Physical Quantity Unit and Dimension

Best Notes on Physical quantity | Scalar quantity | vector quantity | Fundamental quantity | Derived quantity | Unit | Importance of Unit | Property of Unit | Type of Unit | System of Unit | Definition of Basic and Supplementary S.I. units | Advantage of SI units | Some important derived units | Some practical units of length, mass and time | Prefixes used in metric system | Dimension | List of all type of Dimension | Principle of dimensional homogeneity | Uses of Dimensional analysis | Limitations of Dimensional analysis | FAQ

physical quantity unit and dimension

Notes on Physical quantity, Unit and Dimension

Hello everyone, In this post, we are providing best notes of Physical quantity Unit and Dimension – an important topic of Physics for India’s top government exams. This notes is completely free. And it will help students to crack government exams like UPSC CSE, State PCS, SSC CGL, CHSL, CPO, MTS, Railway, EPFO, APFC, CDS, NDA, UGC NET, State pcs, etc.

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Physical Quantity

That quantity which can be measured by an instrument is called physical quantity. Example – length of any object, mass of body, time of any event, velocity, force etc.

Physical Quantity is divided into two parts-

On the bases of unit and measurementOn the bases of direction
and magnitude
1. Fundamental Quantity– The physical quantity which can be treated as independent from other physical quantities is called Fundamental physical quantity.1. Vector Quantity- That physical quantity which has both magnitude and direction. eg.- Force, Velocity, Displacement etc.
2. Derived Physical Quantity- The physical quantity which can be obtained from other physical quantities or the quantity which can be defined in terms of fundamental physical quantity.2. Scalar Quantity- That physical quantity which has only magnitude but no direction. eg.- Time, Mass, Speed.

Definition of Unit:

The chosen standard value used for measuring a physical quantity is called unit.

Why are units important?

To separate out one physical quantity from another we need units.

Basic properties of Unit –

  1. well defined
  2. easy to reproduce
  3. easy to compare
  4. internationally accepted
  5. independent of changes in physical quantity

Types of units-

Fundamental unitsDerived unitsSupplementary units
eg.-1. Length – meterArea – meter2Plane angle – Radian
eg.-2. Mass – KilogramForce – NewtonSolid angle – Steradian
eg.-3. Time – second etc.Volume – meter3 etc.

System of Unit-

Units depend on choice. Each choice of the unit leads to a new system (set) of units. The internationally accepted systems are-

CGS System MKS SystemFPS System
CentimetreMeterFoot
GramKilogramPound
SecondSecondSecond
In SI Units, there are seven fundamental units given in the following table-
Physical QuantitySI UnitSymbol
1. Lengthmetrem
2. Masskilogramkg
3. Timeseconds
4. Electric CurrentampereA
5. TemperaturekelvinK
6. Luminous intensitycandelaCd
7. Amount of substancemolemol
CGS system also known as Metric or Decimal System

Definition of Basic and Supplementary S.I. units

1. Metre (m): It is defined as the distance occupied by 1650763.73 wavelengths in vacuum of the radiations emitted by Krypton-86 atom in the transition between 2p10 and 5d5 states.

OR

One metre is the length of the path travelled by light in vacuum in 1/299,792458 second.

2. Kilogram (kg): It is the mass of a platinum-iridium cylinder (prototype cylinder) having its diameter equal to the height preserved at the International Bureau of Weights and Measures at Sevres near Paris in a Vault. (Height and diameter is 3.9 cm) [Pt +Ir = 90% + 10%]

3. Second (s): It is the duration of 9192, 631,770.0 periods of the radiations corresponding to the transition between two hyper fine levels of the ground state of cesium-133 (Cs113) atom.

OR

It is the time taken by cesium-133 atom to make 9, 192, 631, 770.0 vibrations.

4. Ampere: It is S.I. unit of electric current.It is that constant current which when flows through each of the two long straight parallel conductors of infinite length and negligible area of cross-section placed at one metre apart in vacuum experiences a force of 2 x 10-7 N/m between them. Symbol is ‘A’.

5. Kelvin (K): S.I. unit of thermodynamic temperature. One kelvin (K) is defined as 1/273.16th of the thermodynamic temperature of the triple point of water.

