3} 
Human  10^{2} 
Automobile  10^{3} 
Boeing 747 aircraft  10^{8} 
Moon  10^{23} 
Earth  10^{25} 
Sun  10^{30} 
Galaxy  10^{41} 
Observable universe  10^{55} 
Measurement of Time
To measure any time interval we need a clock. We now use an atomic standard time, which is based on the periodic vibrations produced in cesium atom. This is the basic of cesium clock, sometimes called atomic clock, used in the national standards. Such standards are available in many laboratories. In the cesium available in many laboratories. 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 hyper fine levels of the ground state of cesium133 atom. The vibrations of the cesium atomic clock just as the vibrations of a balance wheel regulate an ordinary wrist watch or the vibrations of a small quartz wristwatch.
The cesium atomic clocks are very accurate. In principle they provide portable standard. The national standard of time interval ‘second’ as well as frequency is maintained through four cesium atomic clocks. A cesium atomic clock is used at the National Physical Laboratories (NPL) New Delhi to maintain the Indian Standard of time.
In our country, the NPL has the responsibility of maintenance and improvement of physical standards, including time, frequency, etc. Note that Indian Standard Time (IST) is linked to this set of atomic clocks. The efficient cesium atomic clocks are so accurate that they impart the uncertainty in time realization as + 1×10^{13} i.e., 1 part in 10^{13. }This implies that the uncertainty gained over time by such a device is less than 1 part 10^{13, }they lose or gain no more than 3 ms in one year. In view of the tremendous accuracy in time measurement, the SI unit of length has been expressed in terms the path length light travels in certain interval of time (1/299, 742,458 of a second).
The time interval of events that we come across in the universe vary over a wide range. The below table gives the range and order of some typical time intervals.
Table 10 – Range and Order of Time Intervals
Event  Time interval (s) 
Life span of unstable particle  10^{24} 
Time required for light to cross nuclear distance  10^{22} 
Period of xrays  10^{19} 
Period of atomic vibrations  10^{15} 
Period of light wave  10^{15} 
Life time of an excited state of an atom  10^{8} 
Period of radio wave  10^{6} 
Period of a sound wave  10^{3} 
Wink of eye  10^{1} 
Time between successive human heart beats  10^{0} 
Travel time for light from the moon to the Earth  10^{0} 
Travel time for light from the Sun to the Earth  10^{2} 
Time period for satellite  10^{4} 
Rotation period of the Earth  10^{5} 
Rotation and revolution period of the moon  10^{6} 
Revolution period of the Earth  10^{7} 
Travel time for light from nearest star  10^{8} 
Average human lifespan  10^{9} 
Age of Egyptian pyramids  10^{11} 
Time since dinosaurs became extinct  10^{15} 
Age of the universe  10^{17} 
Accuracy, Precision of Instruments and Errors in Measurements
Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values also has an error. We shall distinguish between accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured.
The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument. For example, suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, and the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise. Thus every measurement is approximate due to errors in measurement can be broadly classifies as (a) systematic errors and (b) random errors.
Systematic errors
The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are:
Instrumental errors
Instrument errors that arise from the errors 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); in a Vernier calipers the zero mark of the Vernier scale may not coincide with zero mark of the main scale, or simply an ordinary meter scale may be worn off at one end.
Imperfection in experimental technique or procedure
E.g. 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
Personal errors are that 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. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax.
Systematic errors can be minimized by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. For a given setup, these errors may be estimated to a certain extent and the necessary corrections may be applied to the readings.
Random errors
The random errors are those errors, which occur irregularly and hence 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., For example when the same person repeats the same observation, it is very likely that he may get different readings every time.
Least count error
The smallest value that can be measured by the measuring instrument is called its least count. All the readings or measured values are good only up to this value.
The least count error is the error associated with the resolution of the instrument. For example, a Vernier calipers 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 meter 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 observations, the mean value would be very close to the true value of the measured quantity.
