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Dimensions

Every physical quantity has its own identity. Such an identity is nothing but its quality with which it can be distinguished from all the other quantities. Such a unique quality possessed by a quantity is called dimension. Symbolically the dimension is expressed in the characteristic notation which is square bracket [ ]. For example, the dimension of length is expressed as [L] while the dimension of mass is expressed as [M] etc. Similar to the fundamental units, each derived unit also has a unique dimension associated with it. For example, the volume can be mathematically expressed as V = l × b × h where l is the length, b is breadth and h is height. All l, b and h are basically lengths measured in meters. The dimension of each l, b and h is [L] i.e. length. Hence equation of volume can be written in dimensional form as, V = [L][L][L] = [L3]. Any constants existing in the equation are always dimensionless.Thus [L3] is the dimension of the volume which indicates that calculations of volume involves the product of three lengths.Thus it can be said that the complete algebraic formula or relationship to obtain the derived unit from the fundamental units, is nothing but the dimension of the derived unit. The equality sign used to indicate the dimension indicates dimensional nation of the quality and is not related to actual numerical value of the quantity. Thus V = [L3] is the equality in terms of dimensions and should not be mixed up with actual numerical values. 

 

Dimensional Formula of Physical Quantities & Physical Constants

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To express the dimensions of physical quantities in mechanics, the units of length, mass and time are denoted by bracketed capital letters [M],[L] and [T] respectively. If the dimensions of a physical quantity are a in mass, b in length and c in time, then dimensions of that physical, quantity are expressed in the following manner: [Ma Lb Tc]. This is the "dimensional formula" of that quantity. Hence, dimensional formula 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. In other words, the dimensional formula tells us.

1) How and which of the fundamental units have been used to determine the quantity.

2) The nature of the dependence.

The dimensional formulae of physical quantities can be obtained by defining the quantity in terms of other physical quantities of known dimensions and then expressing these physical quantities in terms of fundamental units of mass [M], length [L] and time[T]. For example:

1) Velocity = $\frac{Distance}{Time}$ = $\frac{L}{T}$

                = [L T-1]
                = [M0 L1 T-1]

2) Acceleration = $\frac{Change\ in\ velocity}{Time\ taken}$

                       = $\frac{\frac{L}{T}}{T}$ = [L T-2]

                       = [M0 L1 T-2]

3) Momentum = Mass × Velocity
                     = [M][L T-1]
                     = [M1 L1 T-2]

4) Force = Mass × Acceleration
             = [M][L1 T-2]
             = [M1 L1 T-2]

Dimensional Equation

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Any mathematical equation is expressed using an equality sign to equate left hand side right hand side. A valid mathematical equation, relating various physical quantities must have same dimensions on both right and left side of the equation. It must be dimensionally correct. In fact writing the dimensions on both sides of equation, validity of any mathematical equation can be examined. The equation obtained by replacing each quantity in the original mathematical equation by respective dimensions is called dimensional equation. Such equations help us to find any error, if present in the mathematics equation, relative the various physical quantities. Such equations help us to obtain the dimensions of different derived units from the known valid physical laws and equations. Such an analysis done using dimensional equations is called dimensional analysis. Let us use such a dimensional analysis to derive the dimensions of various mechanical quantities in terms of the fundamental quantities which are length [L], mass [M] and time [T].

Use of Dimensional Analysis

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The dimensional equations have the following uses:

1) Convention of one system of units into another.

2) Checking the correctness of any equation (i.e, the accuracy of various formulae).

3) To derive the relationship between different physical quantities involved in a phenomenon.

4) a) To find the dimensions and units of a given physical quantity in a given system of units.
   
b)
To find the dimensions and units of physical constants or variables in a given equation.    

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