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Unit and Dimension
Physical
Quantity
The quantities which can
be measured are called physical quantities. The physical quantities when
measured, are expressed by magnitude (numerical value) and unit.
The physical quantities
can be divided into two categories; fundamental quantity and derived quantity.
Fundamental Quantity: The basic physical quantity which can be taken
as a standard to measure other physical quantities is known as a fundamental
quantity. In general, seven physical quantities are considered fundamental quantities.
1. What are fundamental Quantities?
They are Mass, Length, Time, Temperature, Luminous
Intensity, electric current, and Amount of Substance. In addition, there are other two
sub-fundamental quantities, which are plane angle and solid angle.
Derived Quantity: A quantity obtained from fundamental quantities
is called derived quantity. Area, volume, density, work, power, etc. are examples of derived physical quantities. Derived physical quantities can be
expressed in terms of fundamental quantities.
Measurement: The process of comparison of an unknown physical
quantity with a known physical quantity is called measurement.
Unit: The known quantity used as the standard for the
measurement is called unit. Unit is of two types: Fundamental unit and derived
unit.
Fundamental Units: The units of fundamental quantities are called
fundamental units. Derived Units: The units of derived quantities are called
derived units.
2. What are the different systems of units in measurement?
System of units:
FPS System: Length is measured in the foot; mass is measured in
pounds and time is measured in seconds.
CGS System: Length is measured in centimeters; mass is
measured in grams and time is measured in seconds.
MKS System: Length is measured in meters; mass is measured in
kilogram and time is measured in the second
SI System: The International System of Units (SI,
abbreviated from the French Système International (d'unités)) is the modern
form of the metric system. It is the only system of measurement with official status in nearly every country in the world.
It comprises a coherent system of units of
measurement starting with seven base units, which are the second (time= s), meter
(length= m), kilogram (mass= kg), ampere (electric current= A), kelvin (thermodynamic
temperature= K), mole (amount of substance, mol), and candela (luminous
intensity, cd).
The system allows for an unlimited number of
additional units, called derived units, which can always be represented as
products of the powers of the base units.
Besides this, there are
other two units called supplementary units. They are radian (unit of plane angle)
and steradian (unit of solid angle)
Download whole pdf: Click here to download
Advantages of SI system
ØSI system has advantages over other system of
units
•SI system is a metric system. Like the CGS system and MKS system, the
multiples and sub-multiples of the unit can be expressed as powers of 10. For
example: multiples and sub-multiples of meter are shown in the picture below
SI system is a coherent system. In the SI system,
dividing and multiplying the base or supplementary unit without introducing
numerical factors, all the derived units can be obtained. For example: 1J work
= 1kgm2s-2.
• (a system of units is said to be coherent if
all the units are either base units or derived from base units without
introducing any numerical factors other than 1.)
•It is a rational system. SI system uses only
one unit for one physical quantity. For example: SI unit of Energy (all types
of energy) is joule.
Dimensions of a physical quantity
•The dimensions of a
physical quantity are defined as the powers of the fundamental quantities which
are involved in the physical quantity.
The dimension of mass is [M], that of length is
[L] and that of time is [T]. Similarly, dimensions of temperature are [K],
dimension of electric current is [A], dimension of luminous intensity is [J]
and the dimension of amount of substance(mole) is [N].
For example, acceleration = velocity time=displacementtime2
Therefore, dimensional equation of acceleration
is [a]=[L][T2] =[LT−2]
Hence the dimensions of acceleration are 1 in
length and -2 in time.
Dimensional formula
ØThe dimensional formula of a physical quantity
is defined as the expression showing how and which basic quantities are
involved in the derived quantity. It is generally written in square bracket [
]. Here, [LT-2] is the dimensional formula of acceleration.
Example: The
dimensional equation for velocity is [v] = [s][t]
=[L][T] =[LT-1]
The hence dimensional
formula for velocity is [LT-1]
Download whole pdf: Click here to download
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Dimensional formula of physical quantities
The dimensional formula and SI units for more than 100 physical
quantities are given in the table below.
ØAn equation obtained by equating a physical
quantity with dimensional formula is called dimensional equation.
Example: dimensional
equation of equation is [v] = [LT-1]
Principle of homogeneity of dimensions
Principle of homogeneity
of dimensions’ states that, “for
the physical relation to be correct, the dimensions of the fundamental
quantities on the left-hand side of the equation is equal to the dimensions of
the fundamental quantities on the right-hand side of the equation.”
For illustration, we
consider an equation A = B + C. For this equation to be correct, dimensions of
A must be equal to dimensions of B and C.
Categories of physical quantities in terms of
dimensional analysis
•Dimensional variable, example:
distance, speed, acceleration, force, work, energy, power etc.
•Dimensional constant, example: Plank’s
constant(h), Stefan’s constant(σ), universal gravitational constant(G) etc.
•Dimensionless variable, example: angle,
refractive index, strain, relative density (specific gravity) etc.
•Dimensionless constant, example: pie(π),
counting numbers etc.
Uses of dimensional analysis
•To check the correctness of
physical relation
•To derive the relation between various physical quantities.
•To convert the value of physical quantity from one physical
quantity from one system of unit into another system of units
•To find the dimensions of constants in the given equation.
Limitations of dimensional analysis
•It does not give any
information about dimensionless constants.
•If the quantity depends on more than three other physical
quantities having dimensions, the formula cannot be derived.
•We cannot derive the formula containing trigonometric functions,
logarithmic functions, exponential functions etc.
•The exact form of relation cannot be formed when there is more
than one part in any relation.
•It gives no information about the physical quantity, whether it is
vector or scalar.
If ‘m’ is the mass, ‘c’ is the velocity of light and x = mc2, then
dimensions of ‘x’ will be:
