# Table of Units and Dimensions of Physical quantities.

Units & Dimensions of Physical quantities in S.I system.

Fundamental Physical Quantities:

 S.No Fundamental Physical Quantity Formula Dimensional Formula S.I Unit of physical quantity 1. Mass Amount of matter in the object M kg 2. Length L meter 3. Time T sec 4. Electric current I or A ampere 5. Amount of substance N mole(mol 6. Luminous intensity J candela(cd) 7. Temperature K or $\theta$ Kelvin

Derived Physical Quantities:

 S.No Derived Physical Quantity Formula Dimensional Formula S.I Unit of physical quantity 1. Area $l\times b$ [$M^0L^2T^o$] $m^2$ 2. Volume $l\times b\times h$ [$M^0L^3T^o$] $m^3$ 3. Density $\frac{M}{V}$ [$M^1L^{-3}T^0$] $kg/m^3$ 4. Specific Gravity $\frac{Density of Substance}{Density of Water}$ [$M^0L^0T^0$] No units 5. Frequency $\frac{no of vibrations}{Time}$ [$M^0L^0T^{-1}$] hertz 6. Angle $\frac{Arc}{radius}$ $M^0L^oT^o$ No units 7. Velocity $\frac{Displacement}{time}$ $M^0L^1T^{-1}$ m/sec 8. Speed $\frac{Distance}{time}$ $M^0L^1T^{-1}$ m/sec 9. Areal velocity $\frac{Area}{time}$ $M^0L^2T^{-1}$ $m^2sec^{-1}$ 10. Acceleration $\frac{Change in velocity }{time}$ $M^0L^1T^{-2}$ $m/sec^2$ 11. Linear momentum $M\times V$ $M^1L^1T^{-1}$ kg m/sec 12. Force $mass\times acceleration$ $M^1L^1T^{-2}$ kg-m/$sec^2$ or Newton 13. Weight w=mg $M^1L^1T^{-2}$ kg-m/$sec^2$ or Newton 14. Moment of force/Torque/Couple $Force\times arm$ $M^1L^2T^{-2}$ kg$m^2sec^{-2}$ 15. Impulse $Force\times time$ $M^1L^1T^{-1}$ kg m/sec or Ns 16. Pressure $\frac{Force}{Area}$ $M^1L^{-1}T^{-2}$ N/$m^2$ or Pa 17. Work $Force\times Distance$ $M^1L^2T^{-2}$ Nm or Joule 18. Kinetic Energy $\frac{1}{2} mv^2$ $M^1L^2T^{-2}$ joule 19. Potential Energy mgh $M^1L^2T^{-2}$ joule 20. Gravitational constant $\frac{Force\times (Length)^2}{(mass)^2}$ $M^{-1}L^3T^{-2}$ $kg^{-1}m^3sec^{-2}$ 21. Gravitational field strength $\frac{Force}{mass}$ $M^0L^1T^{-2}$ $N kg^{-1}$ 22. Gravitational Potential $\frac{Work}{mass}$ $M^0L^2T^{-2}$ $J kg^{-1}$ 23. Force constant (k) $\frac{F}{L}$ $M^1L^0T^{-2}$ $N m^{-1}$ 24. Power $\frac{Work}{time}$ $M^1L^2T^{-3}$ W or J/sec 25. Moment of Inertia ( I ) $Mass\times Distance^2$ $M^1L^2T^{0}$ kg$m^2$ 26. Stress $\frac{Force}{Area}$ $M^1L^{-1}T^{-2}$ N/$m^2$ or Pa 27. Strain $\frac{Change in length}{Origional length}$ $M^0L^0T^0$ No units 28. Modulus of Elasticity $\frac{Stress}{Strain}$ $M^1L^{-1}T^{-2}$ N/$m^2$ or Pa 29. Poission’s Ratio σ =$\frac{Y}{2n}$-1 $M^0L^0T^0$ No units 30. Velocity gradient $\frac{Change in velocity}{Distance}$ $M^0L^0T^{-1}$ $sec^{-1}$ 31. Coefficient of dynamic viscosity $\frac{Tangential stress}{Velocity Gradient}$ $M^1L^{-1}T^{-1}$ kg$m^{-1}sec^{-1}$(or) N-sec/$latex \m^2$ (or)pascal-sec (or)poiseuille 32. Surface Tension $\frac{Force}{Length}$ $M^1L^0T^{-2}$ $kg sec^2$,N/m 33. Angular displacement ($\theta$) $\frac{Arc}{radius}$ $M^0L^oT^o$ no Units 34. Angular velocity(ω) $\frac{Angular displacement}{Time}$ $M^0L^oT^{-1}$ rad/sec 