Dimensional Equations

Mass, time, length, temperature, electric current, luminous intensity, and amount of material are the seven primitive quantities that make up physical quantities. These seven dimensions make up the physical universe. The names of the base amounts can be replaced by symbols. The mechanical quantities mass, time, and length are indicated and represented by [M], [T], and [L], respectively. Other dimensions include K (for temperature), I (for electrical current), cd (for luminance intensity), and mol (for the amount of substance). A physical quantity’s dimensions are the same as the dimensions of its unit. In the following examples, the letters [M], [L], [T], etc. do not indicate its magnitude, but the nature of the unit.

Dimensions of a physical quantity.

To represent derived quantities, appropriate power must be multiplied with the basic values. Dimensions are the raised powers of basic quantities that are used to express unit-derived quantities. To put it another way, the dimensions of a physical quantity are the powers whereby the base quantities are multiplied to form that amount.

Example, dimensional force is

F = [M L T-2]

It’s because the unit of Force is Newton or kg*m/s2

A physical quantity’s dimensions are defined as the powers (or exponents) to which the fundamental units of length, mass, time, and so on should be raised to represent it, or the dimension of the units of a derived physical quantity is defined as the number of times the fundamental units of length, mass, time, and so on appear in the physical quantity.

Dimensional equations and dimensional formulas?

The dimensional formula is a compound statement that explains how and which fundamental quantities are used to create a physical quantity.

Dimensional formula of physical quantities

Below is the SI units and the basic dimensional formula physical quantities.

Physical quantities with the same dimensional formula

A list of various major physical quantities with the same dimensional formula is given below.

A dimensional equation is an equation that contains physical quantities and dimensional formulas. By equating dimensional formulas on the right and left sides of an equation, a dimensional equation is obtained.

The Uses of Dimensional Equation

1. To ensure that a physical relationship is accurate.
2. To calculate the relationship between different physical quantities.
3. To translate the value of a physical quantity from one unit system to another.
4. The dimension of constants in a particular relation must be determined.

Differentiate between dimension variables, dimensionless variables, dimensional constants, and dimensionless constants

The various physical quantities can be categorized into four groups based on the dimensional formula.

• Dimensional variables

Dimensional variables are physical quantities that have dimensions but do not have a constant value.

Eg: velocity, work, and power.

• Dimensionless variables

Dimensionless (non-dimensional) variables are physical quantities that have no dimensions but are variables.

Eg: strain, plane angle

• Dimensional constants

Dimensional constants are physical quantities that have constant values but yet have dimensions.

eg: Planck’s constant (h), Universal gravitational constant (G)

• Dimensionless constants

Dimensionless (non-dimensional) constants are pure numbers like 1,2,3, π etc.

Conclusion

The powers to which the fundamental units are increased to obtain one unit of a physical quantity are called the dimensions of that quantity.

Dimensional analysis is the process of determining the dimensions of physical quantities in order to check their relationships. All quantities in the world can be stated as a function of the fundamental dimensions, which are independent of numerical multiples and constants.