physics

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.

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