An important dimensionless parameter in certain types of fluid flow problems is the Froude number defined as \(V / \sqrt{g \ell}\) where \(V\) is a velocity, \(g\) the acceleration of gravity, and \(\ell\) a length. Determine the value of the Froude number for \(V=10 \mathrm{ft} / \mathrm{s}\) \(g=32.2 \mathrm{ft} / \mathrm{s}^{2},\) and \(\ell=2 \mathrm{ft} .\) Recalculate the Froude number using SI units for \(V, g,\) and \(\ell .\) Explain the significance of the results of these calculations.

Use the given values for \(V=10 \mathrm{ft} / \mathrm{s}\), \(g=32.2 \mathrm{ft} / \mathrm{s}^{2},\) and \(\ell=2 \mathrm{ft}\) to calculate the Froude number using the formula \(V / \sqrt{g \ell}\). Substituting these values into the formula, we get \[ Fr = \frac{10}{\sqrt{32.2 \times 2}} \]\nCalculate the Froude number to give the initial answer.

Convert the given values from Imperial units to SI units and recalculate. The velocity \(V\) is given in feet per second (\(ft/s\)). To convert this to meters per second (\(m/s\)), multiply by approximately 0.3048. This gives \(V = 10 \times 0.3048 = 3.048 m/s\). The acceleration due to gravity \(g\) is given in feet per second squared (\(ft/s²\)). To convert this to meters per second squared (\(m/s²\)), multiply by approximately 0.3048, giving \(g = 32.2 \times 0.3048 = 9.81 m/s²\). The length \(\ell\) is given in feet (\(ft\)). To convert this to meters, multiply by approximately 0.3048, giving \(\ell = 2 \times 0.3048 = 0.6096 m\). Now, we substitute these values into the Froude number formula and calculate a second value for the Froude number using SI units.

Comparing the calculated values of the Froude number using both Imperial and SI units, the interesting thing to note is that they come out to be the same. This is because the Froude number is dimensionless - it does not have any units. This is significant as it allows for the comparison of fluid flow problems across different systems, regardless of the units of measurement used. These values can therefore be compared directly without the need for any conversion factors.