1.4. DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS

The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity. For example, the dimensional formula of the volume is [M° L3 T°], and that of speed or velocity is [M° L T-1]. Similarly, [M° L T–2] is the dimensional formula of acceleration and [M L–3 T°] that of mass density.

A dimensional formula represents a physical quantity in terms of its fundamental dimensions based on the seven base units: length ([L]), mass ([M]), time ([T]), electric current ([A]), thermodynamic temperature ([K]), luminous intensity ([cd]), and amount of substance ([mol]). For instance, the volume of an object can be expressed in terms of length alone because it is calculated as length cubed. Thus, its dimensional formula is [L^3]. Similarly, speed (distance over time) has the formula [L T^{-1}], indicating that it involves length and time.

Think of a building architect who needs to understand how much space is available in a room. Just as the architect uses length measurements in different dimensions to describe the space, scientists use dimensional formulas to categorize and relate physical quantities. For example, just like saying a room is 20 feet long but also relating that to its speed of air circulation during heating or cooling—using formulas allows for identical comparisons.

An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities. For example, the dimensional equations of volume [V], speed [v], force [F] and mass density [ρ] may be expressed as [V] = [M0 L3 T0], [v] = [M0 L T–1], [F] = [M L T–2], [ρ] = [M L–3 T0].

A dimensional equation relates a specific quantity to its dimensional formula by creating an equation where both sides represent the same physical dimensions. For instance, for speed, we write an equation that shows speed ([v]) equal to its dimensional formula ([M° L T^{-1}]). These equations are useful for establishing the relationships between physical quantities and ensuring they are consistent with dimensional analysis.

Imagine you are baking. You know that for every cup of flour (the physical quantity), you also need a specific volume of water. Just like a recipe creates a relationship between quantities, dimensional equations form a bridge to understand how different physical quantities relate to one another through their dimensions, ensuring every measurement fits 'the recipe' of properties in physics.

The dimensional equation can be obtained from the equation representing the relations between the physical quantities. The dimensional formulae of a large number and wide variety of physical quantities, derived from the equations representing the relationships among other physical quantities and expressed in terms of base quantities are given in Appendix 9 for your guidance and ready reference.

To derive a dimensional formula from physical quantities, one typically starts with known equations relating those quantities. For example, force is defined as mass times acceleration (F = m * a). By substituting the dimensional formulas of mass and acceleration into this equation, we can derive the dimensional formula for force: [F] = [M L T^{-2}]. Appendix resources often summarize these relationships for various physical constants and quantities, aiding in quick reference.

Think of it as a detective unraveling clues. Each equation that describes a relationship serves as a clue. By putting these clues together about different physical movement—like how a car accelerates and the forces involved—you can paint the full picture of how dimensions interact in physics.

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