35. Angular acceleration(α) $\frac{Change in angular velocity}{Time}$ $M^0L^oT^{-2}$ rad/$sec^{-2}$ 36. Angular momentum Iω $ML^2T^{-1}$ $kg-m^2 sec^{-1}$ 37. Angular Impulse Iω $ML^2T^{-1}$ $kg-m^2 sec^{-1}$ 38. Temperature $\theta$ or K kelvin or degree Celsius 39. Coefficient of linear expansion(α) $\frac{l_2-l_1}{l_1\times Temp(t_2-t_1)}$ $M^0L^0T^0K^{-1}$ /kelvin 40. Specific heat $\frac{Energy}{Mass\times Temp}$ $M^0L^2T^{-2}K^{-1}$ 41. Latent heat $\frac{Energy}{Mass}$ $M^0L^2T^{-2}$ $joule-kg^{-1}$ 42. Entropy $\frac{Q}\theta$ $M^1L^2T^{-2}K^{-1}$ $J K^{-1}$ 43. Thermal capacity $\frac{H}\theta$ $M^1L^2T^{-2}K^{-1}$ $J K^{-1}$ 44. Gas constant $\frac{PV}{m T}$ $M^0L^2T^{-2}K^{-1}$ $joule-K^{-1}$ 45. coefficient of thermal conductivity $\frac{Qd}{A(\theta_2-\Theta_1)t}$ $M^1L^1T^{-3}K^{-1}$ $W m^{-1}K^{-1}$ 46. Pole strength $Ampere\times meter$ $M^0L^1T^0I$ Am 47. Magnetic Moment $M^0L^2T^0I^1$ $Amp-m^2$ 48. Magnetic flux $\phi$ $ML^2T^{-2}I^{-1}$ weber ;$T-m^{2}$ ;J/Amp 49. Magnetic field,magnetic flux density (B) $MT^{-2}I^{-1}$ Tesla;$J/A-m^{2}$ 50. Permeability of free space $\frac{\mu}{\mu_r}$ $MLT^{-2}I^{-2}$ $NA^{-2}$ 51. Magnetic susceptibilty also called volumetric or bulk susceptibility χm χm = μr − 1 $M^0L^oT^o$ no units 52. Electric Charge $I\times T$ $M^0L^0T^1I^1$ Amp sec , coul 53. Electric potential $\frac{Work}{Charge}$ $M^1L^2T^{-3}I^{-1}$ Volt 54. E.M.F $\frac{Work}{Charge}$ $M^1L^2T^{-3}I^{-1}$ Volt 55. Electric Capacity $\frac{q}{V}$ $M^{-1}L^{-2}T^4I^2$ Farad 56. Electric Resistance $\frac{V}{i}$ $M^1L^2T^{-3}I^{-2}$ Ohm (Ω) or volt/amp 57. Resistivity $\rho$ $\frac{R A}{L}$ $M^1L^3T^{-3}I^{-1}$ Ohm mt (Ω-m) 58. Conductivity $\sigma$ 1/$\rho$ $M^{-1}L^{-3}T^3I$ Siemens/m 59. Permittivity $\varepsilon$ $M^{-1}L^{-3}T^4I^2$ farad/m 60. Electric conductance $\frac{1}{R}$ $M^{-1}L^{-2}T^3I^2$ Siemens (or) mhos 61. Electric power $V\times I$ $M^1L^2T^{-3}I^{-1}$ Watt 62. Electrical Impedance(Z) $\frac{V}{i}$ $M^1L^2T^{-3}I^{-2}$ Ohm (Ω) or volt/amp 63. Electrical admittance 1/Z(Reciprocal of electric impedance) $M^{-1}L^{-2}T^3I^3$ Siemens (or) mhos 64. Self Inductance(L) $\displaystyle v=L\frac{di}{dt}$ $ML^2T^{-2}I{-2}$ weber/amp or Henry 65. Boltzmann’s constant $\frac{Energy}{Temp}$ $M^1L^2T^{-2}K^{-1}$ J/kelvin 66. Stefan’s constant $\frac{E}{At \theta^4}$ $M^1L^0T^{-3}K^{-4}$ $W m^{-2}K^{-4}$ 67. Co-efficient of friction $\mu$ $\mu$=$\frac{F}{N}$,N=Normal reaction dimension less scalar no units 68. Dielectric constant $\varepsilon_r$ It is also called relative permittivity dimension less no units 69. Planck’s constant $E=h\nu$ $ML^2T^{-1}$ J.sec (or) eV.sec 70. Refractive index μ $M^0L^oT^o$ no units 71. Focal length(f) Distance between center of the lens(mirror) to its focus L meter 72. Power of a lens (P) The reciprocal of the focal length of a lens in meters is called power of a lens; p=1/f $L^{-1}$ diaptors 73. Wave number No.of waves/distance $L^{-1}$ $m^{-1}$ 74. Wave length Length of a wave L meter

