1.5.2. Deducing Relation among the Physical Quantities

The method of dimensions can sometimes be used to deduce relation among the physical quantities. For this we should know the dependence of the physical quantity on other quantities (up to three physical quantities or linearly independent variables) and consider it as a product type of the dependence.

Dimensional analysis is a powerful tool in physics. It allows us to find relationships between different physical quantities. When we say a physical quantity depends on others, we typically refer to the idea that one can express it as a product of these quantities raised to some powers. For instance, if we want to find out how the time period of a pendulum is related to its length, mass, and the acceleration due to gravity, we recognize these three quantities as the factors that influence the time period.

Imagine trying to bake a cake. The amount of sugar, flour, and eggs all play a role in determining how sweet and fluffy your cake will be. Similarly, in physics, understanding how different quantities depend on each other enables us to create accurate models.

Let us take an example. Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (l), mass of the bob (m), and acceleration due to gravity (g). Derive the expression for its time period using method of dimensions.

To derive the time period of a simple pendulum (T), we propose a relationship of the form T = k * l^x * g^y * m^z, where k is a dimensionless constant and x, y, z are exponents to be determined. Next, we write the dimensions of each variable: T has dimensions [T], l has dimensions [L], g has dimensions [L T–2], and m has dimensions [M]. By substituting these into our equation, we equate the dimensions to solve for the exponents x, y, and z, leading us to find that T = k * l^(1/2) * g^(-1/2). Ultimately, k is determined through experimentation and equals 2π.

Think of a pendulum like a swing at a playground. If you push it (analogous to gravity), the length of the swing (the string) and how heavy the person on it (mass) influence how long it takes to come back to the starting point. This relationship helps us predict and understand the timings without extensive experimentation each time.

Note that the value of the constant k cannot be obtained by the method of dimensions. Here it does not matter if some number multiplies the right side of this formula, because that does not affect its dimensions.

While dimensional analysis is useful for establishing relationships between physical quantities, it has its limits. The method helps us identify the form of the relationship and the powers involved but cannot provide the numerical constants (like k) that often need empirical determination. This means that while we can say how T scales with l and g, the exact value of T requires experimental data.

Consider a recipe where you know the proportions of ingredients are critical (like 2 parts flour to 1 part sugar) but the total amount of cake you end up with depends on how much of it you bake (the constant). Dimensional analysis tells you how those pieces relate but not how big your cake will actually be!

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