# Limitations of dimensional method

Although dimensional analysis is very useful, it cannot lead us too far because of the following reasons:

- Dimensional method cannot be used to derive equations involving addition and substraction.
- Numerical constants having no dimensions cannot be obtained by method of dimensions.
- Equations using trigonometric, exponential, and logarithmic functions can not be deduced.
- If dimensions are given, physical quantity may not be unique as many physical quantities have same dimensions.
*ML*^{2}*T*^{â€“2}], it may be work or energy or torque. - The method of dimensions cannot be applied to derive formula if in mechanics a physical quantity depends on more than three physical quantities as then there will be less number (=3) of equations than the unknowns (>3). However, we can still check correctness of the given equation dimensionally.
- Even if a physical quantity depends on three physical quantities, out of which two have same dimensions, the formula cannot be derived by the theory of dimensions, e.g., formula for the frequency of a tuning fork
*f*= (*d*/*L*^{2})*v*cannot be derived by theory of dimensions but can be checked.

# Some Important Points

- If name of a unit is kept on the name of scientist, then, e.g., 5 Ampere is wrong, correct is 5 ampere. Also, 5 a is wrong, correct is 5 A.
- Pure numbers are dimensionless.
- All trigonometric ratios, powers, exponential and logarithmic functions are dimensionless.
- All ratios of physical quantities having same dimensional formula are dimensionless, e.g., relative density, relative permeability, dielectric constant, angles, refractive index, etc.
- Dimensions do not depend upon magnitude.
- The dimensions of a physical quantity do not depend on the system of units.
- A physical quantity that does not have any unit must be dimensionless.
- Pure numbers are dimensionless.
- Generally, the symbols of those basic units whose dimension (power) in the dimensional formula is zero are omitted from the dimensional formula.
- Physical quantities are defined as the ratio of two similar quantities are dimensionless.
- If units or dimensions of two physical quantities are same, these need not represent the same physical characteristics.
- Angle is an exceptional physical quantity which though is a ratio of two similar physical quantities (angle = arc/radius) but still requires a unit (degrees or radians) to specify it along with its numerical value.
- Solid angle subtended at a point inside the closed surface is 4
*Ï€*steradian.