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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.
    For example, if the dimensional formula of a physical quantity is [ML2T –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.
    For example, 29835.png cannot be derived by theory of dimensions but its dimensional correctness can be checked.
  • 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/L2)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.
    For example, torque and work have the same units and dimensions but their physical characteristics are different.
  • 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 4π steradian.

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