The drag force $F_D$ on the smooth sphere of fig. falling in a liquid depends on the constant sphere speed $V$, the solid density $\rho_s$, the liquid density $\rho$ and viscosity $\mu$, the sphere diameter $D$, and gravity $g$. Find an expression for $F_D$ using (a) the $M-L-T$ system, and (b) the $F-L-T$ system.

The drag force $F_D$ on a sphere depends on the velocity $V$, the viscosity $\mu$, the density $\rho$, the surface roughness height $e$, the freestream fluctuation intensity $I$ (a dimensionless quantity), and the diameter $D$. Find an expression for $F_D$.

The velocity of propagation of ripples in a shallow liquid depends on the liquid depth $h$, gravity $g$, surface tension $\sigma$, and the liquid density $\rho$. Find an expression for the propagation velocity $V$. See Fig.

Select an appropriate set of variables that influence the drag force $F_D$ on an airfoil and write the final form in terms of dimensionless parameters.

The pressure drop $\Delta p$ in the pipe of Fig. depends on the average velocity, the pipe diameter, the kinematic viscosity, the length $L$ of pipe, the wall roughness height $e$, and the fluid density. Find an expression for $\Delta p$.

Derive an expression for the velocity of liquid issuing from a hole in the side of an open tank if the velocity depends on the height $H$ of the hole from the free surface, the fluid viscosity and density, gravity, and the hole diameter.

Find an expression for the average velocity in a smooth pipe if it depends on the viscosity, the diameter, and the pressure gradient $\partial p / \partial x$.

It is suggested that the velocity of the water flowing from a hole in the side of an open tank depends on the height $H$ of the hole from the surface, the gravity, and the density of the water. What expression relates the variables?

Find an expression for the centrifugal force $F_c$ if it depends on the mass $m$, the angular velocity $\omega$, and the radius $R$ of an impeller.

The normal stress $\sigma$ in a beam depends on the bending moment $M$, the distance $y$ from the neutral axis, and the moment of inertia $I$. Relate $\sigma$ to the other variables if we know that $\sigma$ varies linearly with $y$.

Include gravity $g$ in the list of variables of Example and determine the final expression that results for $h$.

Select $d, \mu$, and $V$ as the repeating variables in Example and find an expression for $F_D$. Show that this is equivalent to the expression of example.

Select $l, V$, and $\rho$ as the repeating variables in Example and find an expression for $F_D$. Show that this is an equivalent form to that of example.

Include the density $\rho$ and viscosity $\mu$ of the surrounding fluid and repeat Problem. This would account for the resistance (drag) of the fluid.