6. Candela (cd): One candela is defined as luminous intensity in a direction at right angle to a surface of 1/600000 square metre area of a black body, at a temperature of freezing platinum under a pressure of 101325 newton per metre square.

In 1979 the new definition of candela was put forward as ‘The candela is the luminous intensity in a given direction of a source that emits monochromatic radiations of frequency 5.4 x 1014 Hz and that has a radiant intensity in that direction of 1/683 watt per steradian.

7. Mole (Mol): One mole is the amount of substance that contains as many elementary units as there are atoms in exactly 0.12 kg of pure carbon-12.

Supplementary Units.

1. Radian: It is the unit of plane angle. It is the measure of the angle subtended at the centre of circle by an arc having length equal to radius of circle.

2л radian = 360°

л radian = 180°

1 radian = 180°/π = 180°×7/22 = 630°/11 =57.3°

2. Steradian (Sr): Steradian is the unit of solid angle. It is the angle subtended at the centre of a sphere by its surface whose area is equal to the square of the radius of the sphere.

Advantages of S.I. units :

1. It is a coherent system of units. (A coherent system is one in which all the derived units can be obtained by multiplying or dividing the given basic units.)

2. It is a rational system of units. (It means it assigns only one unit to a particular physical quantity) e.g. Joule is unit of energy of all types where as in mks system mechanical energy is in Joule, Heat Energy is in calorie and electric energy is in watt hour.

3. It is internationally accepted system of units.

4. It is a metric system of units (i.e., all multiples and submultiples can be expressed in the power of ten)

5. It is absolute system of units, which means there are no gravitational units on the system.

6. It is closely related to cgs system (which means S.I. system can be converted to C.G.S. and vice-versa)

Some important derived units-

Physical QuantityCGS unitsSI unitRelation
1.Forcedynenewton1 newton = 105 dyne
2.work ergjoule1 joule = 107 erg

Some practical units of length, mass and time-

LengthLight year = distance travelled by light in one year in vacuum.
1 LY = 9.46 ⨯ 1015 m
1 Astronomical Unit (A.U.) = 1.5 ⨯ 1011 m
1 Parsec = 3.26 LY = 3.08 ⨯ 1016 m
1 Micron = 10-6 m
1 Angstrom(Å) = 10-10 m
Mass1 Quintal = 102 kg
1 Metric ton = 103 kg
1 Atomic Mass Unit (amu) =1.66 ⨯ 10-27 kg
1 Slug = 14.59 kg
1 Pound = 0.4537 kg
Chandrashekhar limit = 1.4 times the mass of the sun
= 2.8 ⨯ 1030 kg
Time1 Solar day = 86400 sec.
1 Year = 365 ½ solar days
1 Lunar month = 27.3 solar days
Tropical year = It is the year in which total solar eclipse occurs.
Leap year = It is the in which the month of February is of 29 days.

Prefixes used in metric system-

PrefixSymbolMultiplierPrefixSymbolMultiplier
decid10-1decada101
centic10-2hectoh102
millim10-3kilok103
microµ10-6megaM106
nanon10-9gigaG109
picop10-12teraT1012
femtof10-15petaP1015
attoa10-18exaE1018
zeptoz10-21zettaZ1021
yoctoy10-24yottaY1024
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Dimension

Dimension of a physical quantity is defined as the process of representation with the different power of fundamental physical quantity.