# Units & Dimensions Q&A

1. What is a physical quantity?

Ans : Any quantity which is measurable is called physical quantity.

2. Explain the term Fundamental Physical quantity.

Ans: The physical quantity which is independent or which can not be derived from any other physical quantity is called fundamental physical quantity. EX: Mass, Length and Time.

3.Explain the term Derived physical quantity.Give examples.

Ans :The physical quantity which is dependent on other physical quantity or which is derived from other physical quantity is called derived physical quantity. Ex : Area, Electric charge, Magnetic field strength, power etc.

4.How many fundamental quantities are there in C.G.S; F.P.S and M.K.S systems? What are they?

Ans : There are 3 fundamentals quantities in C.G.S; F.P.S and M.K.S systems, they are mass, length and time.

5.How many fundamental quantities are there in S.I systems? What are they?

Ans : In S.I system 7 fundamental quantities are there,they are i) Mass ii)Length iii)Time iv)Electric current v)Intensity of light vi) Thermodynamic temperature vii) Quantity of matter.

6.How many supplementary quantities are there in S.I system? What are they?

Ans : In S.I system there are 2 supplementary quantities, they are i) Plane Angle ii) Solid Angle.

7. What are the units of length in C.G.S ; F.P.S and M.K.S systems.

Ans : The units of length  are cm,foot and meter respectively in C.G.S ; F.P.S and M.K.S systems .

8. what are the units of fundamental quantities in S.I system?

Ans : Mass → kg ; Length → m ; Time → sec ; Electric current → Amp Thermodynamic temperature → kelvin ;

Intensity of light → candela ; Quantity of matter → mole .

9.what are the units of supplementary quantities in S.I system?

Ans : Plane angle → radian ; Solid angle → steradian .

10. Name the physical quantities whose dimensional formula is $M^0L^1T^0$ ?

Ans : The physical quantities are i)Distance or length ii) displacement iii)wave length

11. Name the physical quantities whose dimensional formula is $M^0L^1T^{-1}$ ?

Ans : The physical quantities are i) speed ii) velocity

12.Name the physical quantities whose dimensional formula is $M^1L^2T^{-2}K^{-1}$ ?

Ans : The physical quantities are i)Thermal capacity ii) Entropy

13. Name the physical quantities whose dimensional formula is $M^1L^1T^{-1}$ ?

Ans : The physical quantities are i)Momentum ii) impulse .

14.Name the physical quantities whose dimensional formula is $M^1L^1T^{-2}$ ?

Ans : The physical quantities are i)force ii ) Tension iii) weight .

15.Name the physical quantities whose dimensional formula is $M^1L^2T^{-2}$ ?

Ans : The physical quantities are i) Work ii) Energy iii) Heat iv)Moment of force Iv) Torque .

