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APPLIED PHYSICS
MEASUREMENTSUNITS
AND DIMENSION
AND DIMENSION
The
comparison of any physical quantity with its standard unit is called
comparison of any physical quantity with its standard unit is called
measurement.
Physical Quantities
All the quantities in terms of which laws of physics are described,
and
whose measurement is necessary are called physical quantities.
whose measurement is necessary are called physical quantities.
Units
· A definite amount of a physical
quantity is taken as its standard unit.
quantity is taken as its standard unit.
· The standard unit should be
easily reproducible, internationally accepted.
easily reproducible, internationally accepted.
Fundamental Units
Those physical quantities which are independent to each other are called
fundamental quantities and their units are called fundamental units.
fundamental quantities and their units are called fundamental units.
S.No. Fundamental Quantities Fundamental Units Symbol
1.

Length

metre

m

2.

Mass

kilogram

kg

3.

Time

second

S

4.

Temperature

kelvin

kg

5

Electric
current 
ampere

A

6

Luminous
intensity 
candela

cd

7

Amount
of substance 
mole

mol

Supplementary Fundamental Units
Radian and steradian are two supplementary fundamental units.
It
measures plane angle and solid angle respectively.
measures plane angle and solid angle respectively.
S.No. Supplementary Fundamental Quantities Supplementary
Unit Symbol
Unit Symbol
1

Plane
angle 
radian

rad

2

Solid
angle 
steradian

Sr

Derived Units
Those physical quantities which are derived from fundamental quantities
are called derived quantities and their units are called derived units.
e.g.,
velocity, acceleration, force, work etc.
velocity, acceleration, force, work etc.
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Definitions of Fundamental Units
The seven
fundamental units of SI have been defined as under.
fundamental units of SI have been defined as under.
1.
1 kilogram A cylindrical prototype mass made
of platinum and
1 kilogram A cylindrical prototype mass made
of platinum and
iridium alloys of height
39 mm and diameter 39 mm. It is mass
39 mm and diameter 39 mm. It is mass
of 5.0188 x 1025 atoms of carbon12.
2.
1 metre 1 metre is the distance that
contains 1650763.73
1 metre 1 metre is the distance that
contains 1650763.73
wavelength of orangered light of Kr86.
3.
1 second 1 second is the time in which
cesium atom
1 second 1 second is the time in which
cesium atom
vibrates 9192631770 times in an
atomic clock.
atomic clock.
4.
1 kelvin 1 kelvin is the (1/273.16) part
of the thermodynamics
1 kelvin 1 kelvin is the (1/273.16) part
of the thermodynamics
temperature of the triple point of water.
5.
1 candela 1 candela is (1/60) luminous
intensity of an ideal
1 candela 1 candela is (1/60) luminous
intensity of an ideal
source by an area of cm’ when source is at melting point of
platinum (1760°C).
6.
1 ampere 1 ampere is the electric current
which it maintained
1 ampere 1 ampere is the electric current
which it maintained
in two straight parallel conductor of infinite length and of negligible
crosssection area
placed one metre apart in vacuum will produce
placed one metre apart in vacuum will produce
between them a force 2 x 107 N per metre length.
7.
1 mole 1 mole is the amount of substance
of a system which
1 mole 1 mole is the amount of substance
of a system which
contains a many elementary entities (may be atoms, molecules,
ions, electrons or group of
particles, as this and atoms in 0.012 kg
particles, as this and atoms in 0.012 kg
of carbon isotope 6C12.
Systems of Units
A system of units is the complete set of units, both fundamental and
derived, for all kinds of physical quantities. The common system of
units which
is used in mechanics are given below:
is used in mechanics are given below:
1.
CGS System In this system, the unit of
length is centimetre,
CGS System In this system, the unit of
length is centimetre,
the unit of mass is gram and the unit of time is second.
2.
FPS System In this system, the unit of
length is foot, the unit
FPS System In this system, the unit of
length is foot, the unit
of mass is pound and the unit of time is second.
3.
MKS System In this system, the unit of
length is metre,
MKS System In this system, the unit of
length is metre,
the unit of mass is kilogram and the unit of time is second.
4.
SI System This system contain seven
fundamental
SI System This system contain seven
fundamental
units and two supplementary fundamental
units.
units.
Relationship between Some Mechanical SI Unit
and Commonly
Used Units
Used Units
S.No. Physical Quantity

Unit


1

Length

(a) 1 micrometre = 106 m


(b) 1 angstrom =1010 m





2

Mass

(a) 1 metric ton = 103 kg

(b) 1 pound =
0.4537 kg 
(c)
1 amu = 1.66 x1023 kg 
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3

Volume

(a) 1 litre = 1032 m3


4.