For example Dimensional formula of momentum is MLT-1

List of all type of Dimension

Sr.
no.
Physical QuantityRelation with other Physical QuantitiesDimensional FormulaS.I Unit
1.Arealength × breadth[L] × [L] = [L]2
[M0L2T0]
m2
2.Volumelength × breadth × height [L] × [L] × [L] = [L]3
[M0L3T0]
m3
3.Density mass / volume[M] / [L3] =
[ML-3 T0]
kg/m3
4.Velocitydistance / time[L] / [T] = [M0LT-1 ]m/s
5.Accelerationvelocity / time[LT-1 ] / [T] =
[M0LT-2 ]
m/s2
6.Liner momentummass × velocity[M] × [LT-1 ] =
[MLT-1 ]
kg m/s
7.Angular Velocitydθ/ dt1 / [T] = [M0L0T-1 ]Rad s-1
8.Frequencynumber of rotation/time1 / [T] = [M0L0T-1 ]s-1 hertz (Hz)
9.Angular accelerationdω/dt[1/T] / [T] =
[M0L0T-2 ]
rad/s2
10.CurrentFundamental quantity [M0L0T0 A ]ampere
11.Chargecurrent × time[AT]coulomb
12.Impulse of forceforce × time [MLT-2 ] × [T] =
[MLT-1 ]
N s
13.Momentummass × velocity[M] × [LT-1 ] =
[MLT-1 ]
kg m/s
14.Forcemass × acceleration [M] × [LT-2 ] =
[MLT-2 ]
newton
15.Work, Energyforce × distance [MLT-2 ] × [L] =
[ML2T-2 ]
joule
16.Powerwork / time [ML2T-2 ] / [T] =
[ML2T-3 ]
watt
17.Pressure, Stress,
Modulus of Elasticity
force / area [MLT-2 ] / [L2] =
[ML-1T-2 ]
pascal
18.Moment of Inertiamk2[M] × [L2] =
[ML2T0]
kg m2
19.Torque / Moment of Forceforce × distance [MLT-2 ] × [L] =
[ML2T-2]
Nm or J
20.Angular Momentummomentum × distance [MLT-1 ] × [L] =
[ML2T-1]
kg m2 s-1
21.Planck’s ConstantE = hv,
or h = E / v
[ML2T-2] / [T-1]
[ML2T-1]
J s
22.Coefficient of Viscosity[ML-1T-1]Pas
kg m-1 s-1
23.Surface TensionT = F / ɩ [MLT-2 ] / [L] =
[MT-2 ]
Nm-1
24.Universal Gas
Constant
PV = nRT,
R = PV / nT
[ML2T-2 mol-1 K-1 ]J/mol K
25.Latent HeatL = Q / m[M0L2T-2 ]J/kg
26.Specific Heat[M0L2T-2 K-1]J/ kg K
27.Angelarc / radius [L] / [L] = 1
(dimensionless)
radian (rad)
28.Gravitational constantF = G m1m2 /r2[M-1L3T-2]N m2/kg2
29.Boltzmann constant[ML2T-2K-1]J/K
30.Avogadro numbernumber of mole[M0L0T0mol-1]mol-1
31.Permittivity of free space[M-1L-3T4A2]N m2/c
32.Permeability of free space[MLT-2A-2]Wb/A m

Principle of dimensional homogeneity:

According to this principle the dimensions of each term of a physical equation should be similar. This principle is based on the fact that physical quantities of same kind can be added, subtracted or compared. So force can be added to force not to pressure.

For example, let a physical relation in terms of fundamental quantities.

[MaLbTc] = [MxLyTz]

∴ by principle of homogeneity a = x, b = y, c = z

Uses of Dimensional analysis

The method of studying a physical phenomenon on the basis of dimensions is called dimensional analysis. The main uses of dimensional analysis are as

  1. Convert value of a physical quantity from one unit system to another.
  2. To check the correctness of a physical equation.
  3. Determine the relation among different physical quantities.

Limitations of Dimensional analysis

  1. This method cannot be used to determine relations between more than three physical quantities.
  2. This method cannot be used to derive a relation containing trigonometric or exponential functions.
  3. Many different physical quantities (e.g. modulus of elasticity, stress, pressure) which have the same dimensions, cannot be identified correctly with this method.

FAQ

what is the dimension of the universal gas constant?

[ML2T-2 mol-1 K-1 ]
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what is the dimension of surface tension?

[MT-2 ]
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Which system is known as Metric or Decimal System?

C.G.S system
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Which physical quantities have the same unit?

Work and Energy
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Which physical quantities have no unit?

Strain, specific gravity, Poisson’s ratio etc
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Which physical quantities have the same dimension?

Impulse and momentum
force and tension
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What is the dimension of the temperature gradient?

[M0L-1T0K]
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What is the dimension of heat capacity?

[ML2T-2K-1]
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What is the dimension of Specific heat capacity?

[M0L2T-2K-1]
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What is the dimension of thermal conductivity?

[MLT-3K-1]
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What is the dimension of thermal resistance?

[M-1L-2T3K]
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