Units and Dimensions

definition

Unit and their types

Unit is the quantity of a constant magnitude which is used to measure the magnitudes of other quantities of the same order.
Kinds of unit: (1) Fundamental Unit (2) Derived Unit

definition

CGS, MKS and FPS system of units

1. The Centimetre Gram Second system of units (abbreviated CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time.

2. MKS is the system of units based on measuring lengths in meters, mass in kilograms, and time in seconds. MKS is generally used in engineering and beginning physics, where the so-called cgs system (based on the centimeter, gram, and second) is commonly used in theoretic physics.

3. The foot pound second system or FPS system is a system of units built on the three fundamental units foot for length, pound for either mass and second for time.

example

SI system of units

The International System of Units (SI) defines seven units of measure as a basic set from which all other SI units are derived. The SI base units and their physical quantities are:

1. meter for length
2. kilogram for mass
3. second for time
4. ampere for electric current
5. kelvin for temperature
6. candela for luminous intensity
7. mole for the amount of substance

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Identification of fundamental quantities

A fundamental unit is a unit adopted for measurement of a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no subset quantity can be expressed in terms of the others.

Length (meter)
Mass (kilogram)
Time (second)
Electric current (ampere)
Thermodynamic temperature (kelvin)
Amount of substance (mole)
Luminous intensity (candela)

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Definition of SI units

1. Unit of length: Metre
Definition: The metre is the length of the path travelled by light in vacuum during a time interval of 1 / 299792458 of a second.

2. Unit of Mass: kg,
Definition: The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.

3. Unit of time: Second
Definition: The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the ceasium 133 atom.

4. Current: Ampere
Definition: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to

2 10^{7}

newton per metre of length.

5. Unit of Temperature: Kelvin
Definition: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

6. Unit for amount of substance: mol
Definition: The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12, its symbol is 'mol'.

7. Unit of luminous Intensity: Candela
The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 5401012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

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Identify and give examples of some derived units

The units of all quantities other than fundamental units is called derived unit. Derived units are obtained in terms of fundamental quantities.

Quantity	Definition	Derived Unit	Abbreviation/Symbol
1. Area	length $\times$

breadth

metre

\times

metre

m^{2}


2. Volume	Length $\times$

breadth

\times

height

metre

\times

metre

\times

metre

m^{3}

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Conventions for writing SI units

The conventions followed while writing SI units:

Only units of the SI and those units recognised for use with the SI should be used to express the values of quantities.
All unit names are written in small letters (newton or kilogram) except Celsius.
The unit symbol is in lower case unless the name of the unit is derived from a proper name, in which case the first letter of the symbol is in upper case.
Unit symbols are unaltered in the plural.
Unit symbols and unit names should not be mixed.
Abbreviations such as sec (for either s or second) or mps (for either m/s or meter per second are not allowed.
For unit values more than 1 or less than -1 the plural of the unit is used and a singular unit is used for values between 1 and -1.
A space is left between the numerical value and unit symbol (25 kg, but not 25-kg or 25kg). If the spelled-out name of a unit is used, the normal rules of English are applied.
Unit symbols are in roman type, and quantity symbols are in italic
type with superscripts and subscripts in roman or italic type as
appropriate.

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Identification of units and quantities from each other

It is possible to define a unit for a given physical quantity. For example, length and distance can have the units of meter, inch, miles, light year, etc.
It is possible to estimate the physical quantity from its units. However, it should be noted that multiple quantities can have the same unit and hence this estimate is not accurate. So, if a quantity is given as

1 J

, then it can represent work or some form of energy.

example

Units of Physical quantities arising from certain operation on given physical quantity

Density =

\frac{m a s s}{v o l u m e}

Mass has unit

k g

.
Volume has unit

m^{3}

So the unit of density is

k g / m^{3}

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Conversion of units between two systems of units

Some conversion formulae between systems are listed below:

Q u a n t i t y

S I

C G S

M K S

F P S


Length	1m	1 cm = 0.01 m	1m	1 foot = 0.3048 m
Mass	1 kg	1 g = 0.001 kg	1 kg	1 pound = 0.4536 kg
Time	1 s	1 s	1 s	1 s

Example:
If mass of a man is

200

pounds, its

M K S

weight can be found as follows:

1 p o u n d = 0.4536 k g

200 p o u n d = 0.4536 \times 200 = 90.72 k g

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Prefix for exponents

If it is given that the mass of an object is

1000.2 k g

, then it can also be written in mega-gram as follows.

1 k g = 10^{3} g

1 M g = 10^{6} g

∴ 1 k g = 10^{- 3} M g

1000.24 k g = 1000.24 \times 10^{- 3} M g = 1.00024 M g

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Dimensional Formula

The dimensional formula is defined as the expression of the physical quantity in terms of its basic unit with proper dimensions. For example, dimensional force is

F = [M L T^{- 2}]

It's because the unit of Force is N(newton) or

k g \times m / s^{2}

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Dimensional formula of function of quantities

Let dimensional formulas of two quantities be given by

[A] = [M^{a m} L^{a l} T^{a t}]

and

[B] = [M^{b m} L^{b l} T^{b t}]

Then dimensional formula of

A^{x} B^{y}

is given by

[A^{x} B^{y}] = [M^{x a m + y b m} L^{x a l + y b l} T^{x a t + y b t}]

Example:
Angular momentum of a physical quantity is given by

I = m v r

. Its dimensional formula can be found as:

[m] = [M^{1} L^{0} T^{0}]

[v] = [M^{0} L^{1} T^{- 1}]

[r] = [M^{0} L^{1} T^{0}]

Hence,

[I] = [m v r] = [M^{1} L^{2} T^{- 1}]

shortcut

Conversion between Units using dimensional Analysis

If density of a material is

1

k g / m^{3}

Density has dimensional formula:

[M L^{- 3}]

To convert density of the object in CGS system.
Density=

1000 / 10^{6}

10^{- 3}

g / c c

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Correctness of Physical Equation Using Dimensional Analysis

Checking the correctness of physical equation is based on the principle of homogeneity of dimensions. According to this principle, only physical quantities of the same nature having the same dimensions can be added, subtracted or can be equated. To check correctness of given physical equation, the physical quantities on two side of the equations are expressed in terms of fundamental units of mass, length and time. The powers of

M, L

T

are same on two sides of the equations, then the physical equation is correct otherwise not.

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Establishment of relationship between physical quantities

If all the factors affecting a derived quantity is known, then the function relating it from the quantities can be established using dimensional analysis.
Example: Finding time-period of a simple pendulum (

T

) given it depends on length of the pendulum (

l

) and acceleration due to gravity (

g

).
Dimensional formulae of the quantities are:

[T] = [M^{0} L^{0} T^{1}]

[l] = [M^{0} L^{1} T^{0}]

[g] = [M^{0} L^{1} T^{- 2}]

Let

T = k l^{A} g^{B}

where

k, A, B

are constants.
Then,

[T] = [k] [l^{A}] [g^{B}]

[M^{0} L^{0} T^{1}] = [M^{0} L^{A + B} T^{- 2 B}]

Equating the powers on LHS and RHS,

A + B = 0

- 2 B = 1

Solving,

A = \frac{1}{2}, B = \frac{- 1}{2}

Hence, time-period is given by:

T = k l^{1 / 2} g^{- 1 / 2}

Note:
The established relation between the physical quantities is not unique and hence may or may not be absolutely correct.

TB - UG

Monday, July 31, 2017