Use the symbol M to stand for the dimension mass, L to stand for the dimension length, and T to stand for the dimension time. The notation \([x]\) means “the dimension of \(x\).”

\( \Rightarrow \text{\([m] = M\) and \([g] = LT^{-2}\) are dimensional formulae} \)

The dimensional formula of a product of factors is the product of the dimensional formula of each factor. Thus \([ma] = [m][a]\) \( \Rightarrow [9.8 \cdot \text{m/s}^2] = [9.8] \cdot [\text{m/s}^2] = [m] \cdot [s^{-2}] = LT^{-2} \)

Rayleigh’s method of dimensional analysis identifies the dimensionless products one can form out of the model variables and constants, in this case Δt, m, R, g, and θ. Each dimensionless product takes a form \(\Delta t^\alpha m^\beta R^\gamma g^\delta \theta^\varepsilon\) determined by the Greek letter exponents α, β, γ, δ, and ε or, somewhat more simply, by the form \(\Delta t^\alpha m^\beta R^\gamma g^\delta\)

The key to Rayleigh’s method of finding dimensionless products is to require that the product \(\Delta t^\alpha m^\beta R^\gamma g^\delta\) be dimensionless

\( [\Delta t^\alpha m^\beta R^\gamma g^\delta] = [\Delta t^\alpha] [m^\beta] [R^\gamma] [g^\delta] = [\Delta t]^\alpha [m]^\beta [R]^\gamma [g]^\delta = T^\alpha M^\beta L^\gamma (LT^{-2})^\delta = T^{\alpha - 2\delta} M^\beta L^{\gamma + \delta} \)

the product \(\Delta t^\alpha m^\beta R^\gamma g^\delta\) is dimensionless when \( \begin{cases} \text{T: } \alpha - 2\delta = 0\\ \text{M: } \beta = 0\\ \text{L: } \gamma + \delta = 0 \end{cases} \)

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