Force

(a) 1 dyne = 105 N


(b) 1
kgf = 9.81 N 






(a) 1
kgfm2 = 9.81Nm2 

5.

Pressure

(b) 1 mm of Hg = 133 Nm2


(c) 1 pascal = 1 Nm2







(d) 1 atmosphere pressure = 76
cm of Hg = 1.01 x 105 pascal 



(a) 1 erg =107 J


6.

Work
and energy 
(b) 1
kgfm = 9.81 J 

(c) 1 kWh = 3.6 x 106 J







(d) 1 eV = 1.6 x 1019 J


7.

Power

(d) 1 kgf ms1 =
9.81W 

1 horse
power = 746 W 



Some Practical Units
1.
1 fermi =1015 m
1 fermi =1015 m
2.
1 Xray unit = 1013 m
1 Xray unit = 1013 m
3.
1 astronomical unit = 1.49 x 1011 m (average distance between sun
and earth)
1 astronomical unit = 1.49 x 1011 m (average distance between sun
and earth)
4.
1 light year = 9.46 x 1015 m
1 light year = 9.46 x 1015 m
5.
1 parsec = 3.08 x 1016 m = 3.26 light year
1 parsec = 3.08 x 1016 m = 3.26 light year
Some Approximate Masses
Object

Kilogram

Our
galaxy 
2 x 1041

Sun

2 x 1030

Moon

7 x 1022

Asteroid
Eros 5 x 1015
Eros 5 x 1015
Dimensions
Dimensions
of any physical quantity are those powers which are
of any physical quantity are those powers which are
raised on fundamental units
to express its unit. The expression
to express its unit. The expression
which shows how and which of the base
quantities represent the
quantities represent the
dimensions of a physical quantity, is called the
dimensional formula.
dimensional formula.
Dimensional Formula of Some Physical Quantities
Physical Dimensional MKS
S.No.
Quantity Formula Unit
1

Area

[L2]

metre2

2

Volume

[L3]

metre3

3

Velocity

[LT1]

ms1

4

Acceleration

[LT2]

ms2

5

Force

[MLT2]

newton
(N) 
6

Work or
energy 
[ML2T2]

joule
(J) 
7

Power

[ML2T3]

J s1 or watt

8

Pressure
or stress 
[ML1T2]

Nm2

9

Linear momentum or Impulse [MLT1]

kg ms1


10

Density

[ML3]

kg m3

11

Strain

Dimensionless Unit less


12

Modulus
of elasticity 
[ML1T2]

Nm2

13

Surface
tension 
[MT2]

Nm1

14

Velocity
gradient 
T1

second1

15

Coefficient
of velocity 
[ML1T1]

kg m1s1

16

Gravitational
constant 
[M1L3T2]

Nm2/kg2

17

Moment
of inertia 
[ML2]

kg m2

18

Angular
velocity 
[T1]

rad/s

19

Angular
acceleration 
[T2]

rad/S2

20

Angular
momentum 
[ML2T1]

kg m2S1

21

Specific
heat 
L2T2θ1

kcal kg1K1

22

Latent
heat 
[L2T2]

kcal/kg

23

Planck’s
constant 
ML2T1

Js

24

Universal
gas constant 
[ML2T2θ1]

