Dimensional Analysis
The physical quantities can be expressed in terms of the fundamental quantities and it is written as the product of base quantities raised to some power.
In the expression obtained, the exponent enters into the expression is the dimension of the physical quantity in that base quantity.
Example:
F = m × a
Dimensional formula of force is
= [ M1 L1 T-2 ]
It means the dimension of mass in force is 1, dimension of length is 1 and dimension of time is -2.
Dimensional formula is expressed with the large bracket containing base quantities raised to some power.
Ex. - Dimensional formula of force is
= [ M1 L1 T-2 ]
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Applications of Dimensional Analysis
To check the correctness of a formula.
To derive an expression for a given physical quantity.
To check the correctness of a formula or Principle of Homogeneity
According to the principle of Homogeneity of dimensions, the dimensions of the fundamental quantities ( mass, length and time ) are the same in each and every term on both sides of an expression.
Example :
S = ut + ½ at2
[ L ] = [ L ] + [ L ]
Hence, in the above expression the dimension of each term is [ L ] and same in both sides.
To derive an expression for a given physical quantity
To derive an expression for a physical quantity or to derive the relation between various physical quantities, again the principle of Homogeneity of dimensions, is used.
Example :
Suppose we are trying to find an expression for centripetal force for a particle of mass m moving in a circular path of radius r. The Force F depends upon some power of mass m ( say a ), some power of velocity v ( say b) and on some power of radius of circle r ( say c ).
So we write the equation as
F = k ma vb rc ………(1)
( k is a dimensionless constant )
L.H.S. = [F] = [ M L T-2 ]
R.H.S. = [k ma vb rc ]
= [ M ]a [ LT-1]b [ L ]c
= [ Ma Lb+c T-b ]
Comparing L.H.S. and R.H.S. , we get
a = 1
b + c = 1
b = 2
Hence, c = -1
Putting value of a, b and c in equation 1 we get
Dimensional analysis can also be used to convert one system of units into another.
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Physics Class 11 ( based on new revised syllabus )
1. Physical world
- Units and Measurements, part-1
- Length, mass and time Measurements
- Estimation or Calculation of errors
- Significant Figures and its Rules
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