1.3. DIMENSIONS OF PHYSICAL QUANTITIES

The nature of a physical quantity is described by its dimensions. All the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental or base quantities. W e shall call these base quantities as the seven dimensions of the physical world, which are denoted with square brackets [ ]. Thus, length has the dimension [L], mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd], and amount of substance [mol].

Dimensions of physical quantities give us a way to describe what those quantities represent. Each physical quantity can be related back to fundamental dimensions—like length, mass, and time—using specific symbols enclosed in square brackets. For example, when we say that the dimension of length is [L], it identifies length as a fundamental aspect of the physical world. The same applies to other fundamental quantities like mass ([M]), time ([T]), and several others. Knowing these dimensions allows us to understand the relationships between different physical quantities.

Imagine dimensions as the different ingredients in cooking a recipe. Each ingredient (like flour, sugar, and eggs) represents a fundamental quantity. Just as recipes use these ingredients in certain proportions to create a dish, dimensions combine to describe the properties of physical quantities.

The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. Note that using the square brackets [ ] round a quantity means that we are dealing with ‘the dimensions of’ the quantity.

When we express a derived quantity, like volume or force, we use the base quantities raised to certain powers. For example, the volume of an object can be represented by multiplying its length, width, and height, each represented as [L]. This results in dimensions for volume as [L] x [L] x [L] = [L^3]. This notation highlights how derived quantities relate to fundamental dimensions through their exponents. The square brackets help distinguish between the physical quantity and its dimensional characteristics.

Think of dimensions like the formula for making a cake. Just as you need specific amounts of flour, sugar, and eggs, you need certain amounts of base quantities (like length, mass, time) raised to certain powers to create a derived quantity like volume. The ‘weights’ of each ingredient (the exponents) matter in how the recipe (the physical quantity) turns out.

In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M] and [T]. For example, the volume occupied by an object is expressed as the product of length, breadth and height, or three lengths. Hence the dimensions of volume are [L] × [L] × [L] = [L]³ = [L³]. As the volume is independent of mass and time, it is said to possess zero dimension in mass [M°], zero dimension in time [T°] and three dimensions in length.

In the realm of mechanics, we often express quantities using only the dimensions of length, mass, and time. For instance, when calculating the volume of a cube, we take three lengths (length, breadth, and height), which results in the dimensional formula for volume as [L³]. Since volume does not depend on mass or time, it is considered to have zero dimensions in those categories, denoted as [M°] and [T°]. This framework helps simplify and categorize physical relationships into fundamental concepts.

Imagine using a box to hold toys. The volume of that box tells you how many toys it can hold, which depends only on its dimensions (length, breadth, height), not on the type of toys you put in (mass) or how long you keep the toys inside (time). This conceptual simplification makes it easier for us to understand and predict how objects interact in the physical world.

Similarly, force, as the product of mass and acceleration, can be expressed as Force = mass × acceleration = mass × (length)/(time)². The dimensions of force are [M] [L]/[T]² = [M L T–2]. Thus, the force has one dimension in mass, one dimension in length, and -2 dimensions in time.

Force can be broken down into its components of mass and acceleration. Since acceleration is derived from the length traveled over time squared, we can write the formula for force as Force = mass × (length/time²). This gives us the dimensions of force as [M] for mass, [L] for length, and [T²] for time, thus leading to [M L T⁻²]. This allows us to understand how force behaves relative to the fundamental dimensions.

Consider pushing a car. The amount of force you exert depends on the mass of the car (how heavy it is) and how rapidly you can change its speed (acceleration). The dimensions related to these factors help us quantify and understand the force applied, making it clearer why a more massive vehicle requires more force to accelerate effectively.

Note that in this type of representation, the magnitudes are not considered. It is the quality of the type of the physical quantity that enters. Thus, a change in velocity, initial velocity, average velocity, final velocity, and speed are all equivalent in this context. Since all these quantities can be expressed as length/time, their dimensions are [L]/[T] or [L T–1].

When discussing dimensions, we focus on the type of physical quantity rather than its magnitude. For instance, different types of velocity—whether it is a change in velocity, initial velocity, or final velocity—are all categorized under the same dimension of [L/T]. This means we can treat these quantities as equivalent in terms of their dimensional properties. Understanding this helps simplify equations and analyses in physics, making it easier to apply concepts consistently.

Think of various ways to describe your speed while cycling: it could be your current speed, your average speed over a trip, or how quickly you accelerate. Regardless of which term you use, they all boil down to how far you go in a certain time, represented as a speed dimension of [L/T]. This commonality helps make comparisons and calculations clearer.

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