Use dimensional analysis to show that in a problem involving shallow water waves, both the Froude number and the Reynolds number are relevant dimensionless parameters. The wave speed c of waves on the surface of a liquid is a function of depth h, gravitational acceleration g, fluid density $ho$, and fluid viscosity $\mu$. Manipulate your $\Pi s$ to get the parameters into the following form: $\mathrm{Fr}=rac{c}{\sqrt{g h}}=f(\mathrm{Re}) \quad ext { where } \mathrm{Re}=rac{ ho c h}{\mu}$.

Consider again the model truck example discussed in the section, except that the maximum speed of the wind tunnel is only $50 \mathrm{~m} / \mathrm{s}$. Aerodynamic force data are taken for wind tunnel speeds between $V=20$ and $50 \mathrm{~m} / \mathrm{s}$ - assume the same data for these speeds as those listed in the table. Based on these data alone, can the researchers be confident that they have reached Reynolds number independence?

Water at $20^{\circ} \mathrm{C}$ flows through a long, straight pipe. The pressure drop is measured along a section of the pipe of length $L=1.3 \mathrm{~m}$ as a function of average velocity $V$ through the pipe. The inner diameter of the pipe is $D$ $=10.4 \mathrm{~cm}$. (a) Nondimensionalize the data and plot the Euler number as a function of the Reynolds number. Has the experiment been run at high enough speeds to achieve Reynolds number independence? (b) Extrapolate the experimental data to predict the pressure drop at an average speed of $80 \mathrm{~m} / \mathrm{s}$.

One of the first things you learn in physics class is the law of universal gravitation $F=G rac{m_{1} m_{2}}{r^{2}}$, where F is the attractive force between two bodies, $m_1$, and $m_2$ are masses of the two bodies, r is the distance between the two bodies, and G is the universal gravitational constant equal to $(6.67428 \pm 0.00067) imes 10^{−11}$ [the units of G are not given here]. (a) Calculate the SI units of G. For consistency, give your answer in terms of kg, m, and s. (b) Suppose you don’t remember the law of universal gravitation, but you are clever enough to know that F is a function of G, $m_1, m_2,$ and r. Use dimensional analysis and the method of repeating variables (show all your work) to generate a nondimensional expression for $F = F(G, m_1, m_2, r).$ Give your answer as $\Pi_1= ext { function of }(\Pi_2, \Pi_3, \dots).$ (c) Dimensional analysis cannot yield the exact form of the function. However, compare your result to the law of universal gravitation to find the form of the function (e.g., $\Pi_1 = \Pi_{22}$ or some other functional form).

Define wind tunnel blockage. What is the rule of thumb about the maximum acceptable blockage for a wind tunnel test? Explain why there would be measurement errors if the blockage were significantly higher than this value.

Although we usually think of a model as being smaller than the prototype, describe at least three situations in which it is better for the model to be larger than the prototype.