Physical Quantities and Dimensional Analysis
Physical Quantities
Physical quantities are classified according to two categories: base quantities and derived
quantities
Base quantities: they are self-defined quantities such as length, mass, time etc,
Derived quantities: they are quantities that are derived from basic quantities and are
known by their meanings, such as speed, acceleration, force, and pressure etc.
The International System of Units (SI system)
Specific and uniform standards must be used across the world; quantities determine
dimensions and dimensions are estimated in units. The international system of units SI or
MKSA system consists of 7 base units adapted to 7 physical quantities, as shown in the
following table:
Table I.1: The international system of units (SI system)
Physical quantities | Unit of measurement (SI) | Dimensional symbol |
Length | Meter (m) | L |
Mass | Kilogram (Kg) | M |
Time | Second (s) | T |
Temperature | Kelvin (K) | θ |
Electric current intensity | Ampere (A) | I |
Light intensity | Candela (cd) | J |
Quantity of matter | Mole (mol) | N |
Dimensional Analysis
It is a theoretical tool for interpreting problems based on the dimensions of the physical
quantities involved: length, time, mass, etc. Dimensional analysis makes it possible to:
Check the validity of equations with dimensions
Research into the nature of physical quantities
Search for the homogeneity of physical laws
Determine the unit of a physical quantity based on the essential units (meter, second,kilogram, etc.)
The dimensional equation is represented by the following writing:
[X]=M L T I θ N J
Where
M : Mass (Kg)
L : Length (m)
T :Time (s)
I : Electric current intensity (A)
θ :Temperature (K)
N :Mole (mol)
J : Light intensity (cd)
Where
[π ]=1 , [𝑛𝑜𝑚𝑏𝑟𝑒]=1 , [t ]=T , [ m] =M , [l ]=L , [i ]=I
Some quantities have no dimensions
Example :
The dimensional equation of
Linear speed:

Acceleration:

Force:

Work :

Pressure :

Dimensional Uniformity
Dimensional analysis helps to confirm the validity of physical laws by matching the
dimensions between the two sides of the law. It also helps formulate the final picture of the
mathematical relationship based on the principle of dimensional matching as a condition for
the validity of the relationship, as the unit of the right side of the equation must equal the unit
of the left side of the equation, otherwise the equation is incorrect.
To prove the validity of any equation, the dimensions of the left side must be equal to the dimensions of the right side
Example :
Verify the homogeneity of the following equation:

⇒ 𝐿 = 𝐿 So the equation is homogeneous.