Uses and Limitations of Dimensional Equations Physics Notes

Uses of Dimensional Equations:
The dimensional equations have the three following uses:

  1. For changing the magnitude of a physical quantity from one system of unit to another system of unit.
  2. To check the accuracy of physical relation.
  3. To establish a relationship i.e., formulae between different physical quantities.

1. Conversion of units of physical quantities from one system to another system: The method of dimensional analysis can be used to obtain the value of the physical quantity in some other system, when its value in one system is given.
As discussed earlier, the measurement of a physical quantity is given by:
Q = nu
If the unit of a physical quantity in a system is Ui, and the numerical value is then:
Q = n1u1 …(1)
Similar to the other system if the unit is u2 and magnitude is n2 then:
Q = n2u2 …(2)
From Eqs. (1) and (2)
n1u1 = n2u2 …………..(3)
If a,b,c are the dimensions of a physical quantity in mass, length and time, then:
n1 [M1aL1bT1c] = n2 [M2aL2bT2c]
Here M1, L1, T1 and M2, L2, T2 are the units of mass, length and time in the two systems, then;
n1 = n2\(\left[\frac{\mathrm{M}_{1}}{\mathrm{M}_{2}}\right]^{a}\left[\frac{\mathrm{L}_{1}}{\mathrm{~L}_{2}}\right]^{b}\left[\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}\right]^{c}\)
The equation can be used to find out the value of a physical quantity in the second or the new system, when its value in first system is known.

NCERT Solutions Guru Uses and Limitations of Dimensional Equations Physics Notes

Limitations of Dimensional Equations:
The method of dimensional analysis provides simple and quick solutions to so many physical problems. However, it has a few limitations also:
1. This method does not enable us to determine the value of the constant of proportionality which may be a pure number or a dimensional ratio. The value of the constant has to be determined experimentally or by some other method.

2. This method can not be used to derive the relations, such as S = ut + \(\frac{1}{2}\) at , v2 – u2 = 2as, etc. by usual method. Such relations are called composite relations. Even while deriving such a relation in parts, it does not tell about the nature of the sign (plus or minus) connecting the various terms in relation.

3. This method can not be used to derive a relation in the cases. Where the trigonometric or exponential functions are involved.
Or
r = \(\left(\frac{M}{2 \pi N \rho}\right)^{1 / 3}\)
The value of r, so calculated is of the order of 10-10 m.

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