Dimension, an expression of the character of a derived quantity in relation to fundamental quantities, without regard for its numerical value. In any system of measurement, such as the metric system, certain quantities are considered fundamental, and all others are considered to be derived from them.
Dimensional Formula - definition
The dimensional formula is defined as the expression of the physical quantity in terms of its basic unit with proper dimensions. For example, dimensional force is F=[MLT−2] It's because the unit of Force is N(newton) or kg×m/s2.
Conversion between Units using dimensional Analysis - shortcut
If density of a material is 1kg/m3 Density has dimensional formula: [ML−3] To convert density of the object in CGS system. Density= 1000/106 = 10−3g/cc
Dimensional formula of function of quantities - definition
Let dimensional formulas of two quantities be given by [A]=[MamLalTat] and [B]=[MbmLblTbt] Then dimensional formula of AxBy is given by [AxBy]=[Mxam+ybmLxal+yblTxat+ybt]
Example: Angular momentum of a physical quantity is given by I=mvr. Its dimensional formula can be found as: [m]=[M1L0T0] [v]=[M0L1T−1] [r]=[M0L1T0] Hence, [I]=[mvr]=[M1L2T−1]
Correctness of Physical Equation Using Dimensional Analysis - definition
Checking the correctness of physical equation is based on the principle of homogeneity of dimensions. According to this principle, only physical quantities of the same nature having the same dimensions can be added, subtracted or can be equated. To check correctness of given physical equation, the physical quantities on two side of the equations are expressed in terms of fundamental units of mass, length and time. The powers of M,L & T are same on two sides of the equations, then the physical equation is correct otherwise not.
Establishment of relationship between physical quantities - definition
If all the factors affecting a derived quantity is known, then the function relating it from the quantities can be established using dimensional analysis. Example: Finding time-period of a simple pendulum (T) given it depends on length of the pendulum (l) and acceleration due to gravity (g). Dimensional formulae of the quantities are: [T]=[M0L0T1] [l]=[M0L1T0] [g]=[M0L1T−2]
Let T=klAgB where k,A,B are constants. Then, [T]=[k][lA][gB] [M0L0T1]=[M0LA+BT−2B] Equating the powers on LHS and RHS, A+B=0 −2B=1 Solving, A=21,B=2−1
Hence, time-period is given by: T=kl1/2g−1/2
Note: The established relation between the physical quantities is not unique and hence may or may not be absolutely correct.