16.Name the physical quantities whose dimensional formula is $M^1L^{-1}T^{-2}$?

Ans : The physical quantities are i) pressure ii ) stress iii) Young’s modulus iv) Rigidity modulus v) Bulk modulus .

17.Name the physical quantities whose dimensional formula is $M^0L^0T^{-1}$ ?

Ans : The physical quantities are i) frequency ii) Decay constant iii)Angular velocity .

18 . Name the physical quantities whose dimensional formula is $M^1L^2T^{-1}$ ?

Ans : The physical quantities are i )angular momentum ii )Plank constant .

19. Name the physical quantities whose dimensional formula is $M^1L^0T^{-2}$ ?

Ans : The physical quantities are i )Force constant ii )surface tension .

20. Which physical quantity has negative dimensions in mass ?

Ans : Gravitational constant (G) .

21. State few constants which have dimensions ?

Ans : i) Plnak’s constant (h) ii)Velocity of light in vacuum (c) iii)Permeability of free space ($\mu_0$) iv) Permittivity of free space ($\epsilon_0$) v)Universal gravitational constant (G) vi) Universal gas constant (R)

vii)Boltzmann constant (k) .

22 .which physical quantities have the unit henry ?

Ans : self Inductance and Mutual Inductance have the unit henry .

23. What are the dimensions of  electric conductivity in mass , length and current.

Ans : Electric conductivity has -1,-3 and 2 dimensions  in mass,length and current respectively.

24. What is the unit of electric conductivity in C.G.S and S.I systems?

Ans : It has no unit in C.G.S system ; its unit in S.I system is Siemen/meter or S/m.

25.What are the uses of Dimensional methods?

Ans : To convert units from one system to another. ii )To check the correctness of equations connecting physical quantities iii )To derive the expressions connecting physical quantities.

26. Which is the physical quantity whose S.I unit is Am ?

Ans: Magnetic pole strength.

27. V/m or N/Coulomb are the units of ……. Physical quantity.

Ans : These are the units of Electric field strength.

28.Name five physical quantities which neither have dimensions nor units.

Ans : Refractive Index , specific gravity,susceptibility,dielectric constant,  coefficient of friction.

29. If  V = Xt+Y ; V is the velocity , t is time.What are the dimensional formulas of X and Y ?

Ans : According to principle of homogeneity of dimensions, the dimensions  of M,L and T in every term should be same.

Therefore $M^0L^1T^{-1}$ = X $M^0L^0T^1$ → X = $\frac{M^0L^1T^{-1}}{ M^0L^0T^1}$ ; X → $L^1T^{-2}$ and Y→ $L^1T^{-1}$

30.Which physical quantities does not possess dimensions in mass ?

Ans :Area,volume, velocity, acceleration,angular displacement, angular velocity, angular acceleration.

# Dimensions & Dimensional formulae.

Dimensions: Dimensions of a physical quantity are,the powers to which the fundamental units  are raised to get one unit of the physical quantity.

The fundamental quantities are expressed with following symbols while writing dimensional formulas of derived physical quantities.

Mass →[M] ; Length→[L]; Time→[T]; Electric current →[I] ; Thermodynamic temperature →[K] ;Intensity of light →[cd] ; Quantity of matter →[mol] .

Dimensional Formula :Dimensional formula of a derived physical quantity  is the “expression showing powers to which different fundamental units are raised”.

Ex : Dimensional formula of Force  F →[$M^1L^1T^{-2}$]

Dimensional equation:When the dimensional formula of  a physical quantity is expressed in the form of  an equation by writing the physical quantity on the left hand side and the dimensional formula on the right hand side,then the resultant equation is called Dimensional equation.

Ex: Dimensional equation of Energy is E = [$M^1L^2T^{-2}$] .

Question : How can you derive Dimensional formula of a derived physical quantity.

Ans : We can derive dimensional formula of any derived physical quantity in two ways

i)Using the formula of the physical quantity : Ex: let us derive dimensional formula of Force .

Force F→ma ; substitute the dimensional formula of mass m →[M] ; acceleration →[$LT^{-2}$]

we get F → [M][$L T^{-2}$]; F →[$M^1L^1T^{-2}$] .

ii) Using the units of the derived physical quantity. Ex: let us derive the dimensional formula of momentum.

