2.10 DIMENSIONAL ANALYSIS AND ITS APPLICATIONS
The recognition of concepts of dimensions, which guide the description of physical behaviour is of basic importance as only those physical quantities can be added or subtracted which have the same dimensions. When magnitudes of two or more physical quantities are multiplied, their units should be treated in the same manner as ordinary algebraic symbols. We can cancel identical units in the numerator and denominator. The same is true for dimensions of a physical quantity. Similarly, physical quantities represented by symbols on both sides of a mathematical equation must have the same dimensions.
Example:
Let us consider an equation
\(
\frac{1}{2} m v^{2}=m g h
\)
where \(m\) is the mass of the body, \(v\) its velocity, \(g\) is the acceleration due to gravity and \(h\) is the height. Check whether this equation is dimensionally correct.
Answer: The dimensions of LHS are
\(
\begin{gathered}
\text { [M] }\left[\mathrm{L} \mathrm{T}^{-1}\right]^{2}=[\mathrm{M}]\left[\mathrm{L}^{2} \mathrm{~T}^{-2}\right] \\
=\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2}\right]
\end{gathered}
\)
The dimensions of RHS are
\(
\begin{aligned}
{[\mathrm{M}]\left[\mathrm{L} \mathrm{T}^{-2}\right] } & {[\mathrm{L}]=[\mathrm{M}]\left[\mathrm{L}^{2} \mathrm{~T}^{-2}\right] } \\
&=\left[\mathrm{M} \mathrm{L}^{2} \mathrm{~T}^{-2}\right]
\end{aligned}
\)
The dimensions of LHS and RHS are the same and hence the equation is dimensionally correct.
Summary of dimensions and units:
Table2.6: Important dimensions and units
| Physical Quantity | Dimension | SI Unit |
| Force (F) | \(\left[M^{1} L^{1} T^{-2}\right]\) | Newton |
| Work (W) | \(\left[M^{1} L^{2} T^{-2}\right]\) | joule |
| Power (P) | \(\left[M^{1} L^{2} T^{-3}\right]\) | Watt |
| Gravitational constant (G) | \(\left[M^{-1} L^{3} T^{-2}\right]\) | \(N-m^{2} / k g^{2}\) |
| Angular velocity (ω) | \(\left[T^{-1}\right]\) | radian/s |
| Angular momentum (L) | \(\left[M^{1} L^{2} T^{-1}\right]\) | \(\mathrm{kg}-\mathrm{m}^{2} / \mathrm{s}\) |
| Moment of Inertia (I) | \(\left[M^{1} L^{2}\right]\) | \(\mathrm{kg}-\mathrm{m}^{2}\) |
| Torque \((\tau)\) | \(\left[M^{1} L^{2} T^{-2}\right]\) | N-m |
| Young’s modulus (Υ) | \(\left[M^{1} L^{-1} T^{-2}\right]\) | \(\mathrm{N} / \mathrm{m}^{2}\) |
| Surface Tension (S) | \(\left[M^{1} T^{-2}\right]\) | N/m |
| Co-efficient of Viscocity (η) | \(\left[M^{1} L^{-1} T^{-1}\right]\) | \(\mathrm{N}-\mathrm{s} / \mathrm{m}^{2}\) |
| Pressure (P) | \(\left[M^{1} L^{-1} T^{-2}\right]\) | \(\mathrm{N} / \mathrm{m}^{2}\) (Pascal) |
| Electric Field (E) | \(\left[M^{1} L^{1} I^{-1} T^{-3}\right]\) | V/m |
| Electrical Potential (V) | \(\left[M^{1} L^{2} I^{-1} T^{-3}\right]\) | volt |
| Electric Flux (Ψ) | \(\left[M^{1} T^{3} I^{-1} L^{-3}\right]\) | volt/m |
| Capacitance (C) | \(\left[I^{2} T^{4} M^{-1} L^{-2}\right]\) | farad (F) |
| Permittivity (ε) | \(\left[I^{2} T^{4} M^{-1} L^{-3}\right]\) | \(C^{2} / N-m^{2}\) |
| Permeability (µ) | \(\left[M^{1} L^{1} I^{-2} T^{-3}\right]\) | Newton/A \({ }^{2}\) |
| Magnetic Dipole moment (M) | \(\left[I^{1} L^{2}\right]\) | N-m/T |
| Magnetic Fux (Φ) | \(\left[M^{1} L^{2} I^{-1} T^{-2}\right]\) | Weber (Wb) |
| Magnetic Field (B) | \(\left[M^{1} I^{-1} T^{-2}\right]\) | tesla |
| Inductance (L) | \(\left[M^{1} L^{2} I^{-2} T^{-2}\right]\) | henry |
| Resistance (R) | \(\left[M^{1} L^{2} I^{-2} T^{-3}\right]\) | ohm (Ω) |
| Intensity of wave (I) | \(\left[M^{1} T^{-3}\right]\) | watt \(/ m^{2}\) |
| Specific heat capacity (c) | \(\left[L^{2} T^{-2} K^{-1}\right]\) | \(\mathrm{J} / \mathrm{kg}-\mathrm{K}\) |
| Stephan’s Constant (σ) | \(\left[M^{1} T^{-3} K^{-4}\right]\) | watt \(/ m^{2}-k^{4}\) |
| Electric dipole moment (p) | \(\left[L^{1} I^{1} T^{1}\right]\) | C-m |
| Thermal conductivity (k) | \(\left[M^{1} L^{1} T^{-3} K^{-1}\right]\) | watt \(/ m-k\) |
| Current Density (j) | \(\left[I^{1} L^{-2}\right]\) | ampere \(/ \mathrm{m}^{2}\) |
| Electrical conductivity (σ) | \(\left[I^{2} T^{3} M^{-1} L^{-3}\right]\) | \(\Omega^{-1} m^{-1}\) |
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