Understanding Dimensions in Physics

What Are Dimensions?

Dimensions describe the nature of a physical quantity. Every physical quantity can be expressed using combinations of seven base dimensions:

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

For example, volume is derived from length cubed, so its dimensions are [L3].

Dimensional Formula

The dimensional formula shows how a physical quantity is built from base dimensions. Here are some examples:

  • Volume: \([V] = [L^3]\) (since volume is length × length × length)
  • Speed/Velocity: \([v] = [L T^{-1}]\) (distance per unit time)
  • Force: \([F] = [M L T^{-2}]\) (mass × acceleration, where acceleration is length per time squared)
  • Mass Density: \([\rho] = [M L^{-3}]\) (mass per unit volume)

Dimensional Equation

A dimensional equation sets a physical quantity equal to its dimensional formula. For example:

  • Volume: \([V] = [L^3]\)
  • Speed: \([v] = [L T^{-1}]\)
  • Force: \([F] = [M L T^{-2}]\)

Why Dimensions Matter

Dimensions help us:

  • Check if equations make sense (e.g., both sides must have the same dimensions).
  • Derive relationships between physical quantities.
  • Convert units correctly.

Important NEET Concepts

Here are key ideas often tested in NEET exams:

  1. Dimensional Formulas: Memorize common ones like force \([M L T^{-2}]\), velocity \([L T^{-1}]\), and acceleration \([L T^{-2}]\).
  2. Dimensional Homogeneity: Equations must have the same dimensions on both sides (e.g., \(F = ma\) is dimensionally correct).
  3. Applications in Derivation: Use dimensions to derive unknown formulas or check calculations.
  4. Significant Figures: Retain extra digits in intermediate steps to avoid rounding errors.

Example Problem

Question: What are the dimensions of energy? (Hint: Energy = Force × Distance)

Solution: Force has \([M L T^{-2}]\), and distance has \([L]\). Multiply them:
\([Energy] = [M L T^{-2}] \times [L] = [M L^2 T^{-2}]\).

Final Tip

Always double-check the dimensions in your calculations—it’s a quick way to catch mistakes!