Revision Notes on Unit and Dimensions by cmrohityadav edata

        

Revision Notes on Unit and Dimensions

In order to make the measurement of a physical quantity we have, first of all, to evolve a standard for that measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity.

System of Units:-


(a) C.G.S (Centimeter-Grand-Second) system.

(b) F.P.S. (Foot-Pound-Second) system.

(c) M.K.S. (Meter-Kilogram--Second) system.

(d) M.K.S.A. (Meter-Kilogram-Second-Ampere) unit.

Dimensional Formula:-


Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity.

How to write dimensions of physical quantities:-


(a) Write the formula for that quantity, with the quantity on L.H.S. of the equation.

(b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time.

(c) Substitute M, L and T for mass, length and time respectively.

(d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively.

Characteristics of Dimensions:-


(a) Dimensions of a physical quantity are independent of the system of units.

(b) Quantities having similar dimensions can be added to or subtracted from each other.

(c) Dimensions of a physical quantity can be obtained from its units and vice-versa.

(d) Two different physical quantities may have same dimensions.

(e) Multiplication/division of dimensions of two physical quantities (may be same or different) results in production of dimensions of a third quantity.



PHYSICAL QUANTITY

SYMBOL

DIMENSION

MEASUREMENT UNIT

UNIT


Length

s

L

Meter

m


Mass

M 

M

Kilogram

Kg


Time

t

T

Second

Sec


Electric charge

q

Q

Coulomb

C


luminous intensity

I

C

Candela

Cd


Temperature

T

K

Kelvin

oK


Angle



q

none

Radian

None


Mechanical Physical Quantities (derived)




PHYSICAL QUANTITY

SYMBOL

DIMENSION

MEASUREMENT  UNIT

UNIT













Area

A

L2

square meter

m2


Volume



V

L3

cubic meter 

m3


velocity



v

L/T

meter per second

m/sec


angular velocity



w

T-1

radians per second

1/sec


acceleration



a

LT-2

meter per square second

m/sec2


angular acceleration

a

T-2

radians per square

second 

1/sec2


Force



F

MLT-2

 Newton

Kg m/sec2


Energy



E

ML2T-2

 Joule

Kg m2/sec2


Work



W

       

ML2T-2

Joule

Kg m2/sec2


Heat



Q

     

ML2T-2

Joule

Kg m2/sec2


Torque



t

ML2T-2

Newton meter

Kg m2/sec2


Power



P

ML2T-3

watt  or  joule/sec

Kg m2/sec3


Density



D or ρ

ML-3

kilogram per

cubic meter

Kg/m3


pressure



P

    ML-1T-2

Newton per square meter

Kg m-1/sec2


impulse



J

MLT-1

Newton second

Kg m/sec


Inertia



I

ML2

Kilogram square meter



Kg m2 




luminous 

flux

f



C

lumen (4Pi candle for point source)

cd sr 


illumination



E

CL-2

lumen per

square meter

cd sr/m2


entropy



S

       ML2T-2K-1

joule per degree

Kg m2/sec2K


Volume

rate of flow

Q

L3T-1

cubic meter

per second

m3/sec


kinematic

viscosity

n

L2T-1

square meter

m2/sec


per second




dynamic

viscosity

m



      ML-1T-1



Newton second

per square meter

Kg/m sec        




specific

weight

g

  ML-2T-2

Newton

per cubic meter

Kg m-2/sec2


Electrical Physical Quantities (derived)


Electric

current

I

QT-1

Ampere

C/sec


emf, voltage,

potential

E

ML2T-2Q-1

Volt

Kg m2/sec2C


resistance or 

impedance 

R



ML2T-1Q-2



ohm



Kgm2 /secC2




Electric 

conductivity

s

M-2L-2TQ2

mho 



secC2/Kg m3 




capacitance



C

M-1L-2T2 Q2

Farad

sec2C2/Kgm2


inductance



L

ML2Q-2

Henry

Kg m2 /C2


Current density



J



QT-1L-2



ampere per

square meter

C/sec m2




Charge density

r

QL-3

coulomb per cubic meter

C/m3


magnetic flux,

Magnetic induction

B

MT-1Q-1

weber per

square meter

Kg/sec C


magnetic

intensity

H

QL-1T-1

ampere per meter

C/m sec


magnetic vector

potential

A

MLT-1Q-1

weber/meter

Kg m/sec C


Electric

field intensity

E

MLT-2Q-1

volt/meter or

newton/coulomb

Kg m/sec2 C


Electric displacement

D

QL-2

coulomb per square meter

C/m2


permeability



m

MLQ-2

henry per meter

Kg m/C2


permittivity,

e

T2Q2M-1L-3

farad per meter

sec2C2/Kgm3




dielectric constant

                     

K

    M0L0T0

 None

 None


frequency



f or n

T-1

Hertz

sec-1


angular frequency

W

T-1

radians per second         

sec-1




Wave length



l

L

Meters

M







Principle of homogeneity:-



It states that “ the dimensional formulae of every term on the two sides of a correct relation must be same.”


Types of error:-



(a) Constant errors:- An error is said to be constant error if it affects, every time, a measurement in a similar manner.

(b) Systematic errors:- Errors which come into existence by virtue of a definite rule, are called systematic errors.

(c) Random error or accidental error:- Error which takes place in a random manner and cannot be associated with a systematic cause are called random or accidental errors.

(d) Absolute error:-    

Relative Error:-


 

Percentage Error:-