Unit of Momentum ( p ) → [$kg-m sec^{-1}$] ;

kg is unit of mass → [M] ; is unit of length → [L] ; sec is the unit of time →[T]

Substitute these dimensional formulas in above equation we get p →[$M^1L^1T^{-1}$].

Quantities having no units, can not possess dimensions: Trigonometric ratios, logarithmic functions, exponential functions, coefficient of friction, strain, poisson’s ratio, specific gravity, refractive index, Relative permittivity, Relative permeability. All these quantities neither possess units nor dimensional formulas.

Quantities having units, but no dimensions : Plane angle,angular displacement, solid angle.These  physical quantities possess units but they does not possess dimensional formulas.

Quantities having both units & dimensions : The following quantities  are examples of such quantities.

Area, Volume,Density, Speed, Velocity, Acceleration, Force, Energy etc.

Physical Constants : These are two types

i) Dimension less constants (value of these constants will be same in all systems of units): Numbers, pi, exponential functions are dimension less constants.

ii)Dimensional constants(value of these constants will be different in different systems of units): Universal gravitational constant (G),plank’s constant (h), Boltzmann’s constant (k), Universal gas constant (R), Permittivity of free space($\in_0$) , Permeability of free space ($\mu_0$),Velocity of light (c).

Principle of Homogeneity of dimensions: The term on both sides of a dimensional equation should have same dimensions.This is called principle of Homogeneity of dimensions. (or) Every term on both sides of a dimensional equation should have same dimensions.This is called principle of homogeneity of dimensions.

Uses of Dimensional equations : dimensional equations are used i) to convert units from one system to another,

ii)to check the correctness of the dimensional equations iii)to derive the expressions connecting different physical quantities..

Limitations of dimensional method: The limitations of dimensional metthod,s are

i)The value of dimensionless constants can not be calculated using dimensional methods,

ii) We can not analyze the equations containing trigonometrical, exponential and logarithmic functions using method of dimensions.

iii)If a physical quantity is sum or difference of two or more than two physical quantities, such physical quantities can not be derived with dimensional methods,

iv)If  any equation having dimensional constants like, G, R etc can not be derived using dimensional methods,

v)If any equations is involving more than three fundamental quantities in it, such expressions can not be derived using dimensional methods.

# Units in different systems.

Generally we can use any convenient unit to measure a physical quantity depending on how much magnitude we are measuring or in which system of units we want to measure it.

What kind of unit we should use?

The unit i) must be accepted internationally.

ii) Should be reproducible.

iii) Should be invariable.

iv) Should be easily available.

v) Should be consistent.

vi) Should be large, if the physical quantity to be measured is a big quantity.

Ex: To measure larger lengths we use units like Km, mt etc, to measure large magnitude of time we use units like hour , day ,week, month , year etc.

vii) Should be small if the physical quantity to be measured is small.

Ex: To measure small time we use units like millisecond, microsecond etc

To measure small lengths we use units like millimeter, centimeter etc.

Types of physical Quantities.:

We can broadly divide the physical quantities in to two types i)Fundamental Physical quantities ii)Derived physical quantities.

Fundamental physical quantities: A physical quantity which can exist independently is called Fundamental physical quantity.

Ex: Length, mass and time etc.

Derived physical quantities: A physical quantity which can not exist independently is called derived physical quantity. (Or) A physical quantity which is dependent or derived from any other physical quantity is called derived physical quantity.

Ex : Area, volume, density, speed, acceleration, force, energy etc.

Like the physical quantities we can divide the units in to two types. I)Fundamental units ii)derived units.

Fundamental units : The units of fundamental physical quantities are called fundamental units, (or) The units which are independent or can not derived from any other unit is called fundamental unit.

Ex:­Every unit of length is fundamental unit (irrespective of the system to which it belongs);millimeter, centimeter, meter, kilometer etc.

­ Every unit of time is a fundamental physical quantity ; microsecond, millisecond, second, minute, hour, day etc.

Derived units: The units of derived physical quantities are called derived units. Units of area, volume, speed, density, energy etc are derived units.

