1.2 Dimensions of Physical Quantities

Dimension refers to the fundamental nature of a physical quantity, independent of the specific units used to measure it. For example, physical quantities like length, width, and radius all share the same dimension of length, represented as $[L]$. Dimensional analysis is a tool in physics used to check the validity of equations and understand the relationships between different physical quantities.

Base Quantities and Their Dimensions

The International System of Units (SI) defines seven base quantities, each with a unique dimension.

Supplementary Units→

Derived Quantities and Their Dimensions

Derived quantities are formed by combining base quantities through multiplication or division. Their dimensions are expressed as products of the base dimensions raised to certain powers.

Derived Units→

Types of Physical Quantities

Based on their dimensional properties, physical quantities are categorized into four types:

• Dimensional Variables: Quantities that possess dimensions and have variable magnitudes, such as velocity, force, and acceleration.
• Dimensional Constants: Quantities with dimensions but a constant value, such as the Gravitational constant ($G$), Planck's constant ($h$), and the speed of light in a vacuum ($c$).
• Dimensionless Variables: Quantities that lack dimensions but have variable values, such as angle, strain, and the coefficient of friction.
• Dimensionless Constants: Quantities that have no dimensions and a fixed value, such as pure numbers (1, 2, 3) and mathematical constants like $\pi$ and $e$.

Applications of Dimensional Analysis

Checking the Homogeneity of an Equation

The Principle of Homogeneity states that for an equation to be physically correct, the dimensions of all the terms on both sides must be the same.

Example: Consider the equation of motion, $s=v_{i}t+\frac{1}{2}a{t}^2$.

• Dimensions of displacement ($s$): $[L]$
• Dimensions of initial velocity × time ($v_{i}t$): $[L{T}^{-1}]\times [T]=[L]$
• Dimensions of $\frac{1}{2}\times$ acceleration $\times$ time${}^2$ ($\frac{1}{2}a{t}^2$): $[L{T}^{-2}]\times [T^2]=[L]$

Since all terms have the dimension of length $[L]$, the equation is dimensionally consistent.

Deriving Relationships Between Physical Quantities

Dimensional analysis can be used to deduce the relationship between different physical quantities.

Example: To derive the formula for the period ($T$) of a simple pendulum, assume it depends on its mass ($m$), length ($l$), and the acceleration due to gravity ($g$). Let the relationship be:

$T\propto m^a{l}^b{g}^c$

Substituting the dimensions:

$[T]=[M{]}^a[L{]}^b[L{T}^{-2}{]}^c$

Equating the powers of M, L, and T:

• For M: $0=a$
• For L: $0=b+c$
• For T: $1=-2c$

Solving gives: $a=0$, $c=-\frac{1}{2}$, $b=\frac{1}{2}$

Therefore:

$T\propto \sqrt{\frac{l}{g}}$

This result shows that the period of a simple pendulum is independent of its mass ($a=0$).

Limitations of Dimensional Analysis

1. Dimensional analysis cannot determine the value of dimensionless constants (such as 1, 2, $\pi$, or the constant $k$ in the pendulum formula).
2. It cannot derive formulas that involve trigonometric or exponential functions.
3. It cannot distinguish between physical quantities that share the same dimensions, such as work and torque (both have dimensions $[M{L}^2{T}^{-2}]$).