Absolute Error, Relative Error and Percentage Error
(a) 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 of the quantity under the given conditions of measurement as :
a_{mean = (}a_{1, }a_{2, }a_{3 }………a _{n}) / n……………………………………………………………………….. (2.4)
Or,
a_{mean} =∑^{n}_{i=1} = a i/n …………………………………………………….(2.5)
This is because, as explained earlier, it is reasonable to suppose that individual measurements as likely to overestimate as to underestimate the true value of the quantity.
The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. This is denoted by I Δα I. In absence of any other method of knowing true value, we considered arithmetic mean as the true value. Then the errors in the individual measurement values are,
Δα_{1 }= α_{1}α_{mean }
Δα_{2 }= α_{2}α_{mean}
….. …. ….
….. …. …..
….. …. …..
Δα_{n }= α_{n}α_{mean}
Δα calculated above may be positive in certain cases and negative in some other cases. But absolute error I Δ α I will always be positive.
(b) The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity α. It is represented by Δα_{mean}.
Thus,
Δa_{mean} = (I Δα_{1} I +I Δα_{2} I+I Δα_{3} I+ ……….I Δ α_{n} I / n ………………………………………………………… (2.6)
=∑^{n}_{i=1} I Δα_{1} I/n …………………………………………………………………………………(2.7)
If we do a single measurement, the value we get may be in the range α_{mean }+ Δα_{mean}
i.e., α = α_{mean }+ Δα_{mean}
Or,
a_{mean }– Δa_{mean} ≤ a ≤ a_{mean}± Δa_{mean} ………………………………………………………….(2.8)
This implies that any measurement of the physical quantity a is likely to lie between (Δa _{mean} + Δa_{mean}) and
(Δa_{mean} – Δa_{mean}).
Instead of the absolute error, we often use the relative error or the percentage error (da). The relative error is the ratio of the mean absolute error Δa_{mean} to the mean value a_{mean} of the quantity measured.
Relative error = Δ a_{mean /} a_{mean} ……………………………………………………………………………….. (2.9)
When the relative error is expressed in percent, it is called the percentage error (da).
Thus, Percentage error
δa = (Δa_{mean/} a_{mean}) x 100%
Example 6: Two clocks are being tested against a standard clock located in national laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks are:
Day  Clock  

Clock 1  Clock 2  
Monday  12:00:05  10:15:06 
Tuesday  12:01:15  10:14:59 
Wednesday  11:59:08  10:15:18 
Thursday  12:01:50  10:15:07 
Friday  11:59:15  10:14:53 
Saturday  12:01:30  10:15:24 
Sunday  12:01:19  10:15:11 
If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer?
Le t us now consider an example.
Answer: The range of variation over the seven days of observations is 162 s for clock 1, and 31 s for clock 2. The average reading of clock 1 is much closer to the standard time than the average reading of the clock 2. The important is that a clock’s zero error is not as significant for precision work as its variation, because a ‘zero error’ can always be easily corrected. Hence clock 2 is to be preferred to clock 1.
Example 7 We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71 s and 2.80s. Calculate the absolute errors, relative error or percentage error.
Answer: The mean period of oscillation of the pendulum
T = {(2.63+2.56+2.42+2.71+2.80)s / 5}
=(13.12/5)s
= 2.624 s
= 2.62 s
As the periods are measured to a resolution of 0.01 s, all the times are to second decimal; it is proper to put this mean period also to the second decimal.
The errors in the measurements are
2.63 s 2.62 s = 0.01 s
2.56 s 2.62 s = 0.06 s
2.42 s 2.62 s = 0.20s
2.71 s – 2.62 s = 0.09 s
2.80 s 2.62 s = 0.18 s
Note that the errors have the same units as the quantity to be measured.
