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Dimensions of Physical Quantities

The dimension of the units of a derived physical quantity may be defined as the number of times the fundamental units of mass, length and time appear in the physical quantity.
In mechanics, there are three fundamental quantities, namely, mass, length and time.

The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.

Let the symbols M, L and T respectively denote the units of mass, length and time.
These symbols only indicate the nature of the unit and not its magnitude. Then the dimension of velocity can be written as:
 
Similarly, the dimension of area and density are:
Area = Length
× Breadth = L × L = L2
 
The dimension of velocity has:
  • zero dimension in mass
  • 1dimension in length
  • –1 dimension in time
Area has 2 dimensions of length
  • Zero dimension of mass
  • Zero dimension of time
Density has one dimension of mass, -3 dimensions of length and zero dimension of time. The dimension of the units of a derived physical quantity may be defined as the number of times the fundamental units of mass, length and time appear in the physical quantity.

The expression for velocity obtained above is said to be its dimensional formula. Thus, the dimensional formula for velocity is [M0 L T –1].

It gives the following two information:
  1. Unit of velocity depends upon the unit of length and time and is independent of the unit of mass.
  2. In the unit of velocity, the powers of the unit of length and time are 1 and –1 respectively.
Hence, dimensional formulae of a physical quantity may be defined as the expression that indicates which of the fundamental units of mass, length and time enter into the derived unit of that quantity and with what powers.

If we represent velocity by [V], then an equation such as [V] = [M0 L T –1] is known as dimensional equation.

When a physical quantity is equated with its dimensional formula, the equation obtained is called a dimensional equation.

In general, in the dimensional equation
 
[X] = [Ma Lb Tc]
 
Right hand side represents the dimensional formula of physical quantity X, whose dimension mass, length and time are a, b and c respectively.

From the study of dimensional formulae of physical quantities, we can divide them into four categories:
  1. Dimensional Variables
    The quantities like area, volume, velocity, force, etc. possess dimensions and do not have a constant value. Such quantities are called dimensional variables.
  2. Non-Dimensional Variables
    The quantities like angle, specific gravity, strain, etc. neither possess dimensions nor have they a constant value. Such quantities are called non-dimensional variables.
  3. Dimensional Constants
    The quantities like gravitational constant, Planck’s constant, etc., possess dimensions and also have a constant value. They are dimensional constants.
  4. Non-Dimensional Constant
    The constant quantities which have no dimensions are called non-dimensional constants. These include pure numbers 1, 2, 3, 4….. π , e (=2.718) and all trigonometric functions.




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