UNITS & MEASUREMENTS
Measurement of a physical quantity is the comparison with a standard value of the same kind is called
the unit of that quantity. The process of measurement of a physical quantity involves,
1) selection of unit (u)
2) to find out the no. of times that unit is contained in the given physical quantity. It is called the numerical
value OR magnitude of the quantity (n)
Any measurement (X) can be represented as the product of numerical value and unit
X nu
Fundamental and Derived units
The physical units which can neither be derived from one another, nor they can be further resolved in
to more simpler units are called fundamental units
eg. metre, kg, second
All other physical units which can be expressed in terms of fundamental units are called derived units.
1
eg. ms , kg ms
2
N
System of Units
A complete set of units which is used to measure all kinds of fundamental and derived quantities are
called system of units
1) CGS system - centimetre, gram, second
2) FPS system - foot, pound, second
1 foot = 0.3048 m
1 pound = 0.4536 kg
3) MKS system - metre, kg, second
4) SI system - (International system of units)
Basic SI units Supplementary SI units
Length - metre (m) Plane angle - radian (rad)
Mass - kilogram (kg) Solid angle - steradian (sr)
Time - second (s)
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Temperature - kelvin (K)
Electric current - ampere (A)
Luminous intensity - candela (Cd)
Amount of substance - mole (mol)
SI prefixes for powers of ten
101 - deca (da) 101 - deci (d)
10 2 - hecto (h) 102 - centi (c)
103 - kilo (k) 103 - milli (m)
106 - mega (M) 106 - micro
109 - giga (G) 109 - nano (n)
1012 - tera (T) 1012 - pico (p)
1015 - peta (p) 1015 - femto (f)
1018 - exa (E) 1018 - atto (a)
Some common practical Units
Large distances
1) Light year (ly)
It is the distance travelled by light through vacuum in one year
1 y 9.46 1015 m
2) Astronomical Unit (Au)
It is the average distance between centre of earth and centre of Sun
1 Au 1.496 1011 m
3) Par sec (parallactic sec)
It is the distance at which an arc of length one astronomical unit subtends an angle of 1 second of arc
1 par sec 3.08 1016 m
1 par sec 3.26 y
Large Masses
1) tonne or metric ton = 1000 kg
2) quintal = 100 kg
3) Chandra Shekhar Limit (CSL) = 1.4 times the mass of sun
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Small masses
th
1 12
Atomic mass unit (amu) = It is defined as of the mass of one 6 C - atom
12
1 amu 1.66 1027 kg
Time
1) Solar day - One day (24 hour)
2) Solar year - 365.25 days
3) Lunar month - It is the time taken by the moon to complete one revolution around the earth in its orbit
4) Shake - It is the smallest practical unit of time
1 shake 108 sec
Small Areas
Barn 10 28 m 2
Order of Magnitude
The order of magnitude of a quantity means its value (in suitable power of 10) nearest to the actual
value of that quantity. Consider a no. as a 10b where a is in between 1 & 10, then a is replaced with
10 0 OR 1 if a 5 and with 101 if 5 a 10 . The resulting power of 10 at which the number is reduced
is called its order of magnitude.
Eg. 0.005289 5.289 10 3
5.289 is replaced with 10
10 103 10 2 then its order of magnitude is –2
Dimensional Analysis
The dimensions of a physical quantity are the powers to which the units of base quantities are raised
to represent a derived unit of that quantity. It is denoted with square brackets [ ]
Eg. Force, F = ma = M L T
1 1 2
• The physical quantities can be added or substracted which have the same dimensions
• Special functions such as trigonometric functions, logarithmic functions, and exponential functions
must be dimensionless
• A pure number, ratio of similar physical quantities has no dimension. (Eg. Angle, refractive index,
,...etc)
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Different quantities having same dimension
Work Linear momentum
MLT
1
Energy Im pulse
ML T Surface tension
2 2
Heat
Torque Surface Energy ML0 T 2
Moment of force Spring cons tan t
Dimensional constants : Speed of light (C)
Gravitational constant (G)
Planks constant (h)
Dimensional variables : Area, volume, force,....
Dimensionless constants: Numbers, , .....
Dimensionless variables : Angle, strain, specific gravity, .....
A dimensionally correct equation need not be actually a correct equation, but dimensionally wrong
equation must be wrong
Applications of Dimensional Analysis
1. Conversion of one system of units to another
This is based on the fact that magnitude of a physical quantity remains the same whatever be the
system of units.
Q = nu = constant
n1 u1 n 2 u 2
u1 M1a Lb1T1c u 2 M a2 Lb2 T2c
n1u1
n2
u2
a b c
M L T
n 2 n1 1 1 1
M 2 L 2 T2
Eg. Convert 1 N to dyne (CGS system)
F M1L1T 2 . Here a = 1, b = 1, c = –2
In SI system M1 = kg, L1 = m, T1 = sec
In CGS system M2 = g, L2 = cm, T2 = sec
n1 = 1 n2 = ?
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