The arithmetic mean of all the absolute errors (for the arithmetic mean, we take only the magnitude) is
ΔT_{mean }= (0.01+0.06+0.20+0.09+0.18)s/5
= 0.54 s/5
=0.11 s
That means, the period of oscillation of the simple pendulum is (2.62 + 0.11) s i.e. it lies between (2.62 + 0.11) s and (2.62 – 0.11) s or between 2.73 s and 2.51 s. As the arithmetic mean of all the absolute errors is 0.11 s, there is already an error in the tenth of a second. Hence, there is no point in giving the period to a hundredth. A more correct way will be to write
T = 2.6 + 0.1 s
Note that the last numeral 6 is unreliable, since it may be anything between 5 and 7. We indicate this by saying that the measurement has two significant figures. In this case, the two significant figures are 2, which is reliable and 6, which has an error associated with it.
For this example, the relative error or the percentage error is,
δa = (0.1/2.6)x 100 = 4%
6.2 Combination Errors
If we do an experiment involving several measurements, we must know how the errors in all the measurements combine. For example, density is the ratio of the mass to the volume of the substance. If we have errors in the measurement of mass and of the sizes or dimensions, we must know what the error will be in the density of the substance. To make such estimate, we should learn how errors combine in various mathematical operations. For this, we use the following procedure.
Error of a sum or difference
Suppose two physical quantities A and B have measured values A + ΔA, B + ΔB respectively where DA and DB are their absolute errors. We wish to find the error ΔZ in the sum
Z = A + B,
We have by addition, Z + ΔZ
= (A + ΔA) + (B + ΔB)
The maximum possible error in Z
ΔZ = ΔA + ΔB
For the difference Z = A – B we have
Z + ΔZ = (A + ΔA) – (B + ΔB)
= (AB) + ΔA + ΔB
Or, +ΔZ = +ΔA + ΔB
Hence the rule: 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.
Example 8 The temperature of two bodies measured by a thermometer are t_{1 }= 20^{0}C + 0.5^{0}C and t_{2 = }50^{0}C + 0.5^{0}C, Calculate the temperature difference and error therein.
Answer: t^{’ }= t_{2} –t_{1 = }(50^{0}C + 0.5^{0}C) – (20^{0}C + 0.5^{0} C)
= 30^{0}C + 1^{0}C
Error of a product or a quotient
Suppose Z = AB and the measured values of A and B are A + DA and B + DB Then,
Z + ΔZ = (A + ΔA) (B + ΔB)
= AB + BΔA + AΔB + ΔA ΔB
Dividing LHS by Z and RHS by AB we have,
1 + (ΔZ/Z) = 1 + (ΔA/A) + (ΔB/B) + (ΔA/A) (ΔB/B)
Since ΔA and ΔB are very small, we shall ignore their product.
Hence the maximum relative error,
ΔZ/Z = (ΔA/A) + (ΔB/B)
We can easily verify this is true for division also.
Hence the rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.
Example 9 The resistance R =V/I where V = (100 + 5) V and I = (10 + 0.2) A. Find the percentage error in R.
Answer: The percentage error in V is 5% and in I it is 2%. The total error in R would therefore be 5% + 2% = 7%.
Example 10 Two resistors of resistance R_{1}= (100 + 3) ohm and R_{2} = (200 + 4) ohm are connected in (a) series (b) parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation R = R_{1}+ R_{2} and for (b) 1/R’= 1/R_{1}+1/R_{2}
Answer: (a) The equivalent resistance of series combination
R = R_{1} + R_{2 }= (100 + 3) ohm + (200 + 4) ohm
= 300 + 7 ohm.
(b) The equivalent resistance of parallel combination
R’ =R1R2/(R1+R2) = 200/3 =66.7
Then from 1/R’ = (1/R1)+ (1/R2) we get,
(ΔR’/R’^{2)} = (ΔR_{1}/R^{2}) + (ΔR_{2} /R_{2}^{2})
ΔR’ = R’^{2} (ΔR_{1}/R^{2}) + R’^{2} (ΔR_{2}/R_{2}^{2})
= (66.7/100)^{2 }_{x} 3 + (66.7/200) ^{2 }x 4
= 1.8
Then R’ = 66.7 + 1.8 ohm.