Ex: ­ Every unit of speed is a derived unit ; m/sec, cm/sec, km/hr etc.

­ Every unit of density is a derived unit; kg/m³, gr/cm³ etc.

­ Every unit of acceleration is a derived unit; m/sec², cm/sec², km/hr² etc.

Systems of units: To measure the fundamental physical quantities Length, Mass and time we have three systems of units, they are i) C.G.S System (Metric system)ii)F.P.S System (British system) and iii)M.K.S System. In all these three systems only three physical quantities length, mass and time are considered to be fundamental quantities.

But, in systems International (S.I) system there are seven fundamental physical quantities. Which are i)Length ii)Mass iii)Time iv)Electric current v)Thermo dynamic temperature vi)Luminous intensity vii)Quantity of substance.

In addition to these two more quantities were added as supplementary physical quantities. They are i)Plane angle ii)Solid angle.

Systems,Fundamental physical quantities and their units:In
C.G.S system: Length (centimeter); Mass (gram); Time (second).

F.P.S system :Length (foot);Mass(pound);Time (second).

M.K.S system: Length (meter); Mass (kilogram); Time (second).

S.I System:Length (meter); Mass (kilogram); Time (second); Electric current (ampere); Thermodynamic temperature (kelvin); Intensity of light (candela); Quantity of matter (mole). The units of suplimentary quantities are Plane angle( radian); Solid angle(Steradian).

# What are Units

Metric system

In my previous post, I explained with example how to solve the In problems in physics. To solve problems and to under stand the basics of the Physics it is very important to know what is a physical quantity, types of physical quantities, what is a unit, what are the units of different physical quantities, types of units, symbols of units.

There is one and only branch of science which measures a physical quantity, that branch of science is “Physics”. Measurements have an important role not only in physics but also in every branch of science and everywhere in our day-to-day life.

To measure physical quantities we need units. Let’s try to understand necessity of measurements and units of measurement in Physics.

The information about a physical quantity, by description of its external properties like color, taste etc is incomplete with out knowing its temperature, size (dimensions), which depends on measurement

, i.e. with out measurements it is impossible to know about the external properties of any object. So, it becomes necessary to measure it.

To measure a physical quantity we require a unit. Different physical quantities will have different units.

What is unit? A standard reference of the same physical quantity is essential to measure any physical quantity. That standard which we use to measure a physical quantity is called unit.

Let me put it this way, if we want to measure length of a table, we have to select a standard length (length of our hand), and by comparing the table’s length with the standard length we can measure the length of the table. If the table is 3.5 times that of standard length, i.e. length of our hand then we can write the result as “length of table = 3.5 times the length of our hand. In this example length of hand is taken as standard length to measure the table’s length.

Like that we can define any convenient standard or unit to measure a physical quantity.

But, if we choose standards as in the above example which are not consistent, and can not be reproduced then  errors and confusion in measurements will creep in. To avoid such confusion, instead of taking any undefined reference as a standard, well-defined and universal standards are used. Such a reference taken a standard is generally called a well defined unit or unit. Measurement of every physical quantity will have two parts a number (n) followed by a unit (u).

There fore n u = constant.

Ex: If the length of a table is 1.2 meters.In this measurement number n= 1.2 and unit is meter.

→ length (L)= $n_1u_1$ = 1.2 meters

→ length (L)= )= $n_2u_2$ = 120 centimeters

→ length (L)= $n_3u_3$ = 1200 millimeters

From the above data we can understand that

i)we can measure a physical quantity in different units.what ever may be the unit it’s value is same.

→ L =$n_1u_1$ = $n_2u_2$=$n_3u_3$

If theunit cosen is smaller, the multiple number will be greater.

$u_1$>$u_2$>$u_3$ = $n_1$ <$n_2$<$n_3$

iii)The units(u) of a physical quantity will be reciprocal to the multiple (n)

nu=constant$\Rightarrow$ $n_1u_1$=$n_2u_2$

u$\propto\frac{1}{n}$ or n$\propto\frac{1}{u}$

$\frac{n_1}{ n_2}$= $\frac{u_2}{ u_1}$