Dimensions of Physical Quantities

What Are Dimensions?

Dimensions describe the nature of a physical quantity. Every physical quantity can be expressed in terms of seven fundamental dimensions:

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

Dimensions are written inside square brackets [ ]. For example, the dimension of length is [L].

Dimensional Formula

The dimensional formula shows how a physical quantity is expressed in terms of the fundamental dimensions. The powers (or exponents) of these dimensions define the formula. For example:

  • Volume: Product of length, breadth, and height → \([V] = [L^3]\) or \([M^0 L^3 T^0]\).
  • Force: Mass × acceleration → \([F] = [M L T^{-2}]\).
  • Speed/Velocity: Length divided by time → \([v] = [L T^{-1}]\).

Dimensional Equation

A dimensional equation equates a physical quantity with its dimensional formula. Examples:

  • Volume: \([V] = [M^0 L^3 T^0]\)
  • Speed: \([v] = [M^0 L T^{-1}]\)
  • Force: \([F] = [M L T^{-2}]\)
  • Mass Density: \([\rho] = [M L^{-3} T^0]\)

Key Points to Remember

  1. Dimensions are about the type of quantity, not its magnitude. For example, speed, velocity, and change in velocity all have the same dimensions \([L T^{-1}]\).
  2. In mechanics, most quantities can be described using just [L], [M], and [T].
  3. Dimensional analysis helps check the correctness of equations and derive relationships between physical quantities.

Important Concepts for NEET

Here are high-yield ideas often tested in NEET:

  1. Dimensional Formulas: Memorize the dimensional formulas for common quantities like force \([M L T^{-2}]\), speed \([L T^{-1}]\), and density \([M L^{-3}]\).
  2. Dimensional Homogeneity: Every term in a physical equation must have the same dimensions. This principle is used to verify equations.
  3. Applications of Dimensional Analysis: Deriving relationships between quantities or converting units using dimensions.
  4. Base Quantities: The seven fundamental dimensions (e.g., [L], [M], [T]) form the basis for all derived quantities.

Example to Think About

Why does rounding matter in calculations? If you take the reciprocal of 9.58 (rounded to 0.104) and then take its reciprocal again, you get 9.62 instead of 9.58. To avoid such errors, keep an extra digit in intermediate steps!

Final Tip

When solving problems, always write the dimensions clearly. This helps catch mistakes and ensures your equations make sense!