(Here ΔR is expressed as 1.8 instead of 2 to keep in conformity with the rules of significant figure.
Error in case of a measured quantity raised to power
Suppose Z = A^{2},
Then
ΔZ/Z = ΔA/A + ΔA/A = 2 (ΔA/A)
Hence, the relative error in A^{2 }is two times the error in A.
In general, if Z = A^{p} B^{q} /C^{r}
(ΔZ’/Z) = p (ΔA/A) + q (ΔB/B) + r (ΔC/C)
Hence the rule: The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.
Example 11 Find the relative error in Z, if Z = A^{4}B ^{1/3 }/CD^{3/2}
Answer: The relative error in Z is ΔZ/Z = 4(ΔA/A) + (1/3) (ΔB/B) + (ΔC/C) + (3/2) (ΔD/D).
Example 12 The period of oscillation of a simple pendulum is T = 2π√(L/g) . Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?
Answer : g = 4πL/T^{2}
Here, T = t/n and ΔT =Δt/n . Therefore, ΔT/T = Δt/t. The errors in both L and t are the least count errors, Therefore,
ΔG/g = (ΔL/L) + 2(ΔT/T)
= (0.1/20) +2(1/90) = 0.032
Thus the percentage error in g is,
100(Δg/g) = 100(ΔL/L) + 2 × 100(ΔT/T)
= 3%
Significant Errors
As discussed above, every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures. If we say the period of oscillation of a simple pendulum is 1.62 s, the digits 1 and 6 are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has three significant figures. The length of an object reported after measurement to be 287.5 cm has four significant figures, the digits 2, 8, 7 are certain while 5 is uncertain. Clearly, reporting the result of measurement that includes more digits than the significant digits is superfluous and also misleading since it would give a wrong idea about the precision of measurement.
The rules for determining the number of significant figures can be understood from the following examples. Significant figures indicate, as already mentioned, the precision of measurement which depends on the least count of the measuring instrument. A choice of change of different units does not change the number of significant digits or figures in a measurement. This important remark makes most of the following most of the following observations clear:
(1) For example, the length 2.308 cm has four significant figures. But in certain units, the same value can be written as 0.02308 m or 23.08 mm.
All these numbers have the same number of significant figures (digits 2, 3, 0, 8), namely four. This shows that the location of decimal point is of no consequence in determining the number of significant figures. The example gives the following rules:
 All the non zero 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 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]. However, you can also see the next observation.
 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]
(2) There can be confusion regarding the trailing zero(s). Suppose a length is reported to be 4,700 m, it is evident that the zeroes here are meant to convey the precision of measurement and are therefore, significant. [If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4, 7 m]. Now suppose we change the units, then
4.700 m = 470.0 cm = 4700 mm =0.004700 km.
Since the last number has trailing zero(s) in a number with no decimal, we would conclude erroneously from observation (1) above that the number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures.
(3) To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10). In this notation, every number is expressed as a × 10 ^{b}where a is a number and b is any positive or negative exponent (or power of 10). In order to get an approximate idea of the number, we may 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. When only an estimate is required, the quantity is of the order of 10^{b}. For example, the diameter of the earth is 17 orders of magnitude larger than the hydrogen atom.
It is often customary to write the decimal after the first digit. Now the confusion mentioned in (a) disappears:
4,700 m = 4.7000 x 10^{2 }cm
= 4700 x 10^{3 }mm = 4.700 x 10^{3 }km
The power of 10 is irrelevant to the determination of significant figures. However, all zeroes appearing in the base number in the scientific notation are significant. Each number in this case has four significant figures.
Thus, in the scientific notation, no confusion arises about the trailing zero(s) in the base number a. they are always significant.
(4) The scientific notation is ideal for reporting measurement. But if this is not adopted, we use the rules adopted in the preceding example:
 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.
(5) The digit 0 conveniently put 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.
(6) 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 = 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 =t/n, n is an exact number.
