Table of Units and Dimensions of Physical quantities.

Units & Dimensions of Physical quantities in S.I system.
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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} kgm^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} kgm^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} kgm^{-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 ML^2T^{-1} kg-m^2 sec^{-1}
37. Angular Impulse 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
\varepsilon = \varepsilon_r \varepsilon_0 = (1+\chi)\varepsilon_0
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
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

Multiple choice questions – Units&Dimensions.

1 . 1KWH is  unit of

Ans : 1.Time 2. Power 3. Energy 4. Stress

2. Unit of  Intensity of magnetic induction field  is

Ans : 1.N/Am  2. Tesla   3.Wb/m^{2}    4. All above

3. Which of the following has no units?

Ans : 1. Thermal capacity    2. Magnetic susceptibility 3. Angular acceleration 4. Moment of a magnet

4.Which one of the following units is a fundamental unit?

Ans : 1. watt   2. joule/sec  3. ampere 4. newton

5. 10^5 Fermi is equal to

Ans : 1. 1 meter      2. 100  micron      3.   1angstrom unit 4. 1 mm

6. kg m/sec is the unit of

Ans : 1. Impulse 2. Angular acceleration 3 . Capacity of condenser   4. Acceleration.

7. candela is the unit of

Ans : 1. Magnetic flux   2. Intensity of electric field    3. Luminous intensity 4. Charge

8. If  10 newton = X dynes, the value of  x is

Ans : 1.10^6 2.10^4 3.10^8          4.10^3

9. 1 KWh is equal to

Ans : 1. 360 J    2. 1800 J   3.1800\times10^5J 4. 360\times10^5J

10. Which of the following is a common unit of a physical quantity in M.K.S & S.I systems.

Ans : 1. ampere  2.kelvin   3. mole  4. joule/sec 

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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.