Question

What is the non-dimensional number that results when the ratio of inertia force to gravity force is considered?

• A. Reynolds number
• B. Froude number
• C. Mach number
• D. Euler number

Hint:

Important dimensionless numbers: \[ \boxed{ \begin{aligned} \text{Reynolds Number} &\rightarrow \frac{\text{Inertia}}{\text{Viscous}}\\ \text{Froude Number} &\rightarrow \frac{\text{Inertia}}{\text{Gravity}}\\ \text{Mach Number} &\rightarrow \frac{\text{Velocity}}{\text{Speed of Sound}}\\ \text{Euler Number} &\rightarrow \frac{\text{Pressure}}{\text{Inertia}} \end{aligned} } \]

Answer 1

The Correct Option is B

Step 1: Writing the two forces in dimensional terms.
Inertia force scales as $\rho L^2 V^2$ and gravity force scales as $\rho L^3 g$, where $L$ is a characteristic length, $V$ a characteristic velocity and $g$ the acceleration due to gravity.
Step 2: Forming the ratio.
Dividing inertia force by gravity force cancels the density and one length term, leaving $\dfrac{V^2}{gL}$ as the governing dimensionless group.
Step 3: Identifying the named number.
Taking the square root of this ratio gives $\dfrac{V}{\sqrt{gL}}$, which is exactly how the Froude number is defined, so this inertia to gravity ratio corresponds to the Froude number.
\[ \boxed{\text{Froude number}} \]