7.1 Rules for Arithmetic Operations with Significant Figures
The result of a calculation involving approximate measured values of quantities [i.e. values with limited number of significant figures) must reflect uncertainties in the original measured values. It cannot be more accurate than the original measured values themselves on which the result is based. In general, the final result should not have more significant figures than the original data from which it was obtained. Thus, if mass of an object is measured to be say, 4.237 g (Four significant figures) and its volume is measured to 2.51 cm^{3}, then its density, by mere arithmetic division, is 1.68804780876 g/cm ^{3 }up to 11 decimal places. It would be clearly out of place and irrelevant to mention the calculated value of density to such a precision when the measurements on which the value is based, have much less precision. The following rules for arithmetic operations with significant figures ensure that the final result of a calculation is shown with the precision that is consistent with the precision of the input measured values:
(1) In multiplication or division, the final result should contain as many significant figures as are there in the original number with the least significant figures.
Thus, in the example above, density should be reported to three significant figures.
Density = 4.237 g/2.51 cm^{3}= 1.69 g cm^{3}
Similarly, if the speed of light is given as 3.00 × 10^{8 }m s ^{1 }(three significant figures).
(2) 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, the sum of the numbers 436.32 g, 227.2 g and 0.301 g by mere arithmetic addition is 663.821 g. But the least precise measurement (227.2 g) is correct to only one decimal place. The final result should therefore be rounded off to 663.8 g.
Similarly, three difference in length can be expressed as:
0.307 m – 0.304 m = 0.003 m = 3 x 10^{3 }m.
Note that we should not use rule (1) applicable for multiplication and division and write 664 g s the result in the example of addition and 3.00 × 10^{3} m in the example of subtraction. They do not convey the precision of measurement properly. For addition and subtraction, the rule is in terms of decimal places.
7.2 Rounding off the Uncertain Digits
The result of computation with approximate numbers, which contain more than one uncertain digit, should be rounded off. The rules for rounding off numbers to the approximate significant figures are obvious in most cases. A number 2.746 rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74. The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined in this case) is more than 5, and is left unchanged if the latter is less than 5. But what if the number is 2.745 in which the insignificant digit is 5. Here, the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. Then, 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 since the preceding digit is odd.
In any involved or complex multistep calculation, you should retain, in intermediate steps, one digit more than the significant digits and round off to proper significant figures at the end of calculation. Similarly, a number known to be within many significant figures such as 2.99792458 × 10^{8 }m/s, which is often employed in computations. Finally, remember, that exact numbers that appear in the formula like 2π in T = 2π√(L/g) , have large (infinite) number of significant figures. The value of π = 3.1415926…. is known to a large number of significant figures. You may take the value as 3.142 or 3.14 for π, with limited number of significant figures as required in specific cases.
Example 13 Each side of a cue is measured to be 7.203 m. What are the total surface area and the volume of the cube to appropriate significant figures?
Answer: The number of significant figures in the measured length is 4. The calculated area and the volume should therefore be rounded off to 4 significant figures.
Surface area of the cube = 6(7.203)^{2 }m^{2}
= 311.299254 m^{2}
= 311.3 m^{2}
Volume of the cube = (7.203)^{3 }m^{3}
= 373.714754 m^{3}
= 373.7 m^{3}
Example 14 5.74 g of a substance occupies 1.2 cm^{3}. Express its density by keeping the significant figures in view.
Answer: There are 3 significant figures in the measured mass whereas there are only 2 significant figures in the measured volume. Hence the density should be expressed to only 2 significant figures.
Density = 5.74/1.2 g cm^{3}
= 4.8 g cm^{‑3}
7.3 Rules for Determining the Uncertainty in the Results of Arithmetic Calculations
The rules for determining the uncertainty or error in the number / measured quantity in arithmetic operations can be understood from the following examples.
