Question

Test if the following equation is dimensionally correct:
\[v = \frac{1}{2 \pi}\sqrt{\frac{mgl}{I}};\] 
where h = height, S = surface tension, \[\rho\] = density, P = pressure, V = volume, \[\eta =\] coefficient of viscosity, v = frequency and I = moment of interia.

Answer 1

\[\nu = \frac{1}{2\pi}\sqrt{\left( \frac{mgl}{I} \right)}\]
Frequency, ν = [T⁻¹]

\[\sqrt{\left( \frac{mgl}{I} \right)} = \sqrt{\frac{\left[ M \right] \left[ {LT}^{- 2} \right] \left[ L \right]}{\left[ {ML}^2 \right]}}\]

\[ \Rightarrow \left[ \frac{\left[ {ML}^2 T^{- 2} \right]}{\left[ {ML}^2 \right]} \right]^\frac{1}{2} = \left[ T^{- 1} \right]\]
Since the dimensions of both sides of the equation are the same, the equation is dimensionally correct.