J/molK

Homogeneity Principle
If the
dimensions of left hand side of an equation are equal to the
dimensions of left hand side of an equation are equal to the
dimensions of
right hand side of the equation, then the equation is
right hand side of the equation, then the equation is
dimensionally correct.
This is known as homogeneity
This is known as homogeneity
principle.
Mathematically
[LHS] = [RHS]
[LHS] = [RHS]
Applications of Dimensions
1.
To check the accuracy of physical
equations.
To check the accuracy of physical
equations.
2.
To change a physical quantity
from one system of units to
To change a physical quantity
from one system of units to
another system of units.
3.
To obtain a relation between
different physical quantities.
To obtain a relation between
different physical quantities.
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Significant Figures
In the measured value of a physical quantity, the number of digits
about
the correctness of which we are sure plus the next doubtful digit,
the correctness of which we are sure plus the next doubtful digit,
are called
the significant figures.
the significant figures.
Rules for Finding Significant Figures
1.
All nonzeros digits are
significant figures, e.g., 4362 m has
All nonzeros digits are
significant figures, e.g., 4362 m has
4 significant figures.
2.
All zeros occuring between
nonzero digits are significant
All zeros occuring between
nonzero digits are significant
figures, e.g., 1005 has 4 significant figures.
3.
All zeros to the right of the
last nonzero digit are not significant,
All zeros to the right of the
last nonzero digit are not significant,
e.g., 6250 has only 3 significant
figures.
figures.
4.
In a digit less than one, all
zeros to the right of the decimal point
In a digit less than one, all
zeros to the right of the decimal point
and to the left of a nonzero digit are
not significant,
not significant,
e.g., 0.00325 has only 3 significant figures.
5.
All zeros to the right of a
nonzero digit in the decimal part
All zeros to the right of a
nonzero digit in the decimal part
are significant, e.g., 1.4750 has 5
significant figures.
significant figures.
Significant Figures in Algebraic Operations
(i) In Addition or Subtraction In addition or subtraction of the
numerical values the final result should retain the least decimal
place as in
the various numerical values. e.g.,
the various numerical values. e.g.,
If l1= 4.326 m
and l2 = 1.50 m
and l2 = 1.50 m
Then, l1 + l2 = (4.326
+ 1.50) m = 5.826 m
+ 1.50) m = 5.826 m
As l2 has measured
upto two decimal places, therefore
upto two decimal places, therefore
l1 + l2 = 5.83 m
(ii) In Multiplication or Division In multiplication or division of the
numerical values, the final result should retain the least significant
figures
as the various numerical values. e.g., If length 1= 12.5 m
as the various numerical values. e.g., If length 1= 12.5 m
and breadth b =
4.125 m.
4.125 m.
Then, area A = l x b = 12.5 x
4.125 = 51.5625 m2
4.125 = 51.5625 m2
As l has
only 3 significant figures, therefore
only 3 significant figures, therefore
A= 51.6 m2
Rules of Rounding Off Significant Figures
1.
If the digit to be dropped is
less than 5, then the preceding
If the digit to be dropped is
less than 5, then the preceding
digit is left unchanged. e.g.,
1.54 is rounded off to 1.5.
2.
If the digit to be dropped is
greater than 5, then the
If the digit to be dropped is
greater than 5, then the
preceding digit is raised by one. e.g.,
2.49 is rounded off to 2.5.
3.
If the digit to be dropped is 5
followed by digit other than zero,
If the digit to be dropped is 5
followed by digit other than zero,
then the preceding digit is raised by one.
e.g., 3.55 is rounded off to 3.6.
e.g., 3.55 is rounded off to 3.6.
4.
If the digit to be dropped is 5
or 5 followed by zeros, then the
If the digit to be dropped is 5
or 5 followed by zeros, then the
preceding digit is raised by one, if it is odd
and left unchanged if it is
and left unchanged if it is
even. e.g., 3.750 is rounded off to 3.8 and 4.650 is
rounded off to 4.6.
rounded off to 4.6.
Error
The lack in accuracy in the measurement due to the limit of accuracy of
the instrument or due to any other cause is called an error.
1. Absolute Error
The difference between the true value and the measured value of
a
quantity is called absolute error.
quantity is called absolute error.
2. Mean Absolute Error
The
arithmetic mean of the magnitude of absolute errors in all the
arithmetic mean of the magnitude of absolute errors in all the
measurement is
called mean absolute error.
called mean absolute error.
3. Relative Error The ratio of mean absolute error to the true
value
is called relative
is called relative
4. Percentage Error The
relative error expressed in percentage
relative error expressed in percentage
is called percentage error.
Propagation of Error
(i) Error
in Addition or Subtraction Let x = a + b or x = a – b
in Addition or Subtraction Let x = a + b or x = a – b
If the
measured values of two quantities a and b are
measured values of two quantities a and b are
(a ± Δa and (b ± Δb), then
maximum absolute error in their
maximum absolute error in their
addition or subtraction.
Δx = ±(Δa
+ Δb)
+ Δb)
(ii)
Error in Multiplication or Division Let x = a x b or x = (a/b).
Error in Multiplication or Division Let x = a x b or x = (a/b).
If the
measured values of a and b are (a ± Δa) and (b ± Δb),
measured values of a and b are (a ± Δa) and (b ± Δb),
then maximum relative
error
error