(1) If the length and breadth of a thin rectangular sheet are measured, using a meter scale as 16.2 cm and 10.1 cm respectively, there are three significant figures in each measurement. It means that the length l may be written as
l = 16.2 + 0.1 cm
= 16.2 + 0.6 %
Similarly, the breadth b may be written as
b = 10.1 + 0.1 cm
= 10.1 + 1 %
Then, the error of the product of two (or more) experimental values, using the combination of errors rule, will be
l b = 163.62 cm^{2} + 1.6 %
= 163.2 + 2.6 cm^{2}
This leads us to quote the final result as
l b = 164 + 3 cm^{2}
Here 3 cm^{2 }is the uncertainty or error in the estimation of area of rectangular sheet.
(2) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures.
However, if data are subscribed, the number of significant figures can be reduced. For example, 12.9 g – 7.06 g, both specified to three significant figures, cannot properly be evaluated as 5.84 g but only as 5.8 g, as uncertainties in subtraction or addition combine in a different fashion (smallest number of decimal places rather than the number of significant figures in any of the number added or subtracted).
(3) The relative error of a value of number specified to significant figures depends not only on n but also on the number itself.
For example, thee accuracy in measurement of mass 1.02 g is + 0.01 g whereas another measurement 9.89 g is also accurate to + 0.01 g, the relative error in 1.02 g is,
= (+ 0.01/1.02) x 100%
= + 1%
Similarly, the relative error in 9.89 g is,
= (+ 0.01/9.89) x 100%
= + 0.1%
Finally remember that intermediate results in a multistep computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.
These should be justified by the data and then the arithmetic operations may be carried out; otherwise rounding errors can be build up. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (three) is 0.104, but the reciprocal of 0.104 calculated to three significant figures, is 9.62. However, if we had written 1/9.58 = 0.1044 and then taken the reciprocal to three significant figures, we would have retrieved the original value of 9.58.
This example justifies the idea to retain one more extra digit (than the number of digits in the least precise measurement) in intermediate steps of the complex multistep calculations in order to avoid additional errors in the process of rounding off the numbers.
8.0 DIMENSIONS OF PHYSICAL QUANTITIES
The nature of a physical quantity is described by its dimensions. All the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental or base quantities. We shall call these base quantities as the seven dimensions of the physical world, which are denoted with square brackets [ ]. Thus, length has the dimension [L], mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd], and amount of substance [mo.].
The dimensions of physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. Note that using the square brackets [ ] round a quantity means that we are dealing with the dimensions of the quantity.
In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M], and [T]. For example, the volume occupied by an object is expressed as the product of length, breadth and height, or three lengths. Hence the dimensions of volume are [L] x [L] x [L] = [L] ^{3} = [L^{3}]. As the volume is independent of mass and time, it is said to possess zero dimension in mass [M^{0}], zero dimension in time [T^{0}] and three dimensions in length.
Similarly, force, as the product of mass acceleration, can be expressed as,
Force = mass × acceleration
= mass × (length)/ (time)^{ 2}
The dimensions of force are [M] [L] / [T]^{ 2} = [M L T ^{2}]. Thus, the force has one dimension in mass, one dimension in length, and 2 dimension in time. The dimensions in all other quantities are zero.
Note that in this type of representation, the magnitudes are not considered. It is the quality of the type of the physical quantity that enters. Thus, a change in velocity, initial velocity, average velocity, final velocity, and speed are all equivalent in this context. Since all these quantities can be expressed as length / time, their dimensions are [L]/ [T] or [L/T^{1}].
9.0 DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS
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 is called the dimensional formula of the given physical quantity. For example, the dimensional formula of the volume is [M^{0}L^{3}T^{0}] and that speed or velocity is [M^{0 }L T^{1}]. Similarly, [M^{0 }LT^{2}] is the dimension formula for acceleration and [M L^{3} T^{0}] that of mass density.
