Dimensional Analysis and Its Applications

Dimensions of Physical Quantities

Every physical quantity has dimensions, which describe its nature. These dimensions are based on seven fundamental quantities:

  • Length [L]
  • Mass [M]
  • Time [T]
  • Electric current [A]
  • Thermodynamic temperature [K]
  • Luminous intensity [cd]
  • Amount of substance [mol]

For example:

  • Volume has dimensions \([L^3]\), because it’s length × breadth × height.
  • Force has dimensions \([M L T^{-2}]\), because it’s mass × acceleration (which is length/time²).

Dimensional Formulae and Equations

A dimensional formula shows how a physical quantity is expressed in terms of base dimensions. For example:

  • Speed: \([M^0 L T^{-1}]\)
  • Acceleration: \([M^0 L T^{-2}]\)
  • Mass density: \([M L^{-3} T^0]\)

A dimensional equation equates a physical quantity with its dimensional formula, like:

  • \([V] = [M^0 L^3 T^0]\)
  • \([F] = [M L T^{-2}]\)

Applications of Dimensional Analysis

1. Checking Dimensional Consistency

An equation must have the same dimensions on both sides. For example, the equation for distance traveled:

\[x = x_0 + v_0 t + \frac{1}{2} \alpha t^2\]

All terms have the dimension of length \([L]\), so the equation is dimensionally correct.

Note: A dimensionally correct equation isn’t necessarily right, but a dimensionally wrong equation is definitely wrong!

2. Deducing Relations Between Quantities

Dimensional analysis can help derive relationships. For example, the time period \(T\) of a simple pendulum depends on:

  • Length \(l\)
  • Acceleration due to gravity \(g\)
  • Mass \(m\) (but it turns out \(m\) doesn’t affect \(T\))

Using dimensions, we find:

\[T = k \sqrt{\frac{l}{g}}\]

where \(k\) is a dimensionless constant (later found to be \(2\pi\)).

Important Concepts for NEET

  1. Principle of Homogeneity: All terms in an equation must have the same dimensions.
  2. Dimensional Formulas: Memorize common ones like force \([M L T^{-2}]\), energy \([M L^2 T^{-2}]\), etc.
  3. Checking Equations: Use dimensional analysis to verify if an equation might be correct.
  4. Deriving Relations: Understand how to use dimensions to find relationships between physical quantities.
  5. Limitations: Dimensional analysis can’t determine dimensionless constants or distinguish between quantities with the same dimensions.

Examples

Example 1: Check if \(K = \frac{1}{2}mv^2\) is dimensionally correct.

Left side (kinetic energy \(K\)) has dimensions \([M L^2 T^{-2}]\).

Right side: \([M] \times [L T^{-1}]^2 = [M L^2 T^{-2}]\).

Both sides match, so the equation is dimensionally valid.

Example 2: Why can’t \(K = m^2 v^3\) be correct for kinetic energy?

Dimensions of right side: \([M^2 L^3 T^{-3}]\), which don’t match \([M L^2 T^{-2}]\). So it’s wrong.