As equation obtained equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities. For example, the dimensional equations of volume [V], speed [u], and force [F] and mass density, [ r] may be expressed as,
[V] = [M^{0}L^{3}T^{0}]
[u] = [M^{0 }L T^{1}]
[r] = [M L^{3} T^{0}]
The dimensional equation can be obtained from the equation representing the relations between the physical quantities. The dimensional formulae of a large number and wide variety of physical quantities, derived from the equations representing the relationships among other physical quantities and expressed in terms of base quantities are given below Table for guidance and ready reference:
Table 11 – Dimensional Formula of Physical Quantities
Sr. No.  Physical quantity  Relationship with other physical quantities  Dimensions 

1  Area  Length x breadth  [L^{2}] 
2  Volume  Length x breadth x height  [L^{3}] 
3  Mass density  Mass / Volume  [M]/[L^{3}] or [ML^{3}] 
4  Frequency  1/time period  1/[T] 
5  Velocity, speed  Displacement/time  [L]/[T] 
6  Acceleration  Velocity /time  [LT^{1}]/[T] 
7  Force  Mass x acceleration  [M][LT^{2}] 
8  Impulse  Force x time  [M][LT^{2}][T] 
9  Work, Energy  Work x time  [M][LT^{2}][L] 
10  Power  Work/time  [M][LT^{2}]/[T] 
11  Momentum  Mass x velocity  [M][LT^{1}] 
12  Pressure, stress  Force/area  [MLT^{2}]/[L^{2}] 
13  Strain  Change in dimension/Original dimension  [L]/[L] or [L^{3}]/[L^{3}] 
14  Modulus of elasticity  Stress/strain  [ML^{1}T^{2}]/[M^{0} L^{0}T^{0}] 
15  Surface tension  Force/length  [MLT^{2}]/[L] 
16  Surface energy  Energy/area  [ML^{2}T^{2}]/[L^{2}] 
17  Velocity gradient  Velocity/distance  [LT^{1}]/[L] 
18  Pressure gradient  Pressure/distance  [ML^{1}T^{2}]/[L] 
19  Pressure energy  Pressure x volume  [ML^{1}T^{2}]/[L^{3}] 
20  Coefficient of viscosity  Force/area x velocity gradient  ([MLT^{2}]/([L^{2}][LT^{1} /L]) 
21  Angle, Angular displacement  Arc/radius  [L]/[L] 
22  Trigonometric ration (sin θ, cos θ, tan θ etc.)  Length/length  [L]/[L] 
23  Angular velocity  Angle/time  [L^{0}]/[T] 
Dimensional Analysis and its Applications
The recognition concepts of dimensions, which guide the description of physical behavior is of basic importance as only those physical quantities can be added or subtracted which have the same dimensions. A thorough understanding of dimensional analysis helps us in deducing certain relations among different physical quantities and checking the derivation, accuracy and dimensional consistency or homogeneity of various mathematical expressions. When magnitudes of two or more physical quantities are multiplied, their units should be treated in the same manner as ordinary algebraic symbols. We can cancel identical units in the numerator and denominator. The same is true for dimensions of a physical quantity. Similarly, physical quantities represented by symbols on both sides of a mathematical must have the same dimensions.
10.1 Checking the Dimensional Consistency of Equations
The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. In other words, we can add or subtract similar physical quantities. Thus, velocity cannot be added to force, or an electric current cannot be subtracted from the thermodynamic temperature. This simple principle called the principle of homogeneity of dimensions in an equation is extremely useful in checking the correctness of an equation. If the dimensions of all terms are not same, the equation is wrong. Hence, if we derive an expression for the length (or distance) of an object, regardless of symbols appearing in the original mathematical relation, when all the individual dimensions are simplified, the remaining dimension must be that of length. Similarly, if we derive an equation of speed, the dimensions on both sides of equation, when simplified, must be of length/time or [LT^{1}].
Dimensions are customarily used as preliminary test of the consistency of an equation, when there is some doubt about the correctness of the equation. However, the dimensional consistency does not guarantee correct equations. It is uncertain to the extent of dimensionless quantities or functions. The arguments of special functions, such as the trigonometric, logarithmic and exponential functions must be dimensionless. Apure number, ratio of similar physical quantities, such as angle as ratio (length/length), refractive index as the ration (speed of light in vacuum/speed of light in medium) etc., has no dimensions.
Now we can test the dimensional consistency or homogeneity of the equation
X = x_{0} + v_{0} t + (1/2) a t^{2}
For the distance x travelled by a particle or body in time t which starts from the position x_{0 }with an initial velocity v_{0}at time t = 0 and has uniform acceleration a along the direction of motion.
The dimensions of each term may be written as
[x] = [L]
[x_{0}] = [L]
[v_{0}t] = [LT^{1}][T]
= [L]
[(1/2) a t^{2}] = [LT^{—2}][T^{2}]
= [L]
As each term on the right hand side equation has same dimension, namely that of length, which is same as the dimension of left hand side of the equation, hence this equation is dimensionally correct.
It may be noted that a test of consistency of dimensions tells us no more and no less than a test of consistency of units, but has the advantage that we need not commit ourselves to a particular choice of units, and we need not worry about conversions among multiples and submultiple of the units. It may be borne in mind that if an equation fails this consistency test, it is proved wrong, but it passes, it is not proved right. Thus, dimensionally correct equation need not be actually an exact (correct) equation, but dimensionally wrong (incorrect) or inconsistent equation must be wrong.
Example 15 Let us consider an equation
½ mv^{2} = m g h
where m is the mass of the body, v its velocity, g is the acceleration due to gravity and h is the height. Check whether this equation is dimensionally correct.
Answer: The dimensions of LHS are,
[M] [LT^{1}]^{2} = [M] [L^{2}T^{2}]
= [ML^{2}T^{2}]
The dimensions of RHS are,
[M] [LT^{2}] [L] = [M] [L^{2}T^{2}]
= [ML^{2}T^{2}]
The dimensions of LHS and RHS are same and hence the equation is dimensionally correct.
Example 16 The SI unit of energy is J = kg m^{2}s ^{2}; that of speed v is ms^{1 }and of acceleration a is m s^{1} and of acceleration a is ms^{2}. Which of the formulae for kinetic energy (K) given below can you rule out on the basis of dimensional arguments (m stands for the mass of the body):
(a) K = m^{2}v^{3}
(b) K = (1/2) mv^{2}
(c) K = ma
(d) K = (3/16) mv^{2}
(e) K = (1/2) mv^{2} + ma
Answer: Every correct formula or equation must have the same dimensions on both sides of the equation. Also, only quantities with the same physical dimensions can be added or subtracted. The dimensions of the quantity on the right side are on both sides of the equation [M^{2}L ^{3}T^{3}] for (a); [ML^{2}T^{2}] for (b) and (d); and for (c) [MLT^{2}]. The quantity on the right side of (e) has no proper dimensions since two quantities of different dimensions have been added. Since kinetic energy K has the dimensions of [ML ^{2}T^{2}], formulas (a), (c) and (e) are ruled out. Note that dimensional arguments cannot tell which of the two, (b) or (d), is the correct formula. For this, one must turn to the actual definition of kinetic energy.
Deducing Relation among the Physical Quantities
The method of dimensions can sometimes be used to deduce relation among the physical quantities. For this we should know the dependence of thee physical quantity on other quantities or linearly independent variables) and consider it as a product type of the dependence. See the following example.
Example 17 Consider a simple pendulum, having a bob attached to a string that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (l), mass of the bob (m) and acceleration due to gravity ( g). Derive the expression for its time period using method of dimensions.
Answer: The dependence of the 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^{0}M^{0}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 =1/2, y =1/2, z = 0
Then, T = k l^{1/2 }g^{1/2}
Or. T = k√(l/g)
Note that value of constant k cannot be obtained by the method of dimensions. Here it does not matter if some number multiplies the right side of this formula, because that does not affect its dimensions.
Actually, k = 2π so that T = 2π√(l/g)
Dimensional analysis is very useful in deducing relations among the interdependent physical quantities. However, dimensionless constants cannot be obtained by this method. The method of dimensions can only test the dimensional validity, but not the exact relationship between